Mes premiers remerciements s'adressent à mon directeur de thèse Roman Novikov, qui a guidé mes recherches durant ces trois années. Je le remercie chaleureusement pour son soutien et son aide pendant et hors notre temps de travail ; pour son attention qui m'a aidé à sortir des situations difficiles ; pour nos discussions portant sur des mathématiques et la vie en générale, qui m'ont permis de gagner de l'expérience inestimable.

J'aimerais exprimer ma gratitude à mon directeur de M2 à l'université de Moscou, Alexandre Shananin, qui a guidé mes premiers pas dans la recherche. Je voudrais le remercier de m'avoir fait découvrir la beauté des mathématiques, de m'avoir introduit dans le monde de la recherche et de m'avoir soutenu pendant toute la durée des études jusqu'à présent.

J'exprime ma gratitude à l'égard de Gennadi Henkin pour sa bienveillance et nos nombreuses discussions éclairantes qui étaient une source de motivation pour étendre et approfondir mes connaissances. Je lui suis très reconnaissant de m'avoir expliqué l'importance de garder l'équilibre entre la beauté et la puissance abstraite des sciences pures, et les possibilités impressionnantes et illimités des sciences appliquées. Ce fût un grand plaisir de travailler avec ce grand mathématicien.

Je remercie chaleureusement Colin Guillarmou et Thorsten Hohage d'avoir accepté d'être les rapporteurs de la présente thèse. C'est également un grand honneur de compter parmi les membres du jury François Golse, Laurent Baratchart, Otared Kavian, Vincent Michel et Dimitri Yafaev. Je remercie en particulier Vincent Michel pour avoir corrigé mes fautes de français dans le présent mémoire.

Je tiens à remercier le groupe de problèmes inverses de l'université de Göttingen dirigé par Thorsten Hohage pour l'accueil chaleureux au sein de leur laboratoire en automne 2015 dans le cadre du programme de financement de thèse Gaspard Monge.

J'exprime ma reconnaissance envers tous les membres du CMAP dont j'ai eu l'honneur de faire connaissance. En particulier je voudrais remercier chaleureusement Anne de Bouard et Antonin Chambolle pour leur bienveillance et décisions favorables en tant que présente directrice et ancien directeur du laboratoire, ainsi que Nassera Naar et Alexandra Noiret d'avoir toujours été à l'écoute de mes besoins et d'avoir fourni un soutien administratif de très haut niveau.

J'exprime ma gratitude à la communauté très conviviale de doctorants et stagiaires du laboratoire. Je remercie en particulier Simona, qui m'a toujours proposé avec gentillesse son aide précieuse dans des situations difficiles, ainsi que Luca, Helle, Thi Phong, Adi et Marco pour tous les bienfaits et services amicaux qu'ils m'ont fourni. Ce fût un grand plaisir de partager mon bureau avec Florine, Antoine, Vianney, Gustaw, Etienne, Jean-Bernard, Céline et Pierre ; je les remercie vivement d'avoir crée l'atmosphère amicale et accueillante.

Je souhaite adresser mes remerciements à Federico, Shixu, Kevish et Fedor ainsi qu'aux membres du groupe de travail "Physique Mathématique" pour nos conversations scientifiques et 1 philosophiques très édifiantes qui m'ont apporté beaucoup de plaisir. Je remercie en particulier Kevish pour avoir été mon maître de français personnel et surtout pour son aide à la rédaction des textes.

Je remercie également les membres de l'équipe Real Matrice et en particulier notre capitaine Aymeric pour nos matchs de foot et entraînements, nos piques-niques et nos jolis maillots.

Je tiens à remercier mes amis Dima, Nastia, Sergey et Julie de la confiance, l'appui et le dévouement dont ils ont fait preuve et surtout d'avoir assuré ma tranquillité d'esprit durant ces trois années de thèse.

Enfin, cette thèse doit beaucoup au soutien continu et l'aide inestimable apportés par ma mère Tatiana, mon père Dmitrii et mon frère Serge, même de loin. Je les remercie très affectueusement et de tout coeur pour tout ce qu'ils ont fait pour moi.

7 Proofs of Propositions IV.2 and IV.3 . . . . . . . . . . . . . . . . . . . . .

Table des matières Introduction

Dans la présente thèse nous étudions différents aspects, variantes et cas particuliers du problème de Dirichlet-Neumann inverse et du problème de diffusion inverse pour l'équation de Helmholtz jauge-covariante motivés par des applications. La thèse est basée sur sept articles qui peuvent être divisés sur trois groupes selon la motivation pratique.

Les articles du premier groupe sont motivés par le problème de tomographie acoustique d'un fluide en écoulement. Dans les travaux I, II nous étudions les questions de l'unicité et dans les travaux III, IV, V nous développons un algorithme de reconstruction : Le deuxième groupe a pour motivation la tomographie utilisant des particules élémentaires. D'un point de vue mathématique il s'agit de la généralisation du problème de diffusion inverse classique au cas où seulement le module de l'amplitude de diffusion est connu (il n'y a pas d'information de phase) : VI. Agaltsov A. D., Novikov R. G., Error estimates for phaseless inverse scattering in the Born approximation at high energies, à paraître dans Journal of Geometric Analysis, arxiv.org/abs/1604.06555

Le troisième groupe est motivé par la tomographie d'impédance électrique. D'un point de vue mathématique, il s'agit du problème de Dirichlet-Neumann inverse dans le cas où la géometrie est non-triviale (le problème de Plusieurs modèles de la physique mathématique sont formulés sous forme d'équation aux dérivées partielles sur une certaine variété à bord. Les coefficients de cette équation décrivent certaines caractéristiques d'un phénomène naturel. Les problèmes inverses à valeurs au bord consistent en la détermination de ces coefficients (et, peut-être, de la variété qui est supposée inconnue) à partir des données au bord, qui correspondent aux quantités mesurables.

Dans la présente thèse, nous nous intéressons aux problèmes inverses pour l'opérateur de Helmholtz jauge-covariant. Dans le cas euclidien cet opérateur a deux formes équivalentes :

L A,V = -∆ -2i d j=1 A j (x) ∂ ∂x j + V (x), L A,Q = - d j=1 ∂ ∂x j + iA j (x) 2 + Q(x), (1)
où x = (x 1 , . . . , x d ) ∈ D, ∆ est l'opérateur de Laplace euclidien, A = (A 1 , . . . , A d ), A j , V , Q sont des fonctions assez régulières dans D à valeurs dans M n (C), n ≥ 1, et D est un domaine ouvert dans R d , d ≥ 2. Notons que

L A,V = L A,Q pour Q = V + i d j=1 ∂ ∂x j A j - d j=1 A 2 j . (2)
L'opérateur L A,V survient, en particulier, dans la mécanique des fluides :

1. L'équation L A,V ψ = 0 avec n = 1 modélise la pression acoustique harmonique en temps p(x, t) = Re(ψ(x)e -iωt ) dans un fluide en écoulement. Dans ce cadre,

A = ω c 2 v + i 2 ∇ρ ρ , V = ω 2 c 2 + 2iω αω c , (3)
où c = c(x) désigne la célérité du son, v = v(x) est la vitesse du fluide, ρ = ρ(x) est la densité et α ω = α ω (x) décrit l'absorption à la fréquence ω. Cette équation dans sa forme générale a été récemment élaborée par A. S. Shurup et O. D. Rumyantseva de l'université de Moscou (article soumis à Acoustical Physics).

Pour les cas particuliers, voir, e.g., [START_REF] Roussef | Two-dimensional vector flow inversion by diffraction tomography[END_REF][START_REF] Rychagov | Reconstruction of fluid motion in acoustic diffraction tomography[END_REF][START_REF] Henkin | A multidimensional inverse problem in quantum and acoustic scattering[END_REF][START_REF] Burov | Simulation of a functional solution to the acoustic tomography problem for data from quasipoint transducers[END_REF].

2. L'équation L A,V ψ = 0 avec n ≥ 2, d = 2, survient, en particulier, dans le cadre de tomographie acoustique océanique comme l'équation d'onde en représentation modale pour la pression acoustique dans un fluide en écoulement dans un domaine cylindrique trois-dimensionnel, voir [START_REF] Baykov | Mode tomograpgy of moving ocean[END_REF] et [START_REF] Novikov | Global uniqueness and reconstruction for the multi-channel Gelfand-Calderón inverse problem in two dimensions[END_REF]Séction 2].

De plus, l'opérateur L A,Q apparait souvent dans le cadre de la mécanique quantique :

1. Dans le cas où A j et Q sont des fonctions scalaires réelles l'opérateur L A,Q survient comme le Hamiltonien d'une particule quantique non relativiste sans spin de charge et masse unitaire dans un champ électromagnétique, voir [50, chapître 21-1]. Dans ce cadre, A est le potentiel magnétique et Q est le potentiel électrique.

2. Si n = 2, d = 3 et les matrices A 1 , A 2 , A 3 sont des multiples scalaires réels des matrices de Pauli et Q est une matrice scalaire réelle, l'opérateur L A,Q apparait dans l'équation de Pauli-Schrödinger pour un électron non relativiste dans un champ électromagnétique. Cette équation remonte à [START_REF] Pauli | Zur Quantenmechanik des magnetischen Elektrons[END_REF].

3. De manière plus générale, si A 1 , . . ., A d sont des matrices hermitiennes de trace nulle et Q est une matrice scalaire réelle, l'opérateur (1) survient comme le Hamiltonien d'une particule dans un champ de Yang-Mills externe, voir [START_REF] Schrader | Small -asymptotics for quantum partition functions associated to particles in external Yang-Mills potentials[END_REF].

Notons également que l'équation L A,V u = 0 avec A = i 2 ∇ ln σ, V = 0, décrit le potentiel électrique u dans un domain conducteur de conductivité isotrope σ = σ(x) > 0. Des problèmes inverses à valeurs au bord pour cet opérateur surviennent dans le cadre de tomographie d'impédance éléctrique et ont une longue histoire, voir, e.g., [START_REF] Tihonov | On the uniqueness of the solution of the problem of electric prospecting[END_REF][START_REF] Calderón | On an inverse boundary value problem[END_REF][START_REF]An effectivization of the global reconstruction in the Gelfand-Calderón inverse problem in three dimensions[END_REF][START_REF] Uhlmann | Electrical impedance tomography and Calderón's problem[END_REF][START_REF] Baratchart | Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation[END_REF][START_REF] Kenig | Recent progress in the Calderón problem with partial data[END_REF]. De plus notons que la substitution de jauge ψ = σ 1 2 u mène à l'équation -∆ψ + vψ = 0, où v = -σ -1 2 ∆σ 1 2 . En outre, l'opérateur L A,V apparaît dans un modèle qui décrive le plasma en équilibre thermique [START_REF] Dressler | Stationary solutions of the Vlasov-Fokker-Planck equation[END_REF] et dans les autres modèles stationnaires de type advectiondiffusion dans un milieu mobile, voir, e.g. [START_REF] Probstein | Physicochemical Hydrodynamics : An Introduction[END_REF].

Problème de Dirichlet-Neumann inverse

où f est une fonction donnée sur ∂D, E ∈ [0, +∞) \ E A,V , et E A,V désigne l'ensemble des E ≥ 0 tels que le problème (4) admet une solution unique pour tout f assez régulier, [START_REF]Uniqueness and non-uniqueness in acoustic tomography of moving fluid[END_REF] dans le sens approprié qui sera précisé ci-dessous. Il est connu, en particulier, que dans le cas scalaire auto-adjoint l'ensemble E A,V est discret, voir [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]Théorème 8.37]. L'opérateur de Dirichlet-Neumann Λ A,V est défini comme suit :

Λ A,V f = d j=1
ν j ∂ ∂x j + iA j ψ ∂D , où ψ est la solution de (4), [START_REF] Aharonov | Significance of electromagnetic potentials in quantum theory[END_REF] et ν = (ν 1 , . . . , ν d ) désigne le champ des vecteurs unitaires normaux (extérieurs) à ∂D.

Notons que l'opérateur Λ A,V peut être idéntifié avec son noyau intégral (au sens de Schwartz) Λ A,V (x, y, E), x, y ∈ ∂D :

(Λ A,V f )(x) = ∂D Λ A,V (x, y, E)f (y) dy, x ∈ ∂D. (7)
L'opérateur Λ A,V est considéré comme une donnée mesurable à partir de laquelle il faut déterminer les coefficients A, V dans D.

Ce problème n'a jamais de solution unique. Plus précisément, en conjuguant l'opérateur L A,V par une fonction g assez lisse dans D à valeurs dans GL n (C), on obtient un opérateur L A ,V de même forme :

    
gL A,V g -1 = L A ,V , A j = gA j g -1 + i ∂g ∂x j g -1 , j = 1, . . . , d, Q = gQg -1 , où V , V sont liés à Q, Q par [START_REF] Agaltsov | Finding scattering data for a time-harmonic wave equation with first order perturbation from the Dirichlet-to-Neumann map[END_REF]. [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF] Il s'avère que les opérateurs Λ A,V et Λ A ,V sont conjugés par les valeurs au bord de la fonction g :

Λ A ,V = g| ∂D Λ A,V g -1 | ∂D .
En particulier, si g| ∂D = id n , on a Λ A,V = Λ A ,V . En tenant compte de cette nonunicité, nous considérons le problème suivant :

Problème 1. Trouver les coéfficients A, V dans D (à une transformation de jauge (8) avec g| ∂D = id n près) à partir de l'opérateur Λ A,V correspondant à E fixé.

Notons qu'en accord avec [START_REF] Aktosun | Inverse problems on the line without phase information[END_REF], l'opérateur Λ A,V peut être considéré comme une fonction à 2(d -1) variables réelles, tandis que les coefficients A, V sont des fonctions de d variables réelles. Le problème (1) est donc forméllement non surdéterminé pour d = 2 et surdéterminé pour d ≥ 3.

Le problème 1 peut être consideré comme un problème de Gelfand pour l'opérateur de Helmholtz jauge-covariant à énergie fixée (voir [START_REF] Gelfand | Some problems of functional analysis and algebra[END_REF]) ou comme une généralisation du problème de Calderón de tomographie d'impédance électrique (voir [START_REF] Calderón | On an inverse boundary value problem[END_REF]).

Rappelons que l'opérateur L A,V décrit, en particulier, la pression acoustique dans un fluide en écoulement. Dans ce cadre, l'opérateur Λ A,V correspond aux mesures des ondes acoustiques diffusées associées aux ondes incidentes données. Donc, le problème 1 s'interprète comme le problème de détermination d'un diffuseur acoustique à partir des mesures des ondes diffusées.

Afin d'étudier le problème 1, il est utile d'introduire les notations suivantes :

A = iA 1 dx 1 + • • • + iA d dx d , Ω A = dA + A ∧ A = -i k<l ∂A k ∂x l -∂A l ∂x k + i(A l A k -A k A l ) dx k ∧ dx l . (9)
En géométrie différentielle A et Ω A apparaissent come une 1-forme de connexion et la 2-forme de courbure pour cette connexion. Notons egalement que dans le cas n = 1, d = 3, la forme Ω A est étroitement lié au rotationnel curl A du champ A. Plus précisément, si i, j, k désignent la base canonique de R 3 , on a

Ω A = -i ∂A 2 ∂x 3 -∂A 3 ∂x 2 dx 2 ∧ dx 3 + ∂A 3 ∂x 1 -∂A 1 ∂x 3 dx 3 ∧ dx 1 + ∂A 1 ∂x 2 -∂A 2 ∂x 1 dx 1 ∧ dx 2 , curl A = ∂A 2 ∂x 3 -∂A 3 ∂x 2 i + ∂A 3 ∂x 1 -∂A 1 ∂x 3 j + ∂A 1 ∂x 2 -∂A 2 ∂x 1 k.
On peut montrer que la transformation de jauge (8) mène à la loi de transformation suivante pour Ω A : Ω A = gΩ A g -1 .

Notons également que dans le cas scalaire n = 1, la fonction Q et la 2-forme Ω A sont invariantes par rapport aux transformations [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF]. Dans le cas où l'opérateur L A,Q est le Hamiltonien d'une particule chargée dans un champ électromagnétique, la forme Ω A décrit le champ magnétique corréspondant au potentiel magnétique A. Il découle de ce qui précède que les mesures au bord ne distinguent pas les coefficients d'une même classe de jauge (c.à.d. liés par une transformation [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF]). D'un point de vue physique, cela reflète le principe de relativité de H. Weyl formulé en 1919, selon lequel toutes les configurations du potentiel électromagnétique de la même classe de jauge décrivent la même réalité physique, voir [127, section 1.1]. Notons que la proposition réciproque, à savoir que des configurations de différentes classes de jauge sont physiquement distinguables, est connu comme l'effet Aharonov-Bohm et remonte à [START_REF] Ehrenberg | The refractive index in electron optics and the principles of dynamics[END_REF][START_REF] Aharonov | Significance of electromagnetic potentials in quantum theory[END_REF]. En particulier, même si le champ magnétique Ω A s'anulle, dans le cas d'un domaine D non simplement connexe il est possible que le potentiel magnétique A n'est pas jauge-équivalent à zéro et peut donc influencer le comportement des particules chargées, voir par exemple [START_REF] Aharonov | Significance of electromagnetic potentials in quantum theory[END_REF][START_REF] Yafaev | Scattering by magnetic fields[END_REF][START_REF]Aharonov-Bohm effect revisited[END_REF].

Problèmes direct et inverse de diffusion

Nous allons formuler un analogue du problème 1 dans le cas où D est l'espace entier R d , d ≥ 2. Nous supposons que les coefficients A, V sont assez réguliers et décroissent assez rapidement (par exemple exponentiellement) à l'infini. Nous considérons le problème suivant pour

ψ + A,V = ψ + (x, k), x ∈ R d , k ∈ R d , k 2 = E, pour k fixé :      L A,V ψ + A,V = Eψ + A,V dans R d , ψ + A,V (x, k) = e ikx id n + ψ + sc (x, k), ∂ ∂|x| -ik ψ + sc (x, k) = o(|x| -d-1 2 ), |x| → ∞, (10)
où E ∈ (0, +∞) \ E + A,V , et E + A,V désigne l'ensemble des E > 0 tels que le problème [START_REF] Arians | Geometric approach to inverse scattering for the Schrödinger equation with magnetic and electric potentials[END_REF] admet une solution unique pour tout k ∈ R d , k 2 = E, dans le sens approprié qui sera précisé ci-dessous. Nous appelons E + A,V l'ensemble des résonances réelles positives de l'opérateur L A,V . Il est connu que si A et Q vérifiant (1) et [START_REF] Agaltsov | Finding scattering data for a time-harmonic wave equation with first order perturbation from the Dirichlet-to-Neumann map[END_REF] sont des fonctions scalaires réelles décroissant exponentiellement à l'infini avec leur dérivées partielles, alors E + A,V = ∅, voir [START_REF] Eskin | Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy[END_REF]. On peut montrer que le terme asymptotique principal à l'infini de la fonction ψ + sc s'écrit comme suit : (

) 11
La fonction f = f A,V s'appele l'amplitude de diffusion. Elle est définie sur le produit M E de deux sphères :

M E = (k, l) ∈ R d : k 2 = l 2 = E S d-1 × S d-1 . où x ∈ R d , k ∈ R d , k 2 = E, et en utilisant la formule explicite f A,V (k, l) = (2π) -d R d e -ilx L A,V -L 0,0 ψ + A,V (x, k) dx, (13)
où k, l ∈ R d , k 2 = l 2 = E. Notons que, pour k fixé et A = 0, on peut considèrer [START_REF] Aubin | Nonlinear analysis on manifolds. Monge-Ampère equations[END_REF] et ses versions dérivées par rapport à x 1 , . . ., x d comme une équation intégrale vectorielle pour ψ + A,V (•, k), ∂ ∂x 1 ψ + A,V (•, k), . . ., ∂ ∂x d ψ + A,V (•, k). En outre, notons qu'un algorithme efficace de résolution de l'équation [START_REF] Aubin | Nonlinear analysis on manifolds. Monge-Ampère equations[END_REF] a été proposé par Vainikko [START_REF] Vainikko | Fast solvers of the Lippmann-Schwinger equation[END_REF].

Le problème reciproque de détermination des coefficients A et V à partir de f A,V est connu comme le problème de diffusion inverse. Cependant, il s'avère que l'amplitude de diffusion est invariante par rapport aux transformations de jauge [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF], où g tends vers id n assez rapidement à l'infini. En tenant compte de cette non-unicité, on s'intéresse au problème suivant : Problème 2. Étant donné f A,V sur M E pour E fixé, trouver A, V dans R d à une transformation de jauge (8) près, où g tends vers id n assez rapidement à l'infini.

Notons que l'amplitude de diffusion est une fonction de 2(d -1) variables réelles tandis que les coefficients A et V dépendent de d variables réelles. Donc, le problème 2 est formellement non surdétérminé pour d = 2 et surdéterminé pour d ≥ 3.

Rappelons que dans la mecanique quantique l'opérateur L A,Q de (1), (2) survient en particulier comme le Hamiltonien d'une particule quantique dans un champ externe. Dans ce cadre, |f (k, l)| 2 , k = l, est la densité de probabilité de diffusion d'une particule avec impulsion initiale k dans la diréction l, et la valeur ∂ arg f (k,l) ∂|k| , dénommée déphasage, contribue à un retard de la particule diffusée, voir [30, p. 461-463]. Donc, dans ce cadre, le problème 2 s'interprète comme le problème de détermination du champ extérne à partir des mesures des particules quantiques diffusées. Le problème similaire dans le cas des particules classiques a été aussi étudié, voir Jollivet [START_REF] Jollivet | Inverse scattering at high energies for classical relativistic particles in a long-range electromagnetic field[END_REF] et les réferences qui y sont contenues.

1.4 L'état de l'art 1.4

.1 Problème 1

De très nombreux ouvrages traitent le problème 1 dans le cas où A = 0, voir par exemple [START_REF]Global uniqueness for a two-dimensional inverse boundary value problem[END_REF][START_REF] Astala | Calderón's inverse conductivity problem in the plane[END_REF][START_REF] Bukhgeim | Recovering a potential from Cauchy data in the twodimensional case[END_REF] (unicité, d = 2, n = 1) ; Novikov-Santacesaria [START_REF] Novikov | Global uniqueness and reconstruction for the multi-channel Gelfand-Calderón inverse problem in two dimensions[END_REF][START_REF]Monochromatic recontruction algorithms for two-dimensional multichannel inverse problems[END_REF] (unicité et reconstruction, d = 2, n ≥ 1), Novikov [START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF], Burov et al. [START_REF] Burov | Simulation of a functional solution to the acoustic tomography problem for data from quasipoint transducers[END_REF] (reconstruction et aspects numériques, d = 2, n = 1), Hohage [START_REF] Hohage | On the numerical solution of a three-dimensional inverse medium scattering problem[END_REF] (reconstruction numérique en utilisant les méthodes de type Newton), [START_REF] Sylvester | A global uniqueness theorem for an inverse boundary value problem[END_REF][START_REF] Haberman | Uniqueness in calderón's problem with lipschitz conductivities[END_REF] (unicité, d ≥ 3, n = 1).

La majorité des résultats connus sur le problème 1 dans le cas où A = 0 concernent l'unicité dans le cas des coefficients scalaires (n = 1) en dimension d ≥ 3. Dans ce cas, Sun [START_REF] Sun | An inverse boundary value problem for Schrödinger operators with vector potentials[END_REF] et Panchenko [START_REF] Panchenko | An inverse problem for the magnetic Schrödinger equation and quasi-exponential solutions of nonsmooth partial differential equations[END_REF] ont démontré que si Λ A,V = Λ A ,V et curl A, curl A sont petits, curl A = curl A et Q = Q . Le même résultat sans la condition de petitesse a été obtenu par Nakamura-Sun-Uhlmann pour des coefficients de classe C ∞ [START_REF] Nakamura | Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field[END_REF], Tolmasky (coefficients de classe C 1 ) [START_REF] Tolmasky | Exponentially growing solutions for nonsmooth first-order perturbations of the Laplacian[END_REF], Salo (coefficients continus dans le sens de Dini) [START_REF] Salo | Inverse problems for nonsmooth first order perturbations of the Laplacian[END_REF] et Krupchyk-Uhlmann (coefficients bornés) [START_REF] Krupchyk | Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential[END_REF]. Comme corollaire, l'opérateur Λ A,V détermine A, V uniquement sous condition que les coefficients A, V sont réels et que D est simplement connexe, voir l'article I de la présente thèse.

Le problème de reconstruction en dimension d ≥ 3 a été considérée pour la première fois par Salo [START_REF]Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field[END_REF] qui a proposé un algorithme basé sur les travaux de Novikov [START_REF]Multidimensional inverse spectral problem for the equation -∆ψ+(v(x)-Eu(x))ψ = 0[END_REF] et Nachman [START_REF] Nachman | Reconstructions from boundary measurements[END_REF]. La stabilité de reconstruction de type log log a été démontrée par Tzou [START_REF] Tzou | Stability estimates for coefficients of magnetic Schrödinger equation from full and partial boundary measurements[END_REF].

À notre connaissance, la question de l'unicité en dimension d = 2 a été traitée dans le cas scalaire (n = 1) avec A = 0 pour la première fois par Guillarmou-Tzou [START_REF] Guillarmou | Identification of a connection from Cauchy data on a Riemann surface with boundary[END_REF]. Ils ont démontré que si A et A sont réels et Λ A,V = Λ A ,V , les coefficients A, V et A , V sont liés par une transformation [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF] avec g| ∂D = 1.

Notons également le travail [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF] de Brown-Salo, dans lequel des formules pour la détermination des valeurs tangentes au bord du champ vectoriel A à partir de l'opérateur Λ A,V ont été présentées en dimension d ≥ 2 dans le cas scalaire (n = 1).

Le cas des coefficients matriciels (n ≥ 2) est beaucoup moins étudié. La question de l'unicité a été éxaminé par Eskin-Ralston [START_REF] Eskin | Global uniqueness in the inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials[END_REF][START_REF]Inverse boundary value problems for systems of partial differential equations[END_REF] en dimension d ≥ 3, qui ont démontré que sous la condition de convexité de D, l'égalité des opérateurs Λ A,V et Λ A ,V implique que les coefficients A, V et A , V sont liés par une transformation de jauge [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF] avec g| ∂D = id n .

Notons également un cas particulier très étudié du problème 1, corréspondant à l'opérateur L A,V avec A = i 2 ∇σ σ , σ > 0, V = 0. Comme noté ci-dessus, l'opérateur L A,V survient en tomographie d'impédance électrique. Pour des résultats connus sur ce problème dans le cas euclidien, voir par exemple [START_REF]An effectivization of the global reconstruction in the Gelfand-Calderón inverse problem in three dimensions[END_REF][START_REF] Uhlmann | Electrical impedance tomography and Calderón's problem[END_REF][START_REF] Baratchart | Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation[END_REF][START_REF] Kenig | Recent progress in the Calderón problem with partial data[END_REF]. En outre, une généralisation du problème 1 pour le cas d'une géometrie non euclidienne et qui est supposée inconnue a priori est étudié, en particulier, dans les articles [START_REF] Henkin | On the explicit reconstruction of a Riemann surface from its Dirichlet-to-Neumann operator[END_REF][START_REF]On the reconstruction of conductivity of bordered two-dimensional surface in R 3 from electrical currents measurements on its boundary[END_REF][START_REF] Henkin | Gelfand-calderón inverse problem for anisotropic conductivities on bordered surfaces in R 3[END_REF] (voir aussi les références qui y sont contenues). Nous étudions un problème qui est étroitement lié à une modification du problème 1 pour le cas d'une géometrie inconnue et A = 0 dans l'article VII.

Il est aussi important de noter qu'il existe un grand nombre de travaux qui étudient le problème 4 dans le cas d'information partielle. Ces problèmes sont très importants pour les applications, où il est impossible de mesurer des données sur une partie du bord (par exemple en tomographie medicale), voir par exemple [START_REF] Imanuvilov | Determination of secondorder elliptic operators in two dimensions from partial Cauchy data[END_REF][START_REF] Krupchyk | Inverse problems with partial data for a magnetic Schrödinger operator in an infinite slab or on a bounded domain[END_REF][START_REF] Kenig | Recent progress in the Calderón problem with partial data[END_REF].

Problème 2

En ce qui concerne le problème 2, notons que dans le cas des coefficients à support compact, il est équivalent au problème 1, voir [START_REF]Multidimensional inverse spectral problem for the equation -∆ψ+(v(x)-Eu(x))ψ = 0[END_REF] (A = 0), [START_REF]Inverse scattering problems for Schrödinger operators with magnetic and electric potentials[END_REF] (A = 0), [START_REF]Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF] (A = 0, potentiel de fond non nul) et l'article III de la présente thèse (A = 0, coefficients matriciels, potentiel de fond non nul).

Notons également que le problème 2 dans le cas scalaire avec A = 0 est beaucoup étudié, voir par exemple [START_REF]Multidimensional inverse spectral problem for the equation -∆ψ+(v(x)-Eu(x))ψ = 0[END_REF][START_REF] Hähner | New stability estimates for the inverse acoustic inhomogeneous medium problem and applications[END_REF][START_REF]An iterative approach to non-overdetermined inverse scattering at fixed energy[END_REF] et les références qui y sont contenues. Le cas des coefficients matriciels n ≥ 2 avec A = 0 a été étudié dans [START_REF]Monochromatic recontruction algorithms for two-dimensional multichannel inverse problems[END_REF].

Dans le cas où A = 0 en dimension d ≥ 3, il est connu que si A j et Q sont des fonctions réelles scalaires décroissant exponentiellement à l'infini, l'amplitude de diffusion f A,V à énergie fixée détermine A et Q de (2) à une transformation de jauge [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF] près, voir [START_REF] Novikov | The ∂-equation in the multidimensional inverse scattering problem[END_REF] (pour un petit curl A), [START_REF] Eskin | Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy[END_REF] (A ∈ C d+5 , Q ∈ C d+3 ), [START_REF] Päivärinta | Inverse scattering for the magnetic Schrödinger operator[END_REF] (A ∈ e -γ|x| W 1,∞ , V ∈ e -γ|x| L ∞ , γ > 0). Ce résultat a été géneralisé au cas matriciel dans les ouvrages [START_REF]Inverse scattering problems for the Schrödinger operators with external Yang-Mills potentials[END_REF] (pour A j ∈ C d+5 , Q ∈ C d+4 assez petits, A * j = A j , Q * = Q), [START_REF] Eskin | Global uniqueness in the inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials[END_REF] (pour A j ∈ C d+3 , Q ∈ L ∞ fonctions à support compact). Notons que la condition de décroissance exponentielle est naturelle car il existe un exemple de non unicité pour des coefficients de la classe Schwartz, voir [START_REF]Transparent potentials at fixed energy in dimension two. Fixed-energy dispersion relations for the fast decaying potentials[END_REF].

L'unicité pour le problème 2 en dimension d = 2 dans le cas matriciel n ≥ 1 a été démontrée par Xiaosheng [START_REF] Xiaosheng | Inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials in two dimensions at fixed energy[END_REF], en supposant que A j , Q sont petits et décroissent exponentiellement à l'infini.

Le premier résultat concernant la question de reconstruction pour le problème 2 dans le cas où A = 0, est la formule, dénommée approximation de Born, qui remonte à [START_REF] Faddeev | Uniqueness of the solution of the inverse scattering problem[END_REF] et qui permet de trouver une approximation au coefficient V à partir de f 0,V à énergie fixée :

V (p) = f 0,V (k, l) + O(E -1 2 ), E → +∞, p = k -l, (k, l) ∈ M E , V (p) = (2π) -d R d
e ipx V (x) dx.

Notons que les formules de ce type ne permettent pas de trouver les coefficients plus précisément qu'en O(E -1 2 ). Dans le cas où A = 0, des formules analogiques pour la détermination de curl A à partir de la limite de hautes énergies des données de diffusion ont été obtenues, en particulier, par Shiota [START_REF] Shiota | An inverse problem for the wave equation with first order perturbation[END_REF] (coefficients à support compact), Henkin-Novikov [START_REF] Henkin | A multidimensional inverse problem in quantum and acoustic scattering[END_REF] (coefficients lisses à décroissance rapide à l'infini), Arians [START_REF] Arians | Geometric approach to inverse scattering for the Schrödinger equation with magnetic and electric potentials[END_REF] (le cas de curl A de courte pourtée), Nicoleau [START_REF] Nicoleau | A stationary approach to inverse scattering for Schrödinger operators with first order perturbation[END_REF] (les cas de curl A de courte et longue portée). Dans l'article IV de la présente thèse nous établissons des formules analogiques pour la détermination de A, Q à partir des données de diffusion à l'énergie fixée mais dans la limite des petits coefficients.

En dimension d ≥ 3 la question de reconstruction pour le problème 2 (à une énergie E fixée) a été étudié par Weder-Yafaev [START_REF] Weder | On inverse scattering at a fixed energy for potentials with a regular behaviour at infinity[END_REF] (pour le comportement assez régulier de A, Q à l'infini) et par Salo [START_REF]Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field[END_REF] (pour des coefficients à support compact).

En dimension d = 2, R. Novikov [START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF] a proposé un algorithme qui permet de trouver V à partir de f 0,V avec la précision O(E -N -2

2 ) au sens L ∞ pour V tel que 1 , |α| ≤ N , pour quelque ε > 0 fixé. Cet algorithme est basé sur des idées des articles [START_REF] Manakov | The inverse scattering transform for the time dependent Schrödinger equation and Kadomtsev-Petviashvili equation[END_REF][START_REF] Grinevich | Inverse scattering problem for the twodimensional Schrödinger operator, the ∂-method and nonlinear equations[END_REF][START_REF] Novikov | Construction of two-dimensional Schrödinger operator with given scattering amplitude at fixed energy[END_REF][START_REF]Analogs of multisoliton potentials for the two-dimensional Schroedinger operator and the nonlocal Riemann problem[END_REF]. Nous généralisons cet algorithme bidimensionnel au cas où A = 0 dans l'article IV de la présente thèse.

(1 + |x| 2 ) ε 2 ∂ |α| V ∂x α ∈ L
De plus, notons que des modifications du problème 2 pour le cas des données partielles, quand il est seulement possible de mesurer le module de l'amplitude de diffusion, surviennent dans le cadre de la physique quantique, voir par exemple [START_REF] Chadan | Inverse problems in quantum scattering theory[END_REF][START_REF] Klibanov | Two reconstruction procedures for a 3D phaseless inverse scattering problem for the generalized Helmholtz equation[END_REF][START_REF]Inverse scattering without phase information[END_REF][START_REF]Formulas for phase recovering from phaseless scattering data at fixed frequency[END_REF] et l'article VI de la présente thèse (au moins, pour A = 0).

2 Tomographie acoustique d'un fluide en écoulement.

Résumé des articles I-V

Considérations préliminaires

Le premier groupe d'articles présentées dans le présent ouvrage sont consacrés à la solution du problème de tomographie acoustique de fluide en écoulement. Nous considérons un fluide dans un domaine borné D ⊂ R d , d ≥ 2, avec la célérité du son2 c = c(x), la densité ρ = ρ(x), la vitesse du fluide

v = v(x) et le coefficient d'absorption acoustique α = α(x, ω), α(x, ω) = ω ζ(x) α 0 (x) (14)
à la fréquence fixée ω. Sauf mention contraire, pour des raisons de simplicité, nous supposons que v, c, ρ, α 0 , ζ sont assez lisses dans D ; [START_REF] Baykov | Mode tomograpgy of moving ocean[END_REF] au voisinage de ∂D, les paramètres v, α 0 s'annulent et les paramètres c, ρ sont égaux aux valeurs de fond connues c 0 , ρ 0 .

(

) 16
Décrivons en quelques mots l'expérience tomographique considéré ; pour plus d'informations, voir Burov et autres [START_REF] Burov | Simulation of a functional solution to the acoustic tomography problem for data from quasipoint transducers[END_REF]. Des transducteurs acoustiques de type ponctuel sont placés sur le bord ∂D. Ils peuvent émettre et enregistrer des ondes acoustiques à une fréquence fixée ω appartenant à l'ensemble des fréquences admissibles Ω ⊂ (0, ∞), |Ω| < ∞. Nous considérons l'expérience suivante : on fixe un transducteur qui émit une onde acoustique, cette onde se propage dans le fluide, et on enregistre l'onde diffusée en utilisant les autres transducteurs. Ensuite, on change le transducteur émettant et, peut être, la fréquence des ondes émises, et on répète l'expérience. Le but est de déterminer les paramétres du fluide à partir des donées mesurées. D'un point de vue mathématique, nous considérons le modèle de propagation du son dans un fluide, qui a été envisagé dans différents cas particuliers dans les articles [START_REF] Roussef | Two-dimensional vector flow inversion by diffraction tomography[END_REF][START_REF] Rychagov | Reconstruction of fluid motion in acoustic diffraction tomography[END_REF][START_REF] Henkin | A multidimensional inverse problem in quantum and acoustic scattering[END_REF][START_REF] Burov | Simulation of a functional solution to the acoustic tomography problem for data from quasipoint transducers[END_REF]. Dans le cas général, ce modèle a été élaboré par O. D. Rumyantseva et A. S. Shurup du département d'acoustique de l'université de Moscou (article soumis à Acoustical Physics). Dans ce modèle, la pression acoustique harmonique en temps p(x, t) = Re(ψ(x)e -iωt ), ω > 0, satisfait à l'équation L ω ψ = F dans D, où L ω désigne l'opérateur différentiel suivant :

L ω = -∆ x -2iA ω (x)∇ x -U ω (x), x = (x 1 , . . . , x d ) ∈ D, A ω (x) = ωv(x) c 2 (x) + i 2 ∇xρ(x) ρ(x) , U ω (x) = ω 2 c 2 (x) + 2iω α(x,ω) c(x) , α(x, ω) = ω ζ(x) α 0 (x), (17)
et F décrit des sources du son. Notons que pour des raisons physiques, on a

c ≥ c min > 0, ρ ≥ ρ min > 0, α 0 ≥ 0, v = v, ζ = ζ dans D, pour des constantes c min , ρ min appropriées. ( 18
)
Les définitions (1) et [START_REF] Berezin | The Schrödinger equation, ser. Mathematics and its applications[END_REF]

impliquent que L ω = L A,V , où A = A ω , V = -U ω .
Notons par σ = σ(L • ) l'ensemble des ω ∈ (0, +∞) tels que 0 est une valeur propre de Dirichlet pour L ω dans D.

(

) 19
Pour l'opérateur L ω , ω ∈ σ, on définit la fonction de Green G ω = G ω (x, y) comme la solution du problème

L ω G ω (•, y) = δ y dans R d , ∂ ∂|x| -i ω c 0 G(x, y) = o(|x| -d-1 2 ), |x| → ∞, (20)
pour y ∈ R d fixé, où δ y (x) = δ(x -y), désignant la fonction δ de Dirac. On considère G ω (•, y) comme la pression acoustique à la fréquence ω correspondant à une source du son ponctuelle située en y.

Très souvent le problème de tomographie acoustique est formulé comme suit (voir, e.g., [START_REF] Burov | Simulation of a functional solution to the acoustic tomography problem for data from quasipoint transducers[END_REF]) :

Problème 3. Étant donnés G ω (x, y) pour x, y ∈ ∂D, ω ∈ Ω, Ω ⊂ (0, ∞), |Ω| < ∞, et c 0 , ρ 0 , trouver v, c, ρ, α 0 , ζ dans D.
Le problème 3 survient en tomographie océanique, où on s'intèresse à l'estimation de la température de l'océan et des courrants effectuants le transfert de chaleur. De plus, le problème 3 est motivé par des applications en imagerie médicale, où on s'intèresse à la détemination simultanée du flux sanguin et des charactéristiques scalaires d'un corps humain (par exemple la densité, la célérité du son ou l'absorption) sans danger pour le patient.

Nous considérons également l'opérateur de Dirichlet-

Neumann Λ ω = Λ A,V , A = A ω , V = -U ω , pour l'opérateur L ω = L A,V , ω ∈ σ, défini par les formules suivantes : Λ ω (ψ| ∂D ) = ∂ψ ∂ν | ∂D , L ω ψ = ψ dans D, (21)
où ∂/∂ν désigne la dérivée normale extérieure. Dans le cadre de la tomographie acoustique le problème de Dirichlet-Neumann inverse est formulé comme suit :

Problème 4. Étant donnés Λ ω pour ω ∈ Ω, Ω ⊂ (0, ∞), |Ω| < ∞, et c 0 , ρ 0 , trouver v, c, ρ, α 0 , ζ dans D.
Le problème 4 est une modification du problème 1.

On peut réduire le problème 3 au problème 4 comme suit (voir [START_REF] Berezansky | On uniqueness theorem in inverse problem of spectral analysis for Schrödinger equation[END_REF][START_REF]Multidimensional inverse spectral problem for the equation -∆ψ+(v(x)-Eu(x))ψ = 0[END_REF][START_REF] Nachman | Reconstructions from boundary measurements[END_REF]). Soient A 0 ω et U 0 ω des coefficients définis comme dans [START_REF] Berezin | The Schrödinger equation, ser. Mathematics and its applications[END_REF] en place de L ω ). Il s'avère que les opérateurs Λ ω , Λ 0 ω et les fonctions G ω , G 0 ω sont liés par l'équation intégrale suivante sur ∂D, qui remonte à Nachman [START_REF] Nachman | Reconstructions from boundary measurements[END_REF] (dans le cas où A = 0, n = 1) :

mais correspondant aux paramètres v = 0, α = 0, c = c 0 , ρ = ρ 0 de fond connus. Posons L 0 ω = L A,V , où A = A 0 ω , V = -U 0 ω . On suppose que ω ∈ σ(L 0 • ). Alors,
G ω (x, y) -G 0 ω (x, y) = ∂D ∂D G 0 ω (x, z)(Λ ω -Λ 0 ω )(z, w)G ω (w, y) dz dw, x, y ∈ ∂D, (22)
où la fonction (Λ ω -Λ 0 ω )(z, w) est le noyau intégral de l'opérateur Λ ω -Λ 0 ω défini de même façon que dans la formule [START_REF] Aktosun | Inverse problems on the line without phase information[END_REF]. L'équation [START_REF] Burov | The use of low-frequency noise in passive tomography of the ocean[END_REF] découle des définitions des opérateurs Λ, Λ 0 ω et des fonctions G ω , G 0 ω et de la seconde formule de Green, voir [24, éq. 11]. L'équation [START_REF] Burov | The use of low-frequency noise in passive tomography of the ocean[END_REF] permet de réduire le problème 3 au problème 4, que l'on va étudier dans la présente thèse.

Dans la première partie du présent ouvrage nous étudions les questions d'unicité (articles I, II) et de reconstruction (articles III, IV, V) pour le problème 4, voir la Fig. 1.

Notons que le schéma de reconstruction présenté sur la Fig. 1 a été implementé numériquement dans certains cas particuliers en dimension d = 2 par le groupe de Burov du département acoustique de l'université de Moscou. En particulier, ce schéma dans le cas A = 0 a été étudié numériquement dans [START_REF] Burov | Simulation of a functional solution to the acoustic tomography problem for data from quasipoint transducers[END_REF].

Les résultats de la modélisation numérique pour le cas A = A ont été annoncés dans l'exposé de Shurup-Rumyantseva [START_REF] Shurup | Numerical simulation of the functional approach for recovering vector fields in acoustic tomography[END_REF] (article soumis à Acoustical Physics). En particulier, ils présentent un exemple de reconstruction approchée A du champ vectoriel A (qui figure dans l'opérateur L A,V ) de la Fig. 2 dans le cas des données non bruitées. Sur cette figure, A 1 et A 2 désignent, respectivement, des parties rotationnele (∇ • A 1 = 0) et potentielle (curl A 2 = 0) de A. Notons que l'on utilise des données de sources ponctuelles pour deux fréquences différentes pour la reconstruction de A 2 , voir les articles II et V de la présente thèse. On peut voir que dans le cas des données non bruitées l'algorithme est très précis. Selon la recherche conduite par Shurup-Rumuantseva, l'algorithme est également suffisament stable par rapport au bruit pour des applications pratiques. Plus précisement, nous considérons l'opérateur L A,V de (1) avec A = A et V = V . Les résultats principaux de l'article I peuvent être formulés comme suit : 1), ( 2)). Il est connu que Cependant, nous montrons que la condition A = A, V = V , permet de se débarasser de cette non-unicité et trouver A et V uniquement.

Théorème 1. Soit D ⊂ R d , d ≥ 3, un domaine borné simplement connexe de bord ∂D ∈ C 1 connexe par arcs. Soient A 1 , A 2 ∈ W 1,∞ (D, R d ) et V 1 , V 2 ∈ L ∞ (D, R). Supposons que 0 n'est pas une valeur propre de Dirichlet pour L A 1 ,V 1 et L A 2 ,V 2 dans D, et soient Λ 1 et Λ 2 les opérateurs de Dirichlet-Neumann pour L A 1 ,V 1 et L A 2 ,V 2 , respectivement. Alors, A 1 = A 2 et V 1 = V 2 lorsque Λ 1 = Λ 2 . Théorème 2. Soit D ⊂ R 2 un domaine borné simplement connexe de bord ∂D ∈ C ∞ . Soient A 1 , A 2 ∈ W 2,p (D, R 2 ) et V 1 , V 2 ∈ W 1,p (D, R), p > 2. Supposons que 0 n'est pas une valeur propre de Dirichlet pour L A 1 ,V 1 et L A 2 ,V 2 dans D, et soient Λ 1 et Λ 2 les opérateurs de Dirichlet-Neumann pour L A 1 ,V 1 et L A 2 ,V 2 , respectivement. Alors, A 1 = A 2 et V 1 = V 2 si Λ 1 = Λ 2 .

Pour la démonstration des théorèmes 1 et 2 nous représentons l'opérateur L

A,V comme L A,V = L A,Q , où Q = V + i∇ • A -A • A (cf. (
) (r A ) ( ˆr A ) ( ˆ1 r A ) ( ˆ2 r

Résumé de l'article II

Dans l'article II, d'une part, nous continuons notre étude sur la question d'unicité pour le problème 4 de la sous-section 2.1. Nous montrons que les données mesurables à deux fréquences fixées déterminent la vitesse du fluide, la célérité du son et la densité du fluide si le fluide est supposé non absorbant. Nous montrons également que les données mesurables à trois fréquences fixées déterminent toutes les paramètres du fluide si on suppose que l'absorption n'est pas constante en fonction de fréquence. D'autre part, nous présentons des exemples des fluides qui ne sont pas distinguables à partir des mesures au bord dans le cadre du modèle considéré pour toutes les fréquences. En vue des résultats d'unicité susmentionnés, l'absorption de ces fluides non distinguables ne depend pas de la fréquence (i.e. α(x, ω) = α 0 (x)), ce qui montre que la dépendance à la fréquence est suffisante et nécessaire pour l'identifiabilité du fluide. Rappelons que α est défini dans la formule [START_REF] Barceló | Numerical approximation of the potential in the two-dimesional inverse scattering problem[END_REF]. D'un point de vue mathématique, nous supposons que

c ∈ W 1,∞ (D, R), ρ ∈ C(D) ∪ C 2 (D), v ∈ W 1,∞ (D, R d ), α 0 ∈ C(D), ζ ∈ C(D), ζ = 0, où d ≥ 3, (23)
ou bien que

c ∈ W 2,p (D, R), ρ ∈ W 3,p (D, R), v ∈ W 2,p (D, R d ), α 0 ∈ W 1,p (D, R), ζ ∈ C(D), ζ = 0, où p > 2, d = 2. ( 24
)
Notons que l'on ne requiert pas [START_REF] Berezansky | On uniqueness theorem in inverse problem of spectral analysis for Schrödinger equation[END_REF]. Les résultats principaux de l'article II concernant l'unicité peuvent être formulés comme suit :

Théorème 3. Soit D ⊂ R d (d ≥ 2) un domaine simplement connexe borné de bord ∂D connexe par arcs, où ∂D ∈ C ∞ (d = 2) ou ∂D ∈ C 1 (d ≥ 3). Soient L (j) ω et Λ (j)
ω les opérateurs de [START_REF] Berezin | The Schrödinger equation, ser. Mathematics and its applications[END_REF], [START_REF] Burov | Multifrequency generalization of the Novikov algorithm for the two-dimensional inverse scattering problem[END_REF] [START_REF] Burov | Application of the functional-analytical Novikov algorithm for the purposes of ocean tomography[END_REF] (2) , ρ (1) = Cρ (2) , v (1) = v (2) , où

correspondant aux coefficients c (j) , ρ (j) , v (j) , α (j) 0 , ζ, satisfaisant à (18),
pour d ≥ 3 ou à (18), (24) pour d = 2. Supposons que α (j) 0 = 0, j = 1, 2. Soient ω 1 , ω 2 ∈ [0, +∞) \ (σ(L (1) • ) ∪ σ(L (2) • )), ω 1 = ω 2 . Alors, si Λ (1) ω = Λ (2) ω , ω ∈ {ω 1 , ω 2 }, on a c (1) = c
C = const > 0. Théorème 4. Soit D ⊂ R d (d ≥ 2) un domaine simplement connexe borné de bord ∂D connexe par arcs, où ∂D ∈ C ∞ (d = 2) ou ∂D ∈ C 1 (d ≥ 3). Soient L (j) ω et Λ (j)
ω les opérateurs de [START_REF] Berezin | The Schrödinger equation, ser. Mathematics and its applications[END_REF], [START_REF] Burov | Multifrequency generalization of the Novikov algorithm for the two-dimensional inverse scattering problem[END_REF] correspondant aux coefficients c (j) , ρ (j) , v (j) , α (j) 0 , ζ, satisfaisant à (18), [START_REF] Burov | Application of the functional-analytical Novikov algorithm for the purposes of ocean tomography[END_REF] pour d ≥ 3 ou à (18), [START_REF] Burov | Simulation of a functional solution to the acoustic tomography problem for data from quasipoint transducers[END_REF] (2) , ρ (1) = Cρ (2) , v (1) = v (2) , α (1) = α (2) , où

pour d = 2. Soient ω 1 , ω 2 , ω 3 ∈ [0, +∞) \ (σ(L (1) • ) ∪ σ(L (2) • )) trois fréquences mutuellement différentes. Alors, si Λ (1) ω = Λ (2) ω , ω ∈ {ω 1 , ω 2 , ω 3 }, on a c (1) = c
C = const > 0 et α (j) (x, ω) = ω ζ (j) (x) α (j) 0 (x).
Pour formuler le résultat concernant la non-unicité, nous avons besoin d'introduire quelques notations. Soit

h une fonction réelle supportée dans D, h ∈ C 2 (D), |∇h| 2 < 1 dans D. ( 25
) Posons c (1) ≡ const > 0, ρ (1) ≡ const > 0, v (1) ≡ 0, α (1)
0 ≡ const > -1 2 min x∈D ∆h(x), ( 26
) et c (2) = c (1) (1 -|∇h| 2 ) -1 2 , ρ (2) ≡ const > 0, v (2) = c (1) (1 -|∇h| 2 ) -1 ∇h, α (2) 0 = (1 -|∇h| 2 ) -1 2 α (1) 0 + 1 2 ∆h . (27)
Notons que les paramètres v (j) , c (j) , ρ (j) , α

0 avec ζ (j) = 0 satisfont à ( 18), ( 23), ( 24)

pour j = 1, 2. Λ ω 1 ,. . .,Λ ω N -classes de conjugaison de L ω1 , . . ., L ω N -F , Q ω 1 , . . ., Q ω N v, c, ρ, ζ, α 0 (29) F , f 1 , f 2 , f 3 , ζ, α 0 /c (28)
? j) , ρ (j) , v (j) , α (j) , j = 1, 2, sont définis par [START_REF] Chadan | Inverse problems in quantum scattering theory[END_REF], [START_REF] Chow | On compact complex analytic varieties[END_REF]. Soient L

(j) ω , Λ (j)
ω les opérateurs correspondant au paramètres c (j) , ρ (j) , v (j) , α

(j) 0 et ζ (j) = 0. Alors, Λ (1)
ω = Λ (2) ω pour tout ω ∈ [0, ∞) \ σ, où σ = σ(L (1) • ) = σ(L (2) • ).
Le schèma de démonstration des théorèmes 3 et 4 est donné sur la Fig. 3. Pour ces démonstrations nous considérons les opérateurs L ω 1 , . . .,

L ω N et Λ ω 1 , . . ., Λ ω N , N ≥ 2, de (17), (21) correspondant aux paramètres c, ρ, v, α fixé et tels que ω j ∈ σ(L • ) pour tout j. Il est connu que les opérateurs Λ ω 1 , . . ., Λ ω N déterminent uniquement F , Q ω 1 , . . ., Q ω N dans D, où F = curl v c 2 , Q ω = f 1 -ω 2 f 2 + iωf 3 -2iω 1+ζ α 0 c , (28)
f 1 = ρ 1 2 ∆ρ -1 2 , f 2 = 1 c 2 + v c 2 v c 2 , f 3 = ∇ • v c 2 -v c 2 ∇ ln ρ, (29)
voir [START_REF] Guillarmou | Identification of a connection from Cauchy data on a Riemann surface with boundary[END_REF] (d = 2) et [START_REF] Krupchyk | Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential[END_REF] (d ≥ 3), en tenant compte de ce que L ω = L Aω,-Uω , curl A ω = ωF , et des formules (1), ( 2), [START_REF] Berezin | The Schrödinger equation, ser. Mathematics and its applications[END_REF]. Autrement dit, l'opérateur Λ ω j détermine la classe de conjugaison de l'opérateur L ω j , et les fonctions F , Q ω sont des invariants de cette classe.

Si Q ω est connu pour plusieurs ω, on peut considérer l'expression [START_REF] Colton | Integral equation methods in scattering theory[END_REF] pour

Q ω comme un système déterminant f 1 , f 2 , f 3 , ζ, α 0 /c. Par exemple Re Q ω = f 1 -ω 2 f 2 , et la connaissance de Q ω pour deux ω différents est suffisante pour déterminer f 1 et f 2 à partir du système linéaire.
Ayant trouvé F , f 1 , f 2 , f 3 , on peut déterminer ρ, c, v en utilisant les équations (29) et les valeurs tangentes au bord de v c 2 et ∇ρ, qui sont également uniquement déterminées par l'opérateur Λ ω à une fréquence fixée (voir [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF]).

D'autre part, on peut voir que si ζ = 0, alors Im Q ω = ω(f 3 -2 α 0 c ). Il est donc impossible de trouver f 3 et α 0 c eux-mêmes de cette expression quand Q ω est connu pour plusieurs ω. Cette observation nous amène au théorème 5.

Résumé de l'article III

Dans l'article III nous abordons la question de reconstruction pour le problème 4 de la sous-section 2.1. Nous établissons les formules qui permettent de réduire le problème 4 au problème 2 de la sous-section 1.3. Par souci de concision, nous allons formuler ces résulats dans le cas des données classiques de diffusion.

Revenons au problème 2 de la sous-section 1.3. Rapellons que la fonction ψ + A,V (•, k) satisfait à l'équation de type Lippmann-Schwinger [START_REF] Aubin | Nonlinear analysis on manifolds. Monge-Ampère equations[END_REF]. Cette equation et ses versions dérivées par rapport à x 1 , . . ., x d sont considérées pour k fixé comme une équation vectorielle pour [START_REF] Aubin | Nonlinear analysis on manifolds. Monge-Ampère equations[END_REF] n'admet pas de solution unique.

ψ + A,V (•, k), ∂ ∂x 1 ψ + A,V (•, k), . . ., ∂ ∂x d ψ + A,V (•, k) ∈ L ∞ (R d , M n (C)). Soit E + A,V l'ensemble des k ∈ R d \ 0 tels que l'équation
On désigne par C 1,β (∂D, M n (C)) l'éspace des fonctions de classe C 1 à valeurs dans M n (C) sur ∂D, dont les premières dérivées partielles sont β-höldériennes. On munit cet espace avec la norme

ϕ C 1,β (∂D) = ϕ C 1 (∂D) + max τ max i,j sup x 1 =x 2 |∂ τ ϕ ij (x 1 ) -∂ τ ϕ ij (x 2 )| |x 1 -x 2 | β , où τ est un vecteur unitaire tangent à ∂D, ϕ(x) = (ϕ ij (x)) ∈ M n (C), ∂ k = ∂/∂x k .
f A,V (k, l) = (2π) -d ∂D ∂D e -ilx (Λ A,V -Λ 0,0 )(x, y, E)ψ + A,V (y, k) dy dx, (30)
où k, l ∈ R d \ (0 ∪ E + A,V ), k 2 = l 2 = E, et la fonction ψ + A,V peut être trouvée à partir de l'équation ψ + A,V (x, k) = e ikx id n + ∂D A + (x, y, k)ψ + A,V (y, k) dy, x ∈ ∂D, A + (x, y, k) = ∂D G + (x -z, k)(Λ A,V -Λ 0,0 )(z, y, E) dz, x, y ∈ ∂D, (31)
où k ∈ R d \ (0 ∪ E + A,V ), k 2 = E.
De plus, l'équation (31) pour k fixé est une équation de Fredholm de seconde espèce qui admet la solution unique

ψ + A,V (•, k) ∈ C 1,β (∂D, M n (C)) pour chaque β ∈ (0, 1) fixé.

Nous considérons (31) comme une équation pour trouver ψ +

A,V à partir de l'opérateur Λ A,V -Λ 0,0 . De plus, nous considérons (30) comme une formule explicite pour trouver f A,V à partir de ψ + A,V (x, •), x ∈ ∂D, et de Λ A,V -Λ 0,0 . Notons que du point de vue des applications, l'équation [START_REF] Dolbeault | Surfaces de Riemann de bord donné dans CP n[END_REF] peut être résolue d'une manière stable et efficace pour des A + suffisament petits. Cette situation correspond au cas des petits A et V . Cependant, des applications (par exemple en tomographie de l'océan) amènent au cas, où A est petit et V est proche d'un coefficient connu V 0 tel que :

soit V 0 (x) une matrice diagonale pour chaque x, soit V 0 = V 0 v 0 , où V 0 ∈ M n (C) et v 0 est une fonction scalaire. ( 32
)
Il est possible de modifier le théorème 6 et d'établir des équations stables pour les cas [START_REF]Chaînes holomorphes de bord donné dans CP n[END_REF]. Pour formuler ces résultats, nous avons besoin d'introduire la fonction R + 0,V 0 (x, y, k) comme suit :

R + 0,V 0 (x, y, k) = G + (x -y, k)id n + R d G + (x -z, k)V 0 (z)R + 0,V 0 (z, y, k) dz, (33)
où x, y ∈ R d , k ∈ R d \ 0.
Dans la physique quantique l'équation [START_REF] Santos Ferreira | Determining a magnetic Schrödinger operator from partial Cauchy data[END_REF] est connu comme l'équation de Dyson qui relie les fonctions de Green pour les opérateurs L 0,0 et L 0,V 0 . Notons que les équations de ce type remontent à [START_REF] Dyson | The S matrix in quantum electrodynamics[END_REF] dans le cas particulier de l'électrodynamique quantique et à [START_REF] Schwinger | On Green functions of quantized fields I + II[END_REF] dans un cadre plus géneral. Nous considérons [START_REF] Santos Ferreira | Determining a magnetic Schrödinger operator from partial Cauchy data[END_REF] pour y, k fixés comme une équation intégrale pour [START_REF]Chaînes holomorphes de bord donné dans CP n[END_REF]. Supposons que E n'est pas une valeur propre de Dirichlet pour les opérateurs

R + 0,V 0 (x, y, k) = G + (x -y, k)id n + e ik(x-y) r + 0,V 0 (x, y, k), où r + 0,V 0 (•, y, k) ∈ L ∞ (R d , M n (C)). Notons que l'équation (33) admet une solution unique pour chaque y ∈ R d si et seulement si k ∈ E + 0,V 0 . Notons également que R + 0,0 (x, y, k) = G + (x -y, k)id n . Théorème 7. Soit D ⊂ R d un domaine borné au bord ∂D ∈ C 2 et soient A 1 , . . ., A d , V , V 0 des fonctions höldériennes à valeurs dans M n (C), supportées dans D et telles que V 0 satisfait à (
L A,V et L 0,V 0 dans D, où A = (A 1 , . . . , A d ).
Alors, la formule suivante est valable :

f A,V (k, l) = f 0,V 0 (k, l) +(2π) -d ∂D ψ + 0,V 0 (x, -l)(Λ A,V -Λ 0,V 0 )(x, y, E)ψ + A,V (y, k) dy dx, (34)
où k, l ∈ R d \ 0, k 2 = l 2 = E, k ∈ E + A,V ∪ E + 0,V 0 , et la fonction ψ + peut être trouvée de l'équation ψ + A,V (x, k) = ψ + 0,V 0 (x, k) + ∂D A + 0,V 0 (x, y, k)ψ + A,V (y, k) dy, x ∈ ∂D, A + 0,V 0 (x, y, k) = ∂D R + 0,V 0 (x, z, k)(Λ A,V -Λ 0,V 0 )(z, y, E) dz, x, y ∈ ∂D, (35)
où k ∈ R d \ (0 ∪ E + A,V ∪ E + 0,V 0 ), k 2 = E.
De plus, l'équation [START_REF] Dubrovin | The Schrödinger equation in a periodic field and Riemann surfaces[END_REF] pour k fixé est une équation de Fredholm de seconde espèce qui admet la solution unique

ψ + A,V (•, k) ∈ C 1,β (∂D, M n (C)) pour chaque β ∈ (0, 1) fixé.
Les théorèmes 6, 7 dans le cas A = 0, n = 1 remontent à [START_REF]Multidimensional inverse spectral problem for the equation -∆ψ+(v(x)-Eu(x))ψ = 0[END_REF][START_REF]Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF]. La dérivation des formules [START_REF] Deligne | Quantum fields and strings : A course for mathematicians[END_REF], [START_REF] Dressler | Stationary solutions of the Vlasov-Fokker-Planck equation[END_REF] et des équations [START_REF] Dolbeault | Surfaces de Riemann de bord donné dans CP n[END_REF], [START_REF] Dubrovin | The Schrödinger equation in a periodic field and Riemann surfaces[END_REF] est basée sur les identités de type Alessandrini, qui remontent à [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF]. Plus précisément, nous établissons l'idéntité suivante :

D u 0 (x) L A,V -L 0,V 0 u(x) dx = ∂D u 0 (x) Λ A,V -Λ 0,V 0 (u| ∂D )(x) dx, (36)
pour des fonctions u 0 , u suffisament régulières dans D à valeurs dans [START_REF] Dyson | The S matrix in quantum electrodynamics[END_REF] découle de la seconde formule de Green. Les formules [START_REF] Deligne | Quantum fields and strings : A course for mathematicians[END_REF], [START_REF] Dressler | Stationary solutions of the Vlasov-Fokker-Planck equation[END_REF] et les équations [START_REF] Dolbeault | Surfaces de Riemann de bord donné dans CP n[END_REF], [START_REF] Dubrovin | The Schrödinger equation in a periodic field and Riemann surfaces[END_REF] peuvent être dérivées de la formule [START_REF] Baratchart | Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation[END_REF], de l'équation [START_REF] Aubin | Nonlinear analysis on manifolds. Monge-Ampère equations[END_REF] et de l'identité [START_REF] Dyson | The S matrix in quantum electrodynamics[END_REF].

M n (C) telles que L A,V u = Eu, L 0,V 0 u 0 = Eu 0 et V 0 (x)u 0 (x) = u 0 (x)V 0 (x) dans D. L'identité
Pour démontrer que l'équation [START_REF] Dolbeault | Surfaces de Riemann de bord donné dans CP n[END_REF] est l'équation de Fredholm de seconde espèce (pour la démonstration de [START_REF] Dubrovin | The Schrödinger equation in a periodic field and Riemann surfaces[END_REF] on utilise la même idée), nous la réécrivons dans une forme opérationnelle :

ψ + A,V = e ikx + G + (Λ A,V -Λ 0,0 )ψ + , Λ A,V -Λ 0,0 = N A,V S A,V . (37)
Dans cette formule

G + correspond au noyau intégral G + (x -y, k) (au sens de Schwartz), S A,V envoie une fonction f sur ∂D à la solution ψ du problème (4), (38)
et l'opérateur N A,V est défini comme suit :

(N A,V ψ)(x) = D ∂Γ ∂ν x (x, y, E)(L A,V -L 0,0 )ψ(y) dy, x ∈ ∂D,
où Γ est la fonction de Green pour le problème de Dirichlet pour l'opérateur ∆ + E dans D, et ∂/∂ν x désigne la dérivée normale extérieure.

On peut montrer que les applications suivantes sont linéaires et continues :

C 1,β (∂D) S A,V -→ C 1 (D) N A,V -→ C 2 (∂D) i → C 1,β (∂D) G + -→ C 1,β (∂D), (39)
où i désigne l'inclusion, et on suppose que des fonctions sont à valeurs dans M n (C). En tenant compte de la compacité de l'opérateur i, on voit que (37) est une équation de Fredholm de seconde espèce dans C 1,β (∂D, M n (C)).

Résumé de l'article IV

Dans l'article IV nous continuons à étudier la question de reconstruction pour le problème 4 de la sous-section 1.2 à une énergie E > 0 fixée. Nous établissons un algorithme, qui permet de trouver une solution approchée du problème 2 en dimension d = 2. Cet algorithme est basé sur le problème de Riemann-Hilbert non local (également connu comme le problème Riemann-Hilbert-Manakov) et géneralise l'algorithme de l'ouvrage [START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF] au cas où A = 0.

Notons qu'un algorithme en dimension d ≥ 3 avec n = 1, A = 0, pour des coefficients à support compact a été proposé dans [START_REF]Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field[END_REF] comme une géneralisation de l'algorithme de [START_REF]Multidimensional inverse spectral problem for the equation -∆ψ+(v(x)-Eu(x))ψ = 0[END_REF].

Nous considérons le cas scalaire n = 1 en dimension d = 2. Pour formuler les résultats principaux, nous avons besoin d'introduire quelques notations. Soient (A, V ) un couple de coefficients donnés décroissants à l'infini et soient (A div , V div ) le couple de coefficients decroissants à l'infini, liés aux coefficients A, V par une transformation de jauge [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF] et satisfaisant à l'équation ∇ • A div = 0. Un tel couple de coefficients est déterminé uniquement. Dans le cadre de la physique quantique les couples (A, V ) et (A div , V div ) décrivent la même réalité physique, mais la condition ∇ • A div = 0, dénommé la jauge de Coulomb, permet de se débarasser des degrés de liberté redondants. Notre algorithme permet de trouver (A div , V div ) d'une manière approchée mais efficace à partir de l'amplitude f A,V à E fixé.

Soit E > 0 un nombre fixé qui n'est pas une résonance réelle pour L A,V . L'amplitude de diffusion f A,V peut être considérée comme une fonction sur le produit

T 2 = T × T de deux cercles unité T = {λ ∈ C : |λ| = 1} comme suit : f (e iφ , e iθ ) := f A,V (k, l), φ, θ ∈ [0, 2π), k = E 1 2 (cos φ, sin φ), l = E 1 2 (cos θ, sin θ).
Nous introduisons les opérateurs suivants (voir [START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF]) :

(P ± (λ)u)(λ ) = -πi T u(λ )χ ±i λ λ - λ λ f (λ , λ )|dλ |, (Q ± (x)u)(λ) = πi T h ± (λ, λ )e(λ, λ , x)χ ±i λ λ - λ λ u(λ )|dλ |, e(λ, λ , x) = exp -i √ E 2 (λ -λ )z + (λ -1 -λ -1 z , z = x + iy, (C ± u)(λ) = 1 2πi T u(ξ) ξ -λ(1 ∓ 0) dξ, B(x) = C + Q -(x) -C -Q + (x), où λ, λ ∈ T , x ∈ R 2 , |dλ| = dλ/(iλ)
, χ désigne la fonction de Heaviside, et les fonctions h ± (λ, λ ), λ, λ ∈ T , sont déterminées par l'équation suivante :

h ± (λ, λ ) + (P ± (λ)h ± (λ, •))(λ ) = f (λ, λ ), (λ, λ ) ∈ T 2 . ( 40
)
Le résultat principal de l'article IV consiste en un algorithme qui permet de trouver une approximation A div appr , V div appr aux coefficients A div , V div , et qui peut être formulé comme suit : Algorithme 1. Soit f = f A,V l'amplitude de diffusion pour l'opérateur L A,V à l'énergie fixée E > 0, qui n'est pas une résonance réelle. Alors, on peut trouver une approximation A div appr , V div appr aux coefficients A div , V div en utilisant le schèma suivant :

f -→ h ± -→ µ + -→ µ ± -→ A div appr , V div appr . (41)
Les fonctions h ± , µ + et µ ± sont déterminées successivement en utulisant les équations (40), [START_REF] Eskin | Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy[END_REF] et la formule explicite [START_REF]Inverse scattering problems for Schrödinger operators with magnetic and electric potentials[END_REF] :

µ + (x, λ) + (B(x)µ + (x, •))(λ) = 1, x ∈ R 2 , λ ∈ T, (42) µ
± (x, λ) = µ + (x, λ) + (Q ± (x)µ + (x, •))(λ), x ∈ R 2 , λ ∈ T. (43)
Finalement, les coefficients A div appr , V div appr sont définis comme suit :

A div appr (x) = 1 2 curl ln T µ + (x, ζ)|dζ| , (44)
V div appr (x) = 2|A div appr (x)| 2 + √ E 2π T ∂µ -(x, ζ) ∂z dζ + √ E ∂ ∂ z T µ + (x, ζ) dζ ζ 2 T µ + (x, ζ)|dζ| , (45)
où ∂ ∂z = 1 2 ( ∂ ∂x 1 -i ∂ ∂x 2 ), ∂ ∂z = 1 2 ( ∂ ∂x 1 + i ∂ ∂x 2 ). Théorème 8. Soient E > 0 et x ∈ R 2 fixés. Soit f ∈ C ∞ (T 2 ) une fonction sa- tisfaisant à f L 2 (T 2 ) < 1/(6π)
. Alors l'équation [START_REF]Lectures on linear partial differential equations[END_REF] admet une solution unique h ± ∈ L 2 (T 2 ) et l'équation [START_REF] Eskin | Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy[END_REF] admet une solution unique µ + (x, •) ∈ L 2 (T ). De plus, le dénominateur dans la formule (45) ne s'annule pas, les fonctions A div appr , V div appr sont bornées, décroissent à l'infini et satisfont à ∇ • A div appr = 0. Finalement, l'opérateur L A div appr ,V div appr admet f comme amplitude de diffusion à énergie fixée E. Notons que les conditions requises sur la régularité et la petitesse de f sont surévaluées. Notons également que le calcul des coefficients A div et V div appr aux points différents peut être efféctuée de façon parallèle. De plus, les résultats annoncés dans l'éxposé [START_REF] Shurup | Numerical simulation of the functional approach for recovering vector fields in acoustic tomography[END_REF] témoignent de la convergence rapide de A div appr , V div appr vers A div , V div quand E → +∞. Une étude théorique de la convergence sera menée dans un article ultérieur.

En outre, nous démontrons que la linéarisation de l'algorithme 1 pour le cas des petits coefficients A, V amène aux formules de l'approximation de Born.

Indiquons les idées principales sur lesquelles est basé l'algorithme 1. Nous considérons les solutions de diffusion géneralisées ψ(•, k), k ∈ K E ,

K E = k ∈ C 2 \ R 2 : k 2 = E ,
du problème 10 qui remontent à Faddeev [START_REF]Growing solutions of the Schrödinger equation[END_REF]. Les fonctions ψ(x, k) = e ikx µ(x, k) admettent le développement asymptotique suivant :

µ(x, k(λ)) = µ ± 0 (x) + µ ± 1 (x)λ ±1 + o(|λ| ±1 ), |λ| ±1 → 0, k = (k 1 , k 2 ), k 1 = 1 2 E 1 2 (λ + λ -1 ), k 2 = i 2 E 1 2 (λ -1 -λ). (46)
Notons que une fixation de jauge permet d'obtenir des relations supplémentaires pour les coefficients µ ± 0 . En particulier,

si

A 1 = iA 2 , alors µ - 0 = 1 ; si A 1 = -iA 2 , alors µ + 0 = 1.
Ensuite, en utilisant le développement [START_REF] Faddeev | Uniqueness of the solution of the inverse scattering problem[END_REF] dans la première équation de la formule [START_REF] Arians | Geometric approach to inverse scattering for the Schrödinger equation with magnetic and electric potentials[END_REF] et en égalant les coefficients des mêmes puissances de λ, nous obtenons les équations pour trouver A et V à partir de µ ± 0 et µ ± 1 . Cette relation remonte à [START_REF] Grinevich | Analogs of multisoliton potentials for the two-dimensional Schrödinger operator[END_REF]. Il reste à trouver les coefficients µ ± 0 et µ ± 1 en fonction de l'amplitude f . Il est connu que la fonction µ(x, k(λ)) est uniformément bornée pour x ∈ R 2 , k ∈ K E , continue par rapport à λ ∈ C \ T , et satisfait à l'équation de ∂ :

∂ ∂λ µ(x, k(λ) = r(x, λ)µ x, k(-1/ λ) , λ ∈ C \ (T ∪ 0), (47)
où la fonction r est uniformément petite pour des grands E. De plus, les sauts µ ± (x, λ) = µ(x, k(λ ± 0λ)) sur le cercle T sont liés par l'équation [START_REF]Inverse scattering problems for Schrödinger operators with magnetic and electric potentials[END_REF], où µ + est supposé inconnu, voir [START_REF] Grinevich | Inverse scattering problem for the twodimensional Schrödinger operator, the ∂-method and nonlinear equations[END_REF][START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF].

En imposant la condition de jauge A 1 = iA 2 et en posant r = 0 dans [START_REF]Growing solutions of the Schrödinger equation[END_REF], on se ramène au problème de Riemann-Hilbert non local, pour trouver une approximation µ appr à µ :

Problème 5. Étant donné Q ± , trouver une fonction µ appr (x, k(λ)), x ∈ R 2 , λ ∈ C\T , uniformément bornée, continue, µ appr (x, k(λ)) = 1 + O(|λ| -1 ), λ → ∞, analytique en λ ∈ C \ (T ∪ 0) pour tout x ∈ R 2 , et satisfaisant à la condition de saut µ ±,appr (x, λ) = µ + appr (x, λ) + (Q ± (x)µ + appr (x, •))(λ), x ∈ R 2 , λ ∈ T, µ ±,appr (x, λ) = µ appr (x, k(λ ± 0λ)), λ ∈ T, (48)
où µ + appr est une fonction uniformément bornée continue inconnue.

L'étude des tels problèmes remonte à [START_REF] Manakov | The inverse scattering transform for the time dependent Schrödinger equation and Kadomtsev-Petviashvili equation[END_REF][START_REF] Fokas | On the inverse scattering of the time-dependent Schrödinger equation and the associated Kadomtsev-Petviashvili equation[END_REF][START_REF]Analogs of multisoliton potentials for the two-dimensional Schroedinger operator and the nonlocal Riemann problem[END_REF], et la solution dans le cadre du problème de diffusion consideré remonte à [START_REF] Novikov | Construction of two-dimensional Schrödinger operator with given scattering amplitude at fixed energy[END_REF] (dans le cas "auto-adjoint").

Ayant résolu le problème 5, on détermine des approximations µ ± 0,appr et µ ± 1,appr aux coefficients µ ± 0 et µ ± 1 corréspondant au choix de jauge A 1 = iA 2 à partir de µ ±,appr , en tenant compte de [START_REF] Faddeev | Uniqueness of the solution of the inverse scattering problem[END_REF] et en utilisant la formule de Cauchy pour des fonctions analytiques. Finalement, nous passons de ces coefficients aux coefficients correspondant à la jauge ∇ • A div = 0 en utilisant une transformation de jauge [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF].

Ce schèma de reconstruction a été utilisé dans l'article [START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF], où le cas A = 0 a été considéré. De plus, dans l'article [START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF] il est démontré que la précision de reconstruction est en O(E -N -2

2 ) uniformément, si (1

+ |x| 2 ) ε 2 ∂ |α| Q ∂x α ∈ L 1 , |α| ≤ N , pour quelque ε > 0 fixé.

Résumé de l'article V

Dans l'article V nous établissons des formules pour trouver des paramètres du fluide à partir des fonctions F et Q ω de [START_REF] Colton | Integral equation methods in scattering theory[END_REF] connus pour plusieurs fréquences ω. Notons que pour déterminer les fonctions F , Q ω il suffit de reconstruire l'opérateur L ω à une transformation de jauge (8) près, parce que F et Q ω sont invariants par rapport à ces transformations. L'opérateur L ω , à son tour, peut être trouvé à une transformation de jauge près à partir de l'opérateur Λ ω en utilisant des formules des articles III et IV.

Pour des raisons de simplicité, dans la présente section nous supposons que

D = B 2 R , B 2 R = x ∈ R 2 : |x| < R , R > 0. Nous supposons également que les coefficients c, v, ρ, α 0 , ζ satisfont à (16) et à c ∈ C 2 (D), v ∈ C 2 (D, R 2 ), ρ ∈ C 2,β (D), β ∈ (0, 1], ζ ∈ C(D), α 0 ∈ C(D). (49)
On utilise les notations suivantes :

G(x, y) = 1 2π ln R|x -y| |y| |x -y |y| 2 R 2 |
(fonction de Green-Dirichlet),

P (x, φ) = 1 2π R 2 -|x| 2 R 2 -2R|x| cos(θ -φ) + |x| 2 (noyau de Poisson). Algorithme 2. Soit D = B 2 R , R > 0.
On considère l'opérateur L ω de (17) et on suppose que les conditions (16), ( 18), (49) sont satisfaites. Soient ω 1 , ω 2 , ω 3 ∈ (0, +∞)\σ trois fréquences mutuellement différentes, où σ est défini dans [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF]. Alors, les paramètres c, v, ρ, α 0 , ζ peuvent être trouvés à partir des fonctions

F , Q ω 1 , Q ω 2 , Q ω 3 de (28) comme suit : 1. Déterminer ρ -1
2 comme la solution unique de classe C(D) de l'équation

ρ -1 2 (x) = ρ -1 2 0 + D G(x, y)f 1 (y)ρ -1 2 (y) dy, x ∈ D, f 1 = ω 2 2 Re Q ω 1 -ω 2 1 Re Q ω 2 ω 2 2 -ω 2 1 . (50)
2. Trouver D 0 = x ∈ D : α 0 (x) = 0 de la formule explicite

D 0 = x ∈ D : ω -1 1 Im Q ω 1 = ω -1 2 Im Q ω 2 , et déterminer ζ(x) pour x ∈ D \ D 0 fixé comme la solution positive unique de l'équation ω -1 2 Im Q ω 2 (x) -ω -1 1 Im Q ω 1 (x) ω -1 3 Im Q ω 3 (x) -ω -1 1 Im Q ω 1 (x) = ω 2 ω 1 ζ(x)
-1

ω 3 ω 1 ζ(x) -1 . ( 51
)
3. Définir η ∈ C(D) comme la solution unique de l'équation

η(x) = η 0 (x) + η 1 (x) + D G(x, y)f 1 (y)η(y) dy, x ∈ D, η 0 (x) = D G(x, y)ρ -1 2 (y) f 3 (y) -W (y) • ∇ ln ρ(y) dy, η 1 (x) = -Rρ -1 2 0 2π 0 φ 0 P (x, φ)W (ϑ(θ)) • ϑ ⊥ (θ) dθdφ, (52)
où ϑ(φ) = (cos φ, sin φ) et

f 3 = ω 1 Im Q ω 1 dans D 0 , f 3 = ω ζ 1 ω 2 Im Q ω 2 - ω ζ 2 ω 1 Im Q ω 1 ω ζ 1 -ω ζ 2 dans D \ D 0 , W (x) = 1 2π D (x -y) ⊥ |x -y| 2 F (y) dy, x ∈ D. (53)
4. Trouver c et v dans D et α 0 dans D \ D 0 en utilisant les formules

c -2 = Re Q ω 1 -Re Q ω 2 ω 2 2 -ω 2 1 -∇(ρ 1 2 η) -W 2 dans D, v = c 2 ∇(ρ 1 2 η) -W dans D, α 0 = c 2 • ω -1 2 Im Q ω 2 -ω -1 1 Im Q ω 1 ω ζ 1 -ω ζ 2 dans D \ D 0 . (54)
Notons que l'algorithme 2 permet de trouver tous les paramètres du fluide à partir des donées mesurables pour trois fréquences mutuellement différentes. Nous proposons également un algorithme simplifié qui requiert des données pour deux fréquences différentes pour le cas où il est connu a priori que α 0 = 0. De plus, notons qu'il est possible d'améliorer l'algorithme 2 en utilisant des donées pour des fréquences supplementaires.

La dérivation de l'algorithme 2 suit le schéma présenté sur la Fig. 3. On considère la deuxième formule dans [START_REF] Colton | Integral equation methods in scattering theory[END_REF], dont le membre de gauche est connu pour ω ∈ {ω 1 , . . . , ω N }, comme un système algébrique pour trouver f 1 , f 2 , f 3 , α 0 /c, ζ. Ensuite, on trouve c, ρ, v, α 0 des formules (29) que l'on considère comme des équations différentielles, dont le membre de gauche et les valeurs au bord sont connus. De plus, l'existence et l'unicité des solutions des équations (50) et (52) découlent de l'existence et l'unicité des solutions des équations de Schrödinger équivalentes avec un potentiel de type conductif.

3 Problème de diffusion inverse sans phase. Résumé de l'article VI

Nous étudions une version du problème 2 de la sous-section 1.3 motivée par des applications en physique quantique. Nous considérons l'équation

L 0,v ψ := -∆ψ + v(x)ψ = Eψ, x ∈ R d , d ≥ 2, E > 0, (55)
où v ∈ L ∞ (R d ) est une fonction à support compact. L'équation (55) est l'équation de Schrödinger pour une particule élementaire à l'énergie fixée E interagissant avec un objet macroscopique décrit par le potentiel v. Nous considérons l'amplitude de diffusion f pour l'équation (55) qui est définie par les formules [START_REF] Arians | Geometric approach to inverse scattering for the Schrödinger equation with magnetic and electric potentials[END_REF], [START_REF] Astala | Calderón's inverse conductivity problem in the plane[END_REF]. En physique quantique |f (k, l)| 2 décrit la densité de probabilité de diffusion d'une particule à l'impulsion initiale k dans la direction l |l| = k |k| , et

∂ arg f (k,l) ∂|k|
décrit le retard d'une particule due à la présence du diffuseur v, voir [30, p. 461-463] et [START_REF] Faddeev | Quantum scattering theory for several particle systems[END_REF]. Notons que dans une expérience réelle, il est beaucoup plus facile de mesurer seulement |f | 2 (il suffit de compter le nombre des particules diffusées dans des différentes directions).

Le problème de détermination du diffuseur v à partir de |f | 2 (et ses modifications), denommé le problème de diffusion inverse sans phase, est beaucoup moins étudié que le problème 2 (dans le cas où A = 0, n = 1), voir par exemple [START_REF] Chadan | Inverse problems in quantum scattering theory[END_REF][START_REF] Klibanov | Two reconstruction procedures for a 3D phaseless inverse scattering problem for the generalized Helmholtz equation[END_REF][START_REF]Inverse scattering without phase information[END_REF][START_REF]Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions[END_REF] et les références qui y sont contenues. Notons que ce problème n'admet jamais de solution unique. En particulier, si on désigne par f y l'amplitude de diffusion pour le potentiel

v y (x) = v(x -y) pour y ∈ R d fixé, |f y | 2 = |f | 2 .
Cependant, R. Novikov [START_REF]Formulas for phase recovering from phaseless scattering data at fixed frequency[END_REF] a proposé une modification du problème pluridimensionnel de diffusion inverse sans phase qui permet de se débarasser de la non-unicité et qui n'est pas restrictive d'un point de vue des applications. En suivant [START_REF]Formulas for phase recovering from phaseless scattering data at fixed frequency[END_REF], nous définissons l'ensemble

S = |f | 2 , |f 1 | 2 , . . . , |f m | 2 ,
où f j désigne l'amplitude de diffusion pour le potentiel v j = v + w j et w j est un potentiel de fond connu, j = 1, . . ., m. L'ensemble S est considéré comme une donnée mesurable pour trouver v. Le problème de diffusion inverse sans phase modifié se pose comme suit : Problème 6. Étant donné l'ensemble S des modules d'amplitudes de diffusion à un niveau d'énergie E fixé pour quelques potentiels de fond w 1 , . . ., w n connus, déterminer le potentiel v.

L'étude du problème 6 a été commencé dans l'article [START_REF]Formulas for phase recovering from phaseless scattering data at fixed frequency[END_REF] où des formules de type Born ont été presentées pour la solution approchée du problème 6 en dimension d ≥ 2.

Notons que dans le cas unidimensionnel, un problème analogue avec m = 1 a été posé et étudié par Aktosun-Sacks [START_REF] Aktosun | Inverse problems on the line without phase information[END_REF]. Notons également que dans l'article [START_REF] Aktosun | Inverse problems on the line without phase information[END_REF] il est supposé que S est connu pour tout E > 0.

Dans l'article VI nous continuons à étudier le problème 6. Nous établissons des estimations pour les reconstructions approchées de type Born pour le problème 6, qui sont requises pour le développement des méthodes de reconstruction précises. En particulier, ces estimations sont nécessaires pour géneraliser au cas sans information de phase l'algorihme itératif de [START_REF]An iterative approach to non-overdetermined inverse scattering at fixed energy[END_REF] pour le problème inverse de diffusion avec information de phase.

Notons que les estimations présentées dans l'article VI dépendent de la géometrie de l'ensemble des zéros communs des transformations de Fourier des potentiels de fond w 1 , . . ., w m . Pour des raisons de brièveté, nous allons présenter ces résultats dans les deux cas les plus simples.

Pour formuler des résultats principaux, nous avons besoin d'introduire un potentiel de fond de réference comme une fonction w telle que

w ∈ C comp (R d , R), w(p) ≥ c 1 (1 + |p|) -β , β > d, p ∈ R d , (56)
pour quelque constante c 1 > 0. Dans l'article VI nous proposons une approche pour construire des fonctions w satisfaisant à [START_REF] Grinevich | The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed nonzero energy[END_REF]. Nous avons également besoin d'introduire des notations suivantes :

U w 1 , w 2 = Re U w 1 , w 2 + i Im U w 1 , w 2 , Re U w 1 , w 2 (p) Im U w 1 , w 2 (p) = 1 2 M -1 w 1 , w 2 (p)b w 1 , w 2 (p, E), |p| ≤ 2 √ E, (57)
M w 1 , w 2 (p) = Re w 1 (p) Im w 1 (p) Re w 2 (p) Im w 2 (p) , b w 1 , w 2 (p, E) = |f 1 (p, E)| 2 -|f (p, E)| 2 -| w 1 (p)| 2 |f 2 (p, E)| 2 -|f (p, E)| 2 -| w 2 (p)| 2 , où f (p, E) = f (k E (p), l E (p)), f j (p, E) = f j (k E (p), l E (p)), j = 1, 2, et k E (p)
, l E (p) sont des fonctions à valeurs vectorielles continues par morceaux telles que

k 2 E (p) = l 2 E (p) = E, k E (p) -l E (p) = p, p ∈ B d 2 √ E .
Le premier résultat principal de l'article VI est le théorème suivant :

Théorème 9. Soit v ∈ W n,1 (R d ), n > d, une fonction bornée à support compact. Considérons w 1 = w(• -T 1 ), T 1 ∈ R d \ 0, w 2 = iw 1 , où w est une fonction satisfaisant à (56). Posons u(x, E) = B d r 1 (E) e -ipx U w 1 , w 2 (p, E) dp, x ∈ R d , r 1 (E) = 2τ E α 1 n-d , α 1 = n -d 2(n + β)
, pour quelque τ ∈ (0, 1] fixé, où U w 1 , w 2 est définie dans 57 et β est le nombre de [START_REF] Grinevich | The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed nonzero energy[END_REF]. Alors,

u(•, E) = v + O(E -α 1 ) dans L ∞ (R d ), E → +∞.
Le théorème 9 permet de trouver une approximation u au potentiel inconnu v avec la précision uniforme en O(E -α 1 ) à partir des données de diffusion sans phase à l'énergie E fixée pour le potentiel v et pour les potentiels perturbés v + w 1 et v + w 2 , où w 1 et w 2 sont des potentiels de fond connus, w 1 est réel et w 2 = iw 2 . Notons que le potentiel de fond w 2 est purement imaginaire, ce qui permet de se débarasser des quelques difficultés mathématiques.

Par contre, dans les applications en physique quantique il est souhaitable de n'utiliser que des potentiels de fond réels. Le résultat suivant permet de trouver une approximation au potentiel v à partir des données de diffusion sans phase pour le potentiel v et pour les potentiels perturbés v + w 1 et v + w 2 , où w 1 et w 2 = w 1 (• -y) sont des potentiels de fond réels connus et y ∈ R d \ 0 est fixé.

Pour formuler ce deuxième résultat principal, nous avons besoin d'introduire des notations supplémentaires. On suppose que w 1 = w(• -T 1 ), w 2 = w(• -T 2 ), y = T 2 -T 1 = 0, où w satisfait à [START_REF] Grinevich | The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed nonzero energy[END_REF]. Posons

U ε w 1 , w 2 (p, E) = 1 2 U w 1 , w 2 (p ε -, E) + U w 1 , w 2 (p ε + , E) , p ε ± = p ⊥ + πz(p) y |y| 2 ± ε y |y| 2 , p ⊥ = p -(py) y |y| 2 , p ∈ B 2 √ E ∩ Z ε w 1 , w 2 , Z ε w 1 , w 2 = p ∈ R d : py ∈ (-ε, ε) + πZ , ε ∈ (0, 1), z(p) ∈ Z est le nombre unique tel que |py -πz(p)| < ε. (58) Théorème 10. Soit v ∈ W n,1 (R d ), n > d, une fonction bornée à support compact. Considérons w 1 = w(•-T 1 ), w 2 = w(•-T 2 ), T 1 = T 2 , où w est une fonction satisfaisant à (56). Posons u(x, E) = u 1 (x, E) + u 2 (x, E), x ∈ R d , u 1 (x, E) = B r 2 (E) \Z ε 2 (E) w 1 , w 2 e -ipx U w 1 , w 2 (p, E) dp, u 2 (x, E) = B r 2 (E) ∩Z ε 2 (E) w 1 , w 2 e -ipx U ε w 1 , w 2 (p, E) dp, r 2 (E) = 2τ E α 2 n-d , ε 2 (E) = E -α 2 2 , α 2 = n-d 2 n+β+ n-d 2
, pour quelque τ ∈ (0, 1] fixé, où U w 1 , w 2 est définie dans (57), U ε w 1 , w 2 est définie dans [START_REF] Grinevich | Analogs of multisoliton potentials for the two-dimensional Schrödinger operator[END_REF], et β est le nombre de [START_REF] Grinevich | The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed nonzero energy[END_REF]. Alors,

u(•, E) = v + O(E -α 2 ) dans L ∞ (R d ), E → +∞.
Pour démontrer le théorème 9, nous montrons que U w 1 , w 2 est une bonne approximation pour v dans B r 1 (E) , puis nous utilisons l'analyse de Fourier pour transferer ce résultat dans l'éspace de configuration et etablir une estimation pour l'erreur u(•, E) -v. L'idée est la même pour le théorème 10.

Dans l'article VI nous considérons également le cas, où m = d+1 et w 1 = w(•-T 1 ), w j+1 = w 1 (• -e j ), j = 1, . . ., d -1, où e 1 , . . ., e d désigne la base canonique pour R d . Dans ce cas, l'estimation est meilleure que dans le cas de théorème 10 mais du point de vue des applications, le cadre du théorème 10 est plus prometteur.

4 Problème de valeurs au bord pour une courbe complexe. Résumé de l'article VII Soit (X, g) une variété riemannienne bidimensionnelle compacte au bord lisse ∂X. Sur cette variété on considère le problème de Dirichlet pour l'opérateur de Laplace-Beltrami :

∆ g u = 0 sur X, u| ∂X = u 0 .

Rappelons que dans des coordonées locales x 1 , x 2 sur X dans lesquelles la métrique g est représentée par une matrice (g ij ) de déterminant |g| et d'inverse (g ij ), on a

∆ g u = |g| -1 2 2 i,j=1 ∂ ∂x i |g| 1 2 g ij ∂f ∂x j ).
L'opérateur de Dirichlet-Neumann Λ g pour l'opérateur de Laplace-Beltrami est défini comme suit :

Λ g u 0 = ∂u ∂ν ∂X , où u est la solution de (59), (60)
et ∂/∂ν désigne la dérivée normale éxterieure. On s'intèresse à la détermination de (X, g), en supposant que ∂X et Λ g sont connus.

Il y a deux sources de non-unicité pour ce problème :

Le problème 7 est lié au problème de Calderón sur les surfaces, qui consiste en détermination de la surface conductrice à partir des mesures au bord du courant électrique induit par un potentiel imposé, voir par exemple [START_REF] Henkin | On the explicit reconstruction of a Riemann surface from its Dirichlet-to-Neumann operator[END_REF][START_REF]On the reconstruction of conductivity of bordered two-dimensional surface in R 3 from electrical currents measurements on its boundary[END_REF][START_REF] Henkin | Gelfand-calderón inverse problem for anisotropic conductivities on bordered surfaces in R 3[END_REF].

Lassas et Uhlmann [START_REF] Lassas | On determining a Riemannian manifold from the Dirichlet-to-Neumann map[END_REF] ont démontré que le problème [START_REF] Aktosun | Inverse problems on the line without phase information[END_REF] admet la solution unique (pour Γ de classe C ∞ ). Henkin et Michel [START_REF] Henkin | On the explicit reconstruction of a Riemann surface from its Dirichlet-to-Neumann operator[END_REF] ont considéré la question de charactérisation des données non surdéterminées et ont débuté l'étude de la question de reconstruction.

Il faut souligner que Henkin-Michel [START_REF] Henkin | On the explicit reconstruction of a Riemann surface from its Dirichlet-to-Neumann operator[END_REF] réduisent le problème 7 au problème de détermination d'une courbe complexe dans le plan projectif à partir de son bord comme suit. Soient x, y les coordonnées locales isothermes sur X (voir [START_REF] Springer | Introduction to Riemann surfaces[END_REF]). Posons

z = x + iy, ∂ ∂z = 1 2 ( ∂ ∂x -i ∂ ∂y ), ∂ ∂z = 1 2 ( ∂ ∂x + i ∂ ∂y )
, dz = dx + idy, dz = dx -idy. On peut montrer que si u est la solution du système [START_REF]Analogs of multisoliton potentials for the two-dimensional Schroedinger operator and the nonlocal Riemann problem[END_REF], alors ∂u = ∂u ∂z dz est une 1-forme analytique (i.e. ∂ 2 u ∂z∂z = 0) sur X qui satisfait à

∂u| ∂X = 1 2 Λ g u 0 -iT u 0 (ν * + iτ * ),
où T désigne la dérivée le long du bord ∂X, et ν * , τ * désignent les covecteurs conormal et cotangent unitaires sur ∂X, respectivement. Alors, si u 1 , u 2 , u 3 sont trois fonctions harmoniques sur X telles que ∂u 1 , ∂u 2 , ∂u 3 ne s'annulent en même temps, on peut définir l'application

F : X ∪ Γ → CP 2 , F (z) = (∂u 1 (z) : ∂u 2 (z) : ∂u 3 (z)),
où CP 2 désigne le plan projectif complexe muni des coordonnées homogènes (w 0 : w 1 : w 2 ). Rappelons que des coordonées homogénes sont déterminées à une multiplication par un nombre complexe non nul près. Cela justifie la définition de l'application F . Henkin et Michel montrent que si F | ∂X est un plongement 3 , alors Y = F (X) est une courbe complexe de fermeture compacte de bord γ = F (∂X) et sans composantes fermées, 4 avec un nombre fini de points singuliers, et F : X → Y est sa normalisation. Pour des raisons de simplicité, on suppose que γ ⊂ C 2 , où C 2 = (w 0 : w 1 : w 2 ) ∈ CP 2 : w 0 = 0 avec des coordonnées z 1 = w 1 /w 0 , z 2 = w 2 /w 0 . Donc, on s'est ramené au problème suivant :

Problème 8. Soit X ⊂ CP 2 une courbe complexe compacte au bord γ = ∂X ⊂ C 2 et sans composantes fermées. Étant donnée γ, trouver X .
En utilisant le théorème de structure de Harvey-Shiffman [START_REF] Harwey | A characterization of holomorphic chaines[END_REF], on peut démontrer qu'il existe au plus une courbe complexe 5 X ⊂ CP 2 de bord donné γ ⊂ C 2 et sans Notons que γ est localement le graphe d'une fonction d'une variable réelle tandis que X est localement le graphe d'une fonction analytique d'une variable complexe. En tenant compte de la formule de Cauchy pour des fonctions analytiques, on peut voir que le problème 8 est formellement non surdéterminé.

Dans l'article VII nous proposons un algorithme de reconstruction d'une courbe complexe compacte au bord donné dans le plan projectif. Cet algorithme peut être considéré comme une version constructive du célébre résultat de Harwey-Shiffman [START_REF] Harwey | A characterization of holomorphic chaines[END_REF] sur la détérmination unique d'une chaine holomorphe par son bord. Notre algorithme développe l'algorithme de Henkin-Michel [START_REF] Henkin | On the explicit reconstruction of a Riemann surface from its Dirichlet-to-Neumann operator[END_REF].

Plus précisément, nous proposons des équations explicites pour trouver des polynômes P m mentionnées dans le Théorème 2 de [START_REF] Henkin | On the explicit reconstruction of a Riemann surface from its Dirichlet-to-Neumann operator[END_REF] qui jouent le rôle principal dans la reconstruction. Ces équations dépendent au nombre N -de points à l'infini de la courbe complexe, et nous proposons des équations pour déterminer ce nombre dans les cas N -∈ {0, 1, 2}. Notons que malgré l'encombrante description proposée, il n'y a pas en principe d'obstacle pour la justification et l'implémentation numérique de l'algorithme pour N -≥ 3.

Notre algorithme est basée sur des formules de type Cauchy, qui remontent à [START_REF] Dolbeault | Surfaces de Riemann de bord donné dans CP n[END_REF][START_REF]Chaînes holomorphes de bord donné dans CP n[END_REF], et sur le lemme de Darboux [29, p. 154]. Cet algorithme a été implémenté dans Matlab et a été illustré sur plusieurs exemples, voir l'article VII.

Soit X ⊂ CP 2 une courbe complexe compacte au bord ∂X ⊂ C 2 et sans composante fermée. Posons

C 1 ξ = (z 1 , z 2 ) ∈ C 2 : ξ 0 + ξ 1 z 1 + z 2 = 0}, CP 1 ∞ = (0 : w 1 : w 2 ) ∈ CP 2 , et désignons par N + (ξ) et N -le nombre de points d'intersection de X avec C 1 ξ et CP 1 ∞ , respectivement : N + (ξ) = |X ∩ C 1 ξ |, N -= |X ∩ CP 1 ∞ |.
Il est utile de décrire des intersections de X avec C 1 ξ en utilisant les fonctions h j définies comme suit :

X ∩ C 1 ξ = h j (ξ), -ξ 0 -ξ 1 h j (ξ) ∈ C 2 , j = 1, . . . , N + (ξ) .
On peut vérifier que dans un petit voisinage W ξ * de presque tout ξ * ∈ C 2 les fonctions h j sont analytiques et satisfont à l'équation de Riemann-Burgers (dénommée également l'équation de l'onde de choc), voir, e.g., [START_REF] Darboux | Théorie des surfaces[END_REF] :

∂h j ∂ξ 1 (ξ) = h j (ξ) ∂h j ∂ξ 0 (ξ), ξ ∈ W ξ * . Posons également π 2 (z 1 , z 2 ) = -z 2 . Notons que C \ π 2 (∂X) = ∪ L l=0 Ω l , où Ω l≥0 sont des domaines ouverts connexes, Ω i ∩ Ω j = ∅ pour i = j, Ω l>0 sont bornés et N + (ξ) est constante sur chaque Ω l et N + (ξ) = N -pour ξ ∈ Ω 0 .
Étant donné ∂X, on peut définir les fonctions G k (ξ), k ≥ 0, ξ = (ξ 0 , ξ 1 ) ∈ C 2 , dénommées la transformation de Cauchy-Radon de ∂X et qui remontent à Dolbeault-Henkin [START_REF] Dolbeault | Surfaces de Riemann de bord donné dans CP n[END_REF] :

G k (ξ) = 1 2πi ∂X z k 1 d(ξ 0 + ξ 1 z 1 + z 2 ) ξ 0 + ξ 1 z 1 + z 2 , k ≥ 0. ( 61
)
La formule suivante de [START_REF]Chaînes holomorphes de bord donné dans CP n[END_REF] joue un rôle principal dans la reconstruction de X à partir de γ :

G k (ξ) = h k 1 (ξ) + • • • + h k N + (ξ) (ξ) + P k (ξ), k ≥ 1, ξ ∈ C 2 , (62)
où P k est analytique au voisinage de presque tout ξ = (ξ 0 , ξ 1 ) ∈ C 2 et polynomial en ξ 0 de degré au plus k pour ξ 1 fixé. Nous définissons des moments de Wermer ae ij comme suit (voir [START_REF] Wermer | The hull of a curve in C n[END_REF]) :

ae ij = 1 2πi ∂X z i 1 z j 2 dz 2 , i, j ≥ 0.
Le résultat suivant de l'article VII permet de trouver les polynomes P k necessaires pour la reconstruction de X dans des cas, quand le nombre de points à l'infini N -est égal à 0, 1 ou 2 : Théorème 11. Soit X ⊂ CP 2 \ (0 : 1 : 0) une courbe complexe sans composante fermée de bord lisse γ ⊂ C 2 . Soient P k les fonctions définies dans [START_REF] Haberman | Uniqueness in calderón's problem with lipschitz conductivities[END_REF]. Alors les assertions suivantes sont vraies :

1. Si N -= 0, on a P k = 0 pour tout k ≥ 1.

2. Si N -= 1, on a P 1 (ξ 0 , 0) = c 11 + c 12 ξ 0 , où les constantes c 11 et c 12 satisfont à l'équation suivante pour tout ξ = (ξ 0 , 0), ξ 0 ∈ Ω 0 :

c 11 ∂G 1 ∂ξ 0 (ξ) + c 12 ξ 0 ∂G 1 ∂ξ 0 (ξ) + G 1 (ξ) = G 1 (ξ) ∂G 1 ∂ξ 0 (ξ) -∂G 1 ∂ξ 1 (ξ). ( 63
) 3. Si N -= 2, on a P 1 (ξ 0 , 0) = c 11 + c 12 ξ 0 , P 2 (ξ 0 , 0) = c 21 + c 22 ξ 0 + c 23 ξ 2 0 , où les constantes c 11 , c 12 , c 21 , c 22 , c 23 satisfont à l'équation suivante pour tout (ξ 0 , 0), ξ 0 ∈ Ω 0 : ae 10 (c 2 12 + c 23 ) = ∂G 2 ∂ξ 1 -2 ∂G 1 ∂ξ 1 (G 1 -c 11 -c 12 ξ 0 ) + G 1 (c 22 + 2c 23 ξ 0 ) + ∂G 1 ∂ξ 0 • (G 1 -c 11 -c 12 ξ 0 ) 2 -G 2 + c 21 + c 22 ξ 0 + c 23 ξ 2 0 + G 2 1 -2c 11 G 1 -2c 12 G 1 ξ 0 -G 2 • (-c 12 ). ( 64
)
En supposant que le nombre de points à l'infini N -est égal à 0, 1 ou 2, on peut proposer la méthode suivante pour trouver N -:

Théorème 12. Soit X ⊂ CP 2 \ (0 : 1 : 0) une courbe complexe au bord lisse γ ⊂ C 2 . Choisissons un point ξ * = (ξ * 0 , 0), ξ * 0 ∈ Ω 0 et un voisinage W ξ * de ξ * dans C 2 .
Alors les assertions suivantes sont valables :

1. Si G 1 = 0 dans W ξ * , alors soit N -= 0, soit ∂X le bord d'une courbe complexe compacte dans C 2 , où ∂X désigne ∂X en orientation opposée.

2. S'il existe des constantes complexes c 11 , c 12 telles que

∂ ∂ξ 1 G 1 -P 1 = (G 1 -P 1 ) ∂ ∂ξ 0 (G 1 -P 1 ) dans W ξ * , (65)
où P 1 (ξ 0 , ξ 1 ) = c 11 ξ 0 +c 12 1+c 11 ξ 1 , alors N -≤ 1 et les constantes c 11 , c 12 sont les mêmes que dans le théorème 11.

S'il existe des constantes complexes

a 1 , a 2 , b 1 , b 2 , c 1 , c 2 telles que c 1 + c 2 = ae 10 , ∂ ∂ξ 1 ((G 1 -P 1 ) 2 -G 2 -P 2 ) = ((G 1 -P 1 ) 2 -G 2 -P 2 ) ∂ ∂ξ 0 (G 1 -P 1 ) dans W ξ * , P 1 (ξ 0 , ξ 1 ) = -a 1 ξ 0 +b 1 1-a 1 ξ 1 -a 2 ξ 0 +b 2 1-a 2 ξ 0 , P 2 (ξ 0 , ξ 1 ) = - 2 j=1 a j ξ 0 +b j 1-a j ξ 1 2 + 2a j c j 1-a j ξ 1 , (66)
alors N -≤ 2 et ces constantes sont liées aux constantes du théorème 11 par les relations

a 1 + a 2 = -c 12 , b 1 + b 2 = -c 11 , a 2 1 + a 2 2 = -c 23 , a 1 b 1 + a 2 b 2 = -1 2 c 22 , b 2 1 + b 2 2 + 2a 1 c 1 + 2a 2 c 2 = -c 21 , c 1 + c 2 = ae 10 . (67)
En se basant sur les théorèmes 11, 12, nous établissons l'algorithme suivant, qui permet de trouver X à partir de ∂X : Algorithme 3. Soit X ⊂ CP 2 \ (0 : 1 : 0) une courbe complexe sans composantes fermées au bord lisse γ ⊂ C 2 et telle que N -∈ {0, 1, 2}. Alors, étant donné ∂X, on peut déterminer X de la manière suivante :

1. Trouver N -et N + (ξ) = N -+ G 1 (ξ), en tenant compte que N + (ξ) est constant
sur chaque Ω l≥0 et en utilisant le théorème 12.

(a)

Si N -= 1, trouver le polynome P 1 (ξ 0 , 0) en utilisant le théorème 11.

(b) si N -= 2, trouver les polynomes P 1 (ξ 0 , 0), P 2 (ξ 0 , 0) à l'aide du théorème 11.

3. Soit R > 0 un nombre assez grand tel que

π 2 (γ) ⊂ {|z 2 | < R}. Posons γ R = X ∩ {|z 2 | = R}. Alors γ R = (h j (ξ 0 , 0), -ξ 0 ) : |ξ 0 | = R, 1 ≤ j ≤ N -. (a) Si N -= 1, déterminer h 1 (ξ 0 , 0), |ξ 0 | = R, de la formule explicite : h 1 (ξ 0 , 0) = G 1 (ξ 0 , 0) -P 1 (ξ 0 , 0), |ξ 0 | = R. (b) Si N -= 2, déterminer les polynômes symétriques élémentaires σ 1 (ξ 0 ), σ 2 (ξ 0 ) en h 1 (ξ 0 , 0), h 2 (ξ 0 , 0) pour chaque ξ 0 ∈ C, |ξ 0 | = R,
fixé en utilisant les formules explicites suivantes :

σ 1 := h 1 + h 2 = s 1 , σ 2 := h 1 h 2 = 1 2 (s 2 1 -s 2 ), s 1 (ξ 0 ) := h 1 (ξ 0 , 0) + h 2 (ξ 0 , 0) = G 1 (ξ 0 , 0) -P 1 (ξ 0 , 0), s 2 (ξ 0 ) := h 2 1 (ξ 0 , 0) + h 2 2 (ξ 0 , 0) = G 2 (ξ 0 , 0) -P 2 (ξ 0 , 0). Trouver h 1 (ξ 0 , 0) et h 2 (ξ 0 , 0) (à l'ordre près) pour tout ξ 0 ∈ C, |ξ 0 | = R, fixé
comme les racines du polynôme suivant en t :

t 2 -σ 1 (ξ 0 )t + σ 2 (ξ 0 ) = 0. 4. Soit X R = X ∩ {|z 2 | < R}. Alors ∂X R = γ R , γ R = γ γ R et X R n'a pas de points à l'infini. Définir µ l = N + (ξ 0 , 0), ξ 0 ∈ Ω l , par la formule µ l = G 1 (ξ 0 , 0) + N -, ξ 0 ∈ Ω l .

Trover les sommes des puissances s

k (ξ 0 ) en h k≥1 (ξ 0 , 0) pour chaque ξ 0 ∈ Ω l , |ξ 0 | < R, fixé et pour tout l : s k (ξ 0 ) := h k 1 (ξ 0 , 0) + • • • + h k µ l (ξ 0 , 0) = 1 2πi γ R z k 1 dz 2 z 2 + ξ 0 , 1 ≤ k ≤ µ l .
6. Trouver des polynômes symétriques élémentaires σ k (ξ 0 ) en h j≥1 (ξ 0 , 0), |ξ 0 | < R, ξ 0 ∈ Ω l , pour tout l en utilisant des idéntitées de Newton-Girard qui relient les polynômes élémentaires symmetriques aux sommes des puissances :

σ k (ξ 0 ) = i 1 <•••<i k h i 1 (ξ 0 , 0) • • • h i k (ξ 0 , 0), 1 ≤ k ≤ µ l , σ 1 = s 1 , σ 2 = 1 2 (σ 2 1 -σ 2 ), σ 3 = 1 6 (s 3 1 -3s 1 s 2 + 2s 3 ), e.t.c.
7. En tenant compte du théorème de Viète, trouver h 1 (ξ 0 , 0), . . ., h µ l (ξ 0 , 0) (à l'ordre près), |ξ 0 | < R, ξ 0 ∈ Ω l , comme les racines du polynôme suivant en t : Un exemple de reconstruction d'une courbe complexe est présenté sur la Fig. 4. La courbe complexe sur la Fig. 4 est la partie de la courbe

t µ l -σ 1 (ξ 0 )t µ l-1 + • • • + (-1) µ l σ µ l (ξ 0 ) = 0.
X = (w 0 : w 1 : w 2 ) ∈ CP 2 : w 3 1 = (w 2 -iw 0 )(w 2 + iw 0 )(w 2 -w 0 )
bornée par la courbe réelle

γ = (z 1 (t), z 2 (t)) : t ∈ [0, 2π] , z 2 (t) = 1.9e it , z 1 (t) = 3 (z 2 (t) -i)(z 2 (t) + i)(z 2 (t) -1), z 1 (0) > 0.
Notons que dans cet exemple le nombre de points à l'infini est égal à N -= 2 et les points à l'infini sont (0 : 1 : e ± 2πi 3 ).

Article I

A global uniqueness result for acoustic tomography of moving fluid

A. D. Agaltsov
We consider a model time-harmonic wave equation of acoustic tomography of moving fluid in an open bounded domain in dimension d ≥ 2. We give global uniqueness theorems for related inverse boundary value problem at fixed frequency.

Introduction

Consider the operator

L A,V = -∆ -2iA(x) • ∇ + V (x), (I.1)
where ∆ is the standard Laplacian,

x ∈ D, A ∈ W 1,∞ (D, R d ), V ∈ L ∞ (D, R), D is an open bounded domain in R d (d ≥ 2).
In the present article we study an inverse boundary value problem for the equation L A,V ψ = 0 in D.

As in [START_REF] Agaltsov | Riemann-Hilbert problem approach for two-dimensional flow inverse scattering[END_REF], [START_REF] Roussef | Two-dimensional vector flow inversion by diffraction tomography[END_REF], [START_REF] Rychagov | Reconstruction of fluid motion in acoustic diffraction tomography[END_REF] we consider the equation L A,V ψ = 0 as a model equation for a time-harmonic (e -iωt ) pressure ψ in moving fluid. In this setting

A(x) = ω c 2 (x) v(x), V (x) = - ω 2 c 2 (x)
,

where v is the fluid velocity vector, c is the sound speed, ω is the frequency. Suppose that 0 is not a Dirichlet eigenvalue for operator

L A,V in D. Then the Dirichlet problem L A,V ψ = 0 in D, ψ| ∂D = f, (I.2) is uniquely solvable for ψ ∈ H 1 (D) for any f ∈ H 1/2 (∂D). The Dirichlet-to-Neumann map Λ A,V sends f ∈ H 1/2 (∂D) to Λ A,V f ∈ H -1/2 (∂D) defined by the formula Λ A,V f = ∂ψ ∂ν ∂D + i(A • ν)f, (I.3)
where ν is the unit exterior normal to ∂D and ∂ψ ∂ν ∂D ∈ H -1/2 (∂D) can be defined, in particular, by the following formula :

∂ψ ∂ν ∂D , u = D ∇ψ(x)∇ũ(x) -2iũ(x)A(x)∇ψ(x) + ũ(x)V (x)ψ(x) dx, (I.4)
for u ∈ H 1/2 (∂D) and arbitrary ũ ∈ H 1 (D) with ũ| ∂D = u. Note that since ψ satisfies L A,V ψ = 0, the right hand side of the above formula doesn't depend on the choice of ũ.

The inverse boundary value problem for equation L A,V ψ = 0 in D consists in finding A, V from Λ A,V . In the case when coefficients A, V can be complex-valued there is an obstruction to the unique solvability of this problem caused by the gauge invariance of the map Λ A,V with respect to the gauge transformations

A → A + ∇ϕ, V → V -i∆ϕ + (∇ϕ) 2 + 2A∇ϕ,
where ϕ ∈ W 2,∞ (D, C), ϕ| ∂D = 0, see, e.g., [START_REF] Krupchyk | Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential[END_REF] (d ≥ 3), [START_REF] Guillarmou | Identification of a connection from Cauchy data on a Riemann surface with boundary[END_REF]

(d = 2).
However, in the case of real-valued coefficients A, V there is no gauge non-uniqueness as it was observed, for example, in [START_REF] Agaltsov | Riemann-Hilbert problem approach for two-dimensional flow inverse scattering[END_REF].

In addition, in general case, under some regularity assumptions on ∂D, A and V , the Dirichlet-to-Neumann map Λ A,V uniquely determines the two-form dA and the function q in D and the tangential component of A on ∂D, where

dA = 1≤k<l≤d ∂A l ∂x k - ∂A k ∂x l dx k ∧ dx l , q = V + i∇ • A -A • A, (I.5) and A = (A 1 , . . . , A d ).
In particular, it was shown in [START_REF] Krupchyk | Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential[END_REF] that in dimension d ≥ 3 the map Λ A,V uniquely determines dA and

q in D if A ∈ L ∞ (D, C d ) and V ∈ L ∞ (D, C). And in dimension d = 2 it was shown in [61] that if D is a smooth Riemann surface with boundary (in particular, if D is a planar domain with ∂D ∈ C ∞ ) then the map Λ A,V uniquely determines dA and q provided that A ∈ W 2,p (D, R d ), V ∈ W 1,p (D, C), p > 2.
In addition, concerning the identifiability of tangential components of A on the boundary, it was proved in [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF]

that if ∂D ∈ C 1 (d ≥ 3) or ∂D ∈ C 1,α , α ∈ (0, 1) (d = 2) and if A ∈ C(D, C d ), V ∈ L ∞ (D, C) then Λ A,V uniquely determines A-ν(A•ν)
on ∂D, where ν is the unit exterior normal field to ∂D.

In the present article we combine the aforementioned results in order to obtain the following global uniqueness results in the case when coefficients A, V are real-valued.

Theorem I.1. Let D be a bounded simply connected domain with path connected boundary in

R d (d ≥ 3) with ∂D ∈ C 1 . Let A 1 , A 2 ∈ W 1,∞ (D, R d ) and V 1 , V 2 ∈ L ∞ (D, R). If Λ A 1 ,V 1 = Λ A 2 ,V 2 , then A 1 = A 2 , V 1 = V 2 . Theorem I.2. Let D be a bounded simply connected domain in R 2 with ∂D ∈ C ∞ . Let A 1 , A 2 ∈ W 2,p (D, R d ) and V 1 , V 2 ∈ W 1,p (D, R) with p > 2. If Λ A 1 ,V 1 = Λ A 2 ,V 2 then A 1 = A 2 and V 1 = V 2 .
Theorems I.1 and I.2 are proved in Section 3. In Section 2 we present formulas and equations for finding A, V from dA, q, A -ν(A • ν)| ∂D .

Formulas and equations for finding A, V

In this section we suppose that D is a bounded contractible domain with path connected C 2 boundary in R d (d ≥ 2). By contractibility we mean that there exists

F ∈ C 2 (D × [0, 1], D) such that F 0 ≡ x, F 1 = id D , where F t (x) = F (x, t),
x is some fixed point in D and id D is the identity mapping on D. We also suppose that

A ∈ W 2,∞ (D, R d ), V ∈ L ∞ (D, R).
Given dA, q as in (I.5) in D and A -ν(A • ν) on ∂D, we can find A, V in the following way :

1. Define A = ( A 1 , . . . , A d ) ∈ W 1,∞ (D, R d ) by the formula A k = i<j 1 0 ∂F i t ∂t ∂F j t ∂x k - ∂F j t ∂t ∂F i t ∂x k ∂A j ∂x i • F t - ∂A i ∂x j • F t dt, (I.6)
where k = 1, . . .,

d ; A = (A 1 , . . . , A d ), F t = (F 1 t , . . . , F d t ) and • denotes the composition of maps, i.e. ∂A j ∂x i • F t (y) = ∂A j ∂x i (F t (y)), y ∈ D. 2. Fix x 0 ∈ ∂D. Define ϕ 0 ∈ C 1 (∂D) by the formula ϕ 0 (x) = d k=1 x x 0 A k τ (y) -A k τ (y) dy k , x ∈ ∂D, (I.7)
where

A τ = A -ν(A • ν), A τ = A -ν( A • ν), A τ = (A 1 τ , . . . , A d τ ), A τ = ( A 1 τ , . . . , A d τ )
, ν is the unit exterior normal field to ∂D and integration is over an arbitrary C 1 curve on ∂D linking x 0 to x.

Find the unique generalized solution

ϕ ∈ W 2,∞ (D, R) to ∆ϕ = Im q -∇ • A in D, ϕ| ∂D = ϕ 0 . (I.8) 4.
Coefficients A, V are given by the following formulas :

A = A + ∇ϕ, V = q -i∆ϕ -i∇ • A + A • A + 2 A • ∇ϕ + (∇ϕ) 2 .
This algorithm will be justified in Section 4.

3 Proofs of Theorems I.1, I.2

We will prove Theorems I.1 and I.2 simultaneously. Let

D, A 1 , A 2 , V 1 , V 2 satisfy the conditions of Theorem I.1 (resp. Theorem I.2) and suppose that Λ A 1 ,V 1 = Λ A 2 ,V 2 .
Using Theorem 1.1 of [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF] we obtain that

A 1 -ν(A 1 • ν) | ∂D = A 2 -ν(A 2 • ν) | ∂D , (I.9)
where ν is the unit exterior normal field to ∂D. Using Theorem 1.1 of [START_REF] Krupchyk | Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential[END_REF] (resp. Theorem 1.1 of [START_REF] Guillarmou | Identification of a connection from Cauchy data on a Riemann surface with boundary[END_REF]) we get

dA 1 = dA 2 in D, (I.10) q 1 = q 2 in D, (I.11)
where

dA j = 1≤k<l≤d ∂A l j ∂x k - ∂A k j ∂x l dx k ∧ dx l , q j = V j + i∇ • A j -A j • A j ,
where A j = (A 1 j , . . . , A d j ), j = 1, 2. Since the domain D is simply connected it follows from (I.10) that there exists

ϕ ∈ W 2,∞ (D, R) such that A 1 -A 2 = ∇ϕ in D. (I.12)
In dimension d = 2 it follows from simple connectedness of D and from smoothness of ∂D that ∂D is path connected. Formulas (I.9), (II.30) and path connectedness of ∂D imply that ϕ is constant on ∂D.

Using (II.21), (II.30) we obtain that

V 1 -V 2 = -i∆ϕ -(∇ϕ) 2 + 2A 1 ∇ϕ in D.
Taking the imaginary part of this equation we obtain the equation ∆ϕ = 0 in D.

Since ϕ is constant on ∂D and ϕ ∈ W 2,∞ (D) it follows that ϕ is constant in D. Hence A 1 = A 2 and V 1 = V 2 .
Theorems I.1 and I.2 are proved.

Justification of the algorithm of Section 2

It follows from formula (I.6) that d A = dA. More precisely, if we denote by F * dA the pullback of the form dA by the map F and by ι ∂t we denote the interior product with the vector field

∂ ∂t on {(x, t) ∈ D × [0, 1]}, then d k=1 A k dx k = 1 0 (ι ∂t F * dA) dt,
and the equality d A = dA follows from the Cartan magic formula

L ∂t = d • ι ∂t + ι ∂t • d, where L ∂t is the Lie derivative along ∂ ∂t , d is the exterior derivative on {(x, t) ∈ D × [0, 1]} and • denotes the composition of maps.
Hence we can define ϕ ∈ W 2,∞ (D, R) by the formula

ϕ(x) = x x d k=1 (A k -A k )dx k , x ∈ D,
where x ∈ D is some fixed point and integration is over an arbitrary C 1 curve in D linking x to x. Then ∇ϕ = A -A in D and this implies that ϕ| ∂D differs by constant from ϕ 0 defined in (I.7). We also obtain from (I.5) the equation

V = q -i∆ϕ -i∇ • A + A • A + 2 A • ∇ϕ + (∇ϕ) 2 in D.
Taking into account that V is real-valued and separating the imaginary part in the latter equation we obtain (I.8). Since Im q and ∇ • A belong to L ∞ (D, R), the problem (I.8) is uniquely solvable for ϕ ∈ W 2,∞ (D, R). Thus, the algorithm of Section 2 is justified.

Article II

Uniqueness and non-uniqueness in acoustic tomography of moving fluid

A. D. Agaltsov, R. G. Novikov 1
We consider a model time-harmonic wave equation of acoustic tomography of moving fluid in an open bounded domain in R d , d ≥ 2, with variable sound speed c, density ρ, fluid velocity v and absorption coefficient α. We give global uniqueness results for related inverse boundary value problem for the cases of boundary measurements given for two and for three fixed frequencies. Besides, we also give a non-uniqueness result for this inverse problem for the case of boundary measurements given for all frequencies.

Introduction

We consider a moving fluid in an open bounded domain D ⊂ R d with sound speed c = c(x), density ρ = ρ(x), fluid velocity vector v = v(x) and the sound wave absorption coefficient α = α(x, ω) at fixed frequency ω, where x ∈ D = D ∪ ∂D. For this fluid we consider the following model equation for the time-harmonic (e -iωt ) acoustic pressure ψ :

L ω ψ = 0 in D, L ω = -∆ x -2iA ω (x)∇ x -U ω (x), x = (x 1 , . . . , x d ) ∈ D, (II.1)
where

∆ x = ∂ 2 ∂x 2 1 + • • • + ∂ 2 ∂x 2 d , ∇ x = ∂ ∂x 1 , . . . , ∂ ∂x d , (II.2) A ω (x) = ωv(x) c 2 (x) + i 2 ∇ x ρ(x) ρ(x) , U ω (x) = ω 2 c 2 (x) + 2iω α(x, ω) c(x) , α(x, ω) = ω ζ(x) α 0 (x).
(II.3)

In different particular cases this model was considered, for example, in [START_REF] Agaltsov | Riemann-Hilbert problem approach for two-dimensional flow inverse scattering[END_REF][START_REF] Baykov | Mode tomograpgy of moving ocean[END_REF][START_REF] Burov | Simulation of a functional solution to the acoustic tomography problem for data from quasipoint transducers[END_REF][START_REF] Henkin | A multidimensional inverse problem in quantum and acoustic scattering[END_REF][START_REF] Rumyantseva | Increased resolution of two-dimensional tomography imaging along the transverse coordinate and separate reconstruction of elastic and viscous scatterer characteristics[END_REF][START_REF] Roussef | Two-dimensional vector flow inversion by diffraction tomography[END_REF][START_REF] Rychagov | Reconstruction of fluid motion in acoustic diffraction tomography[END_REF].

In the present work we assume that the fluid parameters c, ρ, v, α are such that

A ω and V ω are sufficiently regular on D for any ω > 0, (II.4)

c ≥ c min > 0, ρ ≥ ρ min > 0, α 0 ≥ 0 in D
for some constants c min , ρ min .

(II.5)

For simplicity we consider equation (II.1) assuming that

ω ∈ σ(L z ), (II.6)
where σ(L z ) consists of all z ∈ C such that 0 is a Dirichlet eigenvalue for operator L z in D.

(II.7)

For equation (II.1), under assumptions (II.4), (II.6), we consider the Dirichlet-to-Neumann boundary map Λ ω defined by the relation

Λ ω (ψ| ∂D ) = ∂ψ ∂ν | ∂D + i(A ω • ν)ψ| ∂D , (II.8)
fulfiled for all sufficiently regular solutions ψ of (II.1) in D, where ν is the unit exterior normal to ∂D. We consider Λ ω as all possible boundary measurements for the model described by equation (II.1) at fixed ω.

In the present work we consider the following inverse boundary value problem for equation (II.1) :

Problem II.1. Given boundary data Λ ω for some frequencies ω, find fluid parameters c, ρ, v, α in D.

Under the assumption that

A ω ≡ 0 on ∂D, (II.9)

Problem II.1 is closely related with the inverse scattering problem for the equation

L ω ψ = 0 on R d , (II.10)
where

A ω ≡ 0, U ω ≡ ω 2 c 2 0 on R d \ D,
where c 0 can be considered as the mean value of c on ∂D. By scattering data for equation (II.10) we mean, first of all, the scattering amplitude ; see, e.g., [START_REF] Agaltsov | Riemann-Hilbert problem approach for two-dimensional flow inverse scattering[END_REF][START_REF] Agaltsov | Finding scattering data for a time-harmonic wave equation with first order perturbation from the Dirichlet-to-Neumann map[END_REF][START_REF] Henkin | A multidimensional inverse problem in quantum and acoustic scattering[END_REF][START_REF]Multidimensional inverse spectral problem for the equation -∆ψ+(v(x)-Eu(x))ψ = 0[END_REF] for definitions of the scattering amplitude.

Due to results going back to [START_REF]Multidimensional inverse spectral problem for the equation -∆ψ+(v(x)-Eu(x))ψ = 0[END_REF], it is known that the inverse boundary value problems and the inverse scattering problems for operators like L ω are, actually, equivalent. For simplicity of exposition, in the present work we do not formulate the inverse scattering version of Problem II.1 in detail.

Note also that Problem II.1 at fixed ω is closely related with inverse boundary value and inverse scattering problems for the Schrödinger equation in magnetic field at fixed energy. The reason is that the operator L ω at fixed ω is closely related with the magnetic Schrödinger operator at fixed energy.

In the present work we are mainly focused on Problem II.1 and its inverse scattering version for the case when v ≡ 0 in D, or, in other words, we are focused on the acoustic tomography of moving fluid in the framework of the wave propagation model (II.1), (II.3).

As regards results given in the literature on this acoustic tomography of moving fluid, see, e.g., [START_REF] Agaltsov | Finding scattering data for a time-harmonic wave equation with first order perturbation from the Dirichlet-to-Neumann map[END_REF][START_REF]A global uniqueness result for acoustic tomography of moving fluid[END_REF][START_REF] Agaltsov | Riemann-Hilbert problem approach for two-dimensional flow inverse scattering[END_REF][START_REF] Baykov | Mode tomograpgy of moving ocean[END_REF][START_REF] Roussef | Two-dimensional vector flow inversion by diffraction tomography[END_REF][START_REF] Rychagov | Reconstruction of fluid motion in acoustic diffraction tomography[END_REF] and references therein including the case when v ≡ 0.

In particular, in [START_REF] Agaltsov | Riemann-Hilbert problem approach for two-dimensional flow inverse scattering[END_REF] a Riemann-Hilbert problem approach to the inverse scattering version of Problem II.1 at fixed frequency ω was developed for the case when ρ ≡ const, α ≡ 0, d = 2.

In addition, in [START_REF]A global uniqueness result for acoustic tomography of moving fluid[END_REF] global uniqueness theorems for Problem II.1 at fixed frequency ω were proved for the case when ρ ≡ const, α ≡ 0, d = 2 or d ≥ 3.

Note that in the present work we use, in particular, results developed in the literature for the case of the inverse boundary value problem for the Schrödinger equation in magnetic field at fixed energy ; see [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF][START_REF] Guillarmou | Identification of a connection from Cauchy data on a Riemann surface with boundary[END_REF][START_REF] Krupchyk | Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential[END_REF] and references therein.

Note also that in the present work we use global uniqueness results on the Dirichlet problem for some linear and non-linear perturbations of the Laplace equation in D ; see systems (II.28), (II.34) of Section 3 and related results of [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF].

The main results of the present work can be summarized as follows : (A1) We show that the boundary data Λ ω given for two different frequencies ω = ω 1 , ω 2 uniquely determine the coefficients c, ρ, v under the assumptions that ω 1 , ω 2 ∈ σ(L ω ) and α ≡ 0, see Theorems II. (1) , ρ (1) , v (1) , α

(1) 0 and different coefficients c (2) , ρ (2) , v (2) , α

(2) 0 such that σ(L (1) ω ) = σ(L (2) ω ) and Λ (1) ω = Λ (2) ω for ω ∈ C\σ(L (i) ω )
for the case when ζ ≡ 0, see Theorem II.5 of Section 2. We recall that in the aforementioned results σ(L ω ) is defined by (II.7). The uniqueness results (A1), (A2) can be considered as results on global identifiability in acoustic tomography of moving fluid, whereas the non-uniqueness result (B) can be considered as a result on principal non-identifiability in this tomographical problem.

Note also that in these results the determination of ρ is considered modulo the transformations ρ → Cρ, where C is a positive constant.

The main results of the present work are presented in detail in the next section.

Main results

Let W n,p (D, C) denote the standard Sobolev space consisting of complex-valued functions which are n times differentiable in L p (D), where n ≥ 0, p ≥ 1 (including p = ∞) and D is the domain of Section 1. We consider also W n,p (D, R), W n,p (D, R d ) and W n,p (D, C d ) defined in the standard way.

We say that D is simply connected iff D is path connected and each continuous loop in D is contractible. In addition, we say that a set S in R d is path connected iff each pair of points in S can be joined by a continuous path in S. See, e.g., [START_REF] Dubrovin | The Schrödinger equation in a periodic field and Riemann surfaces[END_REF] in connection with these definitions.

In the present work we assume mainly that

c ∈ W 1,∞ (D, R), c > 0, ρ ∈ C( D) ∪ C 2 (D), ρ > 0, v ∈ W 1,∞ (D, R d ), α 0 ∈ C( D), ζ ∈ C( D), ζ = 0, α 0 , ζ are real-valued, for d ≥ 3, (II.11) c ∈ W 2,p (D, R), c > 0, ρ ∈ W 3,p (D, R), ρ > 0, v ∈ W 2,p (D, R d ), α 0 ∈ W 1,p (D, R), ζ ∈ C( D), ζ = 0, α 0 , ζ are real-valued, where p > 2, d = 2.
(II.12)

Let L ω , σ(L ω ) and Λ ω be defined as in Section 1.

In the present work we obtain, in particular, the following global uniqueness resutls for Problem II. ω correspond to coefficients c (j) , ρ (j) , v (j) , α (j) , where c (j) , ρ (j) , v (j) satisfy (II.11), α (j) ≡ 0, j = 1, 2. Let ω 1 ,

ω 2 ∈ [0, +∞) \ (σ(L (1) ω ) ∪ σ(L (2) ω )), ω 1 = ω 2 . Then the coincidence of the boundary data Λ (1) ω = Λ (2)
ω for ω ∈ {ω 1 , ω 2 } implies that c (1) = c (2) , ρ (1) = Cρ (2) , v (1) = v (2) , where C = const > 0.

Theorem II.2. Let D be an open bounded simply connected domain in R 2 with path connected C ∞ boundary ∂D. Let L (j) ω and Λ (j) ω correspond to coefficients c (j) , ρ (j) , v (j) , α (j) , where c (j) , ρ (j) , v (j) satisfy (II.12), α (j) ≡ 0, (2) , ρ (1) = Cρ (2) , v (1) = v (2) , where C = const > 0.

j = 1, 2. Let ω 1 , ω 2 ∈ [0, +∞) \ (σ(L (1) ω ) ∪ σ(L (2) ω )), ω 1 = ω 2 . Then the coincidence of the boundary data Λ (1) ω = Λ (2) ω for ω ∈ {ω 1 , ω 2 } implies that c (1) = c
Theorem II.3. Let D be an open bounded simply connected domain in R d , d ≥ 3, with path connected C 1 boundary ∂D. Let L (j) ω and Λ (j) ω correspond to coefficients c (j) , ρ (j) , v (j) , α (j) 0 , ζ (j) satisfying (II.11), j = 1, 2. Let ω 1 , ω 2 , ω 3 ∈ (0, +∞) \ (σ(L (1) ω ) ∪ σ(L (2) ω ))
be three pairwise different frequencies. Then the coincidence of the boundary data (2) , ρ (1) = Cρ (2) , v (1) = v (2) , α (1) = α (2) , where C = const > 0 and α

Λ (1) ω = Λ (2) ω for ω ∈ {ω 1 , ω 2 , ω 3 } implies that c (1) = c
(j) (x, ω) = ω ζ (j) (x) α (j) 0 (x). Theorem II.4. Let D be an open bounded simply connected domain in R 2 with path connected C ∞ boundary ∂D. Let L (j) ω and Λ (j) ω correspond to coefficients c (j) , ρ (j) , v (j) , α (j) 0 , ζ (j) satisfying (II.12), j = 1, 2. Let ω 1 , ω 2 , ω 3 ∈ (0, +∞) \ (σ(L (1) ω ) ∪ σ(L (2) ω ))
be three pairwise different frequencies. Then the coincidence of the boundary data (2) , ρ (1) = Cρ (2) , v (1) = v (2) , α (1) = α (2) , where C = const > 0 and α We set c (1) ≡ const > 0, ρ (1) ≡ const > 0,

Λ (1) ω = Λ (2) ω for ω ∈ {ω 1 , ω 2 , ω 3 } implies that c (1) = c
(j) (x, ω) = ω ζ (j) (x) α
v (1) ≡ 0, α (1)
0 ≡ const > -1 2 min x∈D ∆h(x), (II.14) and c (2) = c (1) (1 -|∇h| 2 ) -1/2 , ρ (2) ≡ const > 0, v (2) = c (1) (1 -|∇h| 2 ) -1 ∇h, α (2) 0 = (1 -|∇h| 2 ) -1/2 (α (1) 0 + 1 2 ∆h).
(II.15)

Note that for these fluid parameters c (j) , ρ (j) , v (j) , α

0 with ζ (j) ≡ 0 the conditions (II.4), (II.5) are fulfiled for both cases j = 1 and j = 2.

In the present work, in addition to global uniqueness results of Theorems II.1, II.2, II.3, II.4 we give also the following non-uniqueness result.

Theorem II.5. Let D be an open bounded domain in R d , d ≥ 2, with smooth boundary. Let h satisfy (II.13), h ≡ 0, and c (j) , ρ (j) , v (j) , α (j) , j = 1, 2, be defined by (II.14), (II.15). Let L (j) ω and Λ (j)

ω correspond to coefficients c (j) , ρ (j) , v (j) , α (j) 0 with ζ (j) ≡ 0. Then Λ (1) ω = Λ (2) ω for all ω ∈ C \ σ, where σ = σ(L (1) z ) = σ(L (2) z ).
Theorem II.5 is proved in Section 5.

3 Proofs of Theorems II.1 and II. Note that :

L (j) ω = d k=1 1 i ∂ ∂x k + A (j),k ω 2 + q (j) ω , (II.16)
where

A (j) ω = (A (j),1 ω , . . . , A (j),d ω ) = ωv (j) (c (j) ) 2 + i 2 ∇ρ (j) ρ (j) , q (j) ω = - ω 2 (c (j) ) 2 + i∇ • ω (c (j) ) 2 v (j) + i 2 ∇ρ (j) ρ (j) - ω 2 (c (j) ) 4 (v (j) ) 2 + 1 4 (ρ (j) ) -2 (∇ρ (j) ) 2 - iωv (j) ∇ρ (j) (c (j) ) 2 ρ (j) ,
(II.17) ∇• is the standard divergence, j = 1, 2. By Theorem 1.1 of [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF] we have that the tangential components of the fields A

(1) ω and A

ω on ∂D are equal. And, as a corollary, tangential components of ∇ρ (1) ρ (1) and ∇ρ (2) ρ (2) on ∂D are equal, (II.18)

tangential components of v (1) (c (1) ) 2 and v (2) (c (2) ) 2 on ∂D are equal.
(II. [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF] Using (II.18) and the path connectedness of ∂D we obtain ln ρ (2) -ln ρ (1) = ln C on ∂D, ρ (2) | ∂D = Cρ (1) | ∂D for some positive constant C > 0.

(II.20)

Using Theorem 1.1 of [START_REF] Krupchyk | Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential[END_REF] and the simple connectedness of D we get :

q (2) ω -q (1) ω = 0 in D (II.21)
and

A (2) ω -A (1) ω = ∇ϕ ω in D, (II.22)
where

ϕ ω ∈ W 2,∞ (D, C), ω ∈ {ω 1 , ω 2 }.
Separating the real and the imaginary parts of (II.21) we get :

ω 2 1 (c (1) ) 2 - 1 (c (2) ) 2 + (v (1) ) 2 (c (1) ) 4 - (v (2) ) 2 (c (2) ) 4 + ∇ρ (2) 2ρ (2) 2 - ∇ρ (1) 2ρ (1) 2 -∇ • ∇ρ (2) 2ρ (2) + ∇ • ∇ρ (1)
2∇ρ (1) = 0, (II. [START_REF] Burov | Application of the functional-analytical Novikov algorithm for the purposes of ocean tomography[END_REF] where ω ∈ {ω 1 , ω 2 } ;

∇ • v (1) (c (1) ) 2 - v (2) (c (2) ) 2 - ∇ρ (1) ρ (1) v (1) (c (1) ) 2 + ∇ρ (2) ρ (2) v (2) (c (2) ) 2 = 0.
(II.24)

Using (II.23) and the assumptions that ω 1 , ω 2 ≥ 0, ω 1 = ω 2 , we obtain

∇ρ (2) 2ρ (2) 2 - ∇ρ (1) 2ρ (1) 2 -∇ • ∇ρ (2) 2ρ (2) + ∇ • ∇ρ (1) 2ρ (1) = 0, (II.25) 1 (c (1) ) 2 - 1 (c (2) ) 2 + (v (1) ) 2 (c (1) ) 4 - (v (2) ) 2 (c (2) ) 4 = 0. (II.26) Let u (j) = 1 2 ln ρ (j) , j = 1, 2. (II.27)
Due to (II.20), (II.25) and (II.27), we have ∆u (2) -(∇u (2) ) 2 = ∆u (1) -(∇u (1) ) 2 in D, u (2) = u (1) + 1 2 ln C on ∂D.

(II.28)

Since u (1) , u (2) ∈ C( D) ∩ C 2 (D), it follows from Theorem 10.1 of [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF] that

u (2) = u (1) + 1 2 ln C in D and, consequently, ρ (2) = Cρ (1) in D. (II.29)
Further, taking the real part of (II.22) and using (II. [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF]) we obtain

v (2) (c (2) ) 2 - v (1) (c (1) ) 2 = ∇β ω in D (II.
(c (1) ) 2 = v (2) (c (2) ) 2 in D.
(II.35)

Finally, using (II.26), (II.35) we obtain

c (2) = c (1) and v (2) = v (1) in D. (II.36)
This completes the proof of Theorem II.1.

Proof of Theorem II.2

In a similar way with the proof of Theorem II.1, we have formulas (II. [START_REF] Berezansky | On uniqueness theorem in inverse problem of spectral analysis for Schrödinger equation[END_REF])-(II.20) for d = 2.

Let µ (j) = -i 2 ln ρ (j) , (II.37) and L (j) ω = e -iµ (j) L (j) ω e iµ (j) , (II. [START_REF] Erickson | Greedy optiman homotopy and homology generators[END_REF] where e iµ (j) , e -iµ (j) denote the multiplication operators by the functions e iµ (j) , e -iµ (j) , j = 1, 2. Using (II.37), (II.38) one can see that

σ( L (j) z ) = σ(L (j) z ), j = 1, 2, (II.39) and Λ (1) ω = Λ (2) ω , (II.40)
where Λ

(1)

ω , Λ (2)
ω are the Dirichlet-to-Neumann maps for operators

L (1) ω , L (2)
ω in D, respectively, ω ∈ {ω 1 , ω 2 }.
By direct computation we obtain that

L (j) ω = 2 k=1 1 i ∂ ∂x k + A (j),k ω 2 + q (j) ω , (II.41)
where

A (j) ω = ( A (j),1 ω , A (j),2 ω ) = ω (c (j) ) 2 v, q (j) ω = q (j)
ω .

(II.42)

Note that the fields A

(1)

ω , A (2)
ω do not contain the imaginary part in contrast with

A (1) ω , A (2)
ω . Now, using (II.39), (II.40), (II.42) and also Theorem 1.1 of [START_REF] Guillarmou | Identification of a connection from Cauchy data on a Riemann surface with boundary[END_REF] and simple connectedness of D we get the equalities (II.23), (II.24) for d = 2, ω ∈ {ω 1 , ω 2 }, and also the equality

A (2) ω -A (1) ω = ∇ ϕ ω in D, (II.43)
where

ϕ ω ∈ W 2,∞ (D, R), ω ∈ {ω 1 , ω 2 }.
In a similar way with the proof of Theorem II.

Proofs of Theorems II.3 and II.4

Note that under the assumptions of Theorems II.3, II.4, we have that L (j) ω is given by (II.16), where

A (j) ω = (A (j),1 ω , . . . , A (j),d ω ) = ωv (j) (c (j) ) 2 + i 2 ∇ρ (j) ρ (j) , q (j) ω = - ω 2 (c (j) ) 2 + i∇ • ω (c (j) ) 2 v (j) + i 2 ∇ρ (j) ρ (j) - ω 2 (c (j) ) 4 (v (j) ) 2 + 1 4 (ρ (j) ) -2 (∇ρ (j) ) 2 - iωv (j) ∇ρ (j) (c (j) ) 2 ρ (j) -2iω 1+ζ (j) α (j) 0 c (j) , (II.45)
where j = 1, 2.

Proof of Theorem II.3

In a similar way with the proof of Theorem II.1, we have formulas (II.18)-(II. [START_REF] Burov | The use of low-frequency noise in passive tomography of the ocean[END_REF], where in (II.21), (II.22) the functions q

(1) ω , q (2) ω , A (1) ω , A (2)
ω are defined as in (II.45) and ω ∈ {ω 1 , ω 2 , ω 3 } . Now, separating the real and imaginary parts of (II.21) we obtain equality (II.23) and also the equality

∇ • v (1) (c (1) ) 2 - v (2) (c (2) ) 2 - ∇ρ (1) ρ (1) v (1) (c (1) ) 2 + ∇ρ (2) ρ (2) v (2) (c (2) ) 2 +2ω ζ (2) α (2) 0 (c (2) ) 2 -2ω ζ (1) α (1) 0 (c (1) ) 2 = 0, (II.46)
where ω ∈ {ω 1 , ω 2 , ω 3 }.

Using (II.46) and the assumption that ω 1 , ω 2 , ω 3 are positive and mutually different frequencies we obtain, in particular, that (II.24) holds.

In a similar way with the proof of Theorem II.1, proceeding from (II. [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF])-(II.24) we obtain (II.36).

Next, in order to show that α

(2)

0 (x) = α (1) 0 (x) for fixed x ∈ D we consider two cases : (a) ζ (1) (x) = ζ (2) (x) ; (b) ζ (1) (x) = ζ (2) (x).
For the case (a) using (II.46) and the assumption that ω 1 , ω 2 , ω 3 are positive and mutually different frequencies, in addition to (II. [START_REF] Burov | Simulation of a functional solution to the acoustic tomography problem for data from quasipoint transducers[END_REF], we obtain also that α (j) 0 (c (j) ) 2 = 0 at point x, j = 1, 2, (II.47)

and, as a corollary, α

0 (x) = α (2) 0 (x) = 0. (II.48) (1)
For the case (b) using (II.46) and the assumption that ω 1 , ω 2 , ω 3 are positive and mutually different frequencies, in addition to (II. [START_REF] Burov | Simulation of a functional solution to the acoustic tomography problem for data from quasipoint transducers[END_REF], we obtain also that

α (2) 0 (c (2) ) 2 - α (1) 0 (c (1) ) 2 = 0 at point x.
(II.49)

Using (II.36) and (II.49) we obtain

α (2) 0 (x) = α (1) 0 (x). (II.50)
Finally, the result that α (2) = α (1) in D follows from (II.48) for the case (a) and from (II.50) for the case (b).

This completes the proof of Theorem II.3.

Proof of Theorem II.4

In a similar way with the proofs of Theorems II. Finally, using (II.46) we complete the proof of Theorem II.4 in a completely similar way to the proof of Theorem II.3.

5 Proof of Theorem II.5

Let µ = ω c (1) h. (II.51)
One can check by direct computation that

e -iµ L (1) ω e iµ = -∆ -2iA (2) ω ∇ -U (2) ω , (II.52)
where

A (2) ω = ω c (1) ∇h, U (2) ω = ω 2 (c (1) ) 2 (1 -|∇h| 2 ) + 2iω c (1) (α (1) 0 + 1 2 ∆h), (II.53)
and e iµ , e -iµ denote the multiplication operators by the functions e iµ , e -iµ , respectively. Using (II.52) one can see that

L (2) ω = e -iµ
σ(L (1) z ) = σ(L (2) z ), (II.56)
and Λ (1) ω = Λ (2) ω for all ω ∈ C \ σ, (II.57)

where σ = σ(L (1) z ) = σ(L (2)
z ). This completes the proof of Theorem II.5.

Article III

Finding scattering data for a time-harmonic wave equation with first order perturbation from the Dirichlet-to-Neumann map

A. D. Agaltsov
We present formulas and equations for finding scattering data from the Dirichlet-to-Neumann map for a time-harmonic wave equation with first order perturbation with compactly supported coefficients. We assume that the coefficients are matrix-valued in general. To our knowledge, these results are new even for the general scalar case.

Introduction

We consider the following equation :

Lψ def == -∆ψ -2i d j=1 A j (x)∂ x j ψ + V (x)ψ = Eψ, x ∈ D ⊂ R d , (III.1) where x = (x 1 , . . . , x d ), ∂ x j = ∂/∂x j , ∆ = ∂ 2 x 1 + • • • + ∂ 2 x d , E ∈ C, d = 2, 3, D is a bounded open domain in R d with ∂D ∈ C 2 , (III.2)
A 1 , . . ., A d , V are sufficiently regular M n (C)-valued functions on D and M n (C) is the set of n × n complex matrices. We also assume that E is not a Dirichlet eigenvalue for operator L in D.

(III.3)

Note that equation (III.1) can be written in the form

d j=1 -i∂ x j + A j (x) 2 ψ + v(x)ψ = Eψ, (III.4)
where

v(x) = V (x) - d j=1 A 2 j (x) + i d j=1 ∂ x j A j (x). (III.5)
For equation (III.1) (or (III.4)) we consider the maps Φ(E), Λ(E) such that

Φ(E)(ψ| ∂D ) = ∂ψ ∂ν ∂D , Λ(E)(ψ| ∂D ) = ∂ψ ∂ν + i d j=1 ν j A j ψ ∂D ,
for all sufficiently regular solutions ψ of (III.1) in D = D ∪ ∂D, for example for all In a similar way with [START_REF] Isayev | Reconstruction of a potential from the impedance boundary map[END_REF], assumption (III.3) can be dropped by considering an appropriate Robin-to-Robin map instead of the Dirichlet-to-Neumann map.

ψ ∈ C 1 (D, M n (C)) ∩ C 2 (D, M n (C))
Note that Λ(E) is invariant with respect to the gauge transformations

A j → gA j g -1 + i(∂ x j g)g -1 , j = 1, . . . , d, v → gvg -1 , (III.6)
where g is a sufficiently regular M n (C)-valued function on D with det g(x) = 0, x ∈ D and g(x) = Id n on ∂D, where Id n is the identity n × n matrix. Note also that Φ(E) is invariant with respect to the gauge transformations (III.6) under the additional assumption that

d j=1 ν j ∂ x j g = 0 on ∂D. Furthermore, if d j=1 ν j A j = 0 on ∂D (in particular, if A 1 , . . ., A d have compact supports in D), then Λ(E) = Φ(E). Besides, if A 1 , . . ., A d are
known on ∂D, one can easily compute Λ(E) given Φ(E) and vice versa.

For n = 1 equation (III.4) can be considered as a Schrödinger equation at fixed energy E with magnetic potential A = (A 1 , . . . , A d ) and electric potential v, see, e.g., [START_REF] Novikov | The ∂-equation in the multidimensional inverse scattering problem[END_REF][START_REF] Eskin | Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy[END_REF][START_REF]Inverse scattering problems for Schrödinger operators with magnetic and electric potentials[END_REF].

Equation (III.4) for n ≥ 2 with Hermitian matrices A 1 , . . ., A d and with scalar matrix v can be considered as a Schrödinger equation for a particle in an external Yang-Mills field, see [START_REF] Schrader | Small -asymptotics for quantum partition functions associated to particles in external Yang-Mills potentials[END_REF][START_REF]Semiclassical asymptotics, gauge fields and quantum chaos[END_REF][START_REF] Taylor | Semiclassical spectra of gauge fields[END_REF][START_REF]Inverse scattering problems for the Schrödinger operators with external Yang-Mills potentials[END_REF].

Besides, equation (III.1) for n = 1 is a model equation for the time-harmonic (e -iωt ) acoustic pressure ψ in a moving fluid, see, e.g., [START_REF] Roussef | Two-dimensional vector flow inversion by diffraction tomography[END_REF][START_REF] Rychagov | Reconstruction of fluid motion in acoustic diffraction tomography[END_REF]. In this setting

E = ω c 0 2 , A j (x) = ω c 0 u j (x), V (x) = 1 -r 2 (x) ω c 0 2
, where j = 1, . . ., d, c 0 is a reference sound speed, r is a scalar index of refraction, u = (u 1 , . . . , u d ) is a normalized fluid velocity vector.

In addition, for n ≥ 2, d = 2, equation (III.1) arises as a wave equation in the mode representation for a time-harmonic acoustic pressure ψ in a moving fluid in a three-dimensional cylindrical domain of finite height and with base D, see [START_REF] Baykov | Mode tomograpgy of moving ocean[END_REF].

We also consider equation (III.1) in the entire space :

Lψ ≡ -∆ψ -2i d j=1 A j (x)∂ x j ψ + V (x)ψ = Eψ, x ∈ R d , (III.7)
where A 1 , . . ., A d , V are sufficiently regular M n (C)-valued functions with sufficient decay at infinity. There are scattering functions ψ + , f and Faddeev-type generalized scattering functions ψ, h and ψ γ , h γ associated with equation (III.7).

Functions ψ + , f can be defined as follows :

ψ + (x, k) = e ikx Id n + R d G + (x -y, k)× × -2i d j=1 A j (y)∂ y j + V (y) ψ + (y, k) dy (III.8) G + (x, k) = -(2π) -d R d e iξx dξ ξ 2 -k 2 -i0 , (III.9) where x ∈ R d , k ∈ R d \ 0, k 2 = E ; f (k, l) = (2π) -d R d e -ilx -2i d j=1 A j (x)∂ x j + V (x) ψ + (x, k) dx, (III.10)
where

k ∈ R d \ 0, l ∈ R d \ 0, k 2 = l 2 = E.
Actually, we consider (III.8) and its differentiated versions, where ∂ x j , j = 1, . . ., d, are applied to both sides of (III.8), as a system of coupled linear integral equations for ψ + , ∂ x j ψ + , j = 1, . . ., d.

Functions ψ and h are defined as follows :

ψ(x, k) = e ikx Id n + R d G(x -y, k)× × -2i d j=1 A j (y)∂ y j + V (y) ψ(y, k) dy, (III.11) G(x, k) = e ikx g(x, k), g(x, k) = -(2π) -d R d e iξx dξ ξ 2 + 2kξ , (III.12) where x ∈ R d , k ∈ C d \ R d , k 2 = E ; h(k, l) = (2π) -d R d e -ilx -2i d j=1 A j (x)∂ x j + V (x) ψ(x, k) dx, (III.13)
where

k, l ∈ C d \ R d , Im k = Im l, k 2 = l 2 = E.
In a similar way with (III.8), we consider (III.11) and its differentiated versions as a system of coupled linear integral equations for ψ, ∂ x j ψ, or, more precisely, for µ, ∂ x j µ, j = 1, . . ., d, where ψ = e ikx µ.

Finally, functions ψ γ and h γ are defined as follows :

ψ γ (x, k) = ψ(x, k + i0γ), h γ (k, l) = h(k + i0γ, l + i0γ), (III.14) where x ∈ R d , k, l ∈ R d \ 0, k 2 = l 2 = E, γ ∈ S d-1 ,
∈ (0, +∞). Function h(k, l), k, l ∈ C d \ R d , Im k = Im l, k 2 = l 2 = E, is considered as the scattering data S E for equation (III.7) at fixed E ∈ C \ (0, +∞).
In a similar way with the map Λ(E), scattering data S E is invariant with respect to gauge transformations (III.6), where g is a sufficiently regular M n (C)-valued function on R d decaying fast enough at infinity with det g(x) = 0 for x ∈ R d , see, e.g., [START_REF] Agaltsov | Riemann-Hilbert problem approach for two-dimensional flow inverse scattering[END_REF] for the case n = 1.

Let D be a fixed domain satisfying (III.2). Let

A 1 , . . ., A d , V ∈ C 0,α c (D, M n (C)) for some 0 < α ≤ 1, (III.15)
where C 0,α c (D, M n (C)) denotes the space of M n (C)-valued component-wise Höldercontinuous functions with compact support in D. As it was noted above, in the case of coefficients satisfying (III.15) the maps Φ(E) and Λ(E) are the same. For coefficients A 1 , . . ., A d , V satisfying (III.15) we consider the Dirichlet-to-Neumann map Φ(E) for equation (III.1) and the scattering data S E for equation (III.7). In the latter case we define coefficients A 1 , . . ., A d , V outside of D by zero matrices.

Problem III.1. Given Φ(E) at fixed E (or for E in some fixed set) find A 1 , . . ., A d , V of (III.1) (modulo gauge transformations (III.6)).

More precisely, we develop the approach of [START_REF]Multidimensional inverse spectral problem for the equation -∆ψ+(v(x)-Eu(x))ψ = 0[END_REF] (where this approach was suggested for n = 1, A 1 , . . ., A d ≡ 0) and reduce Problem III.1 to the following inverse scattering problem for equation (III.7) :

Problem III.2. Given S E at fixed E (or for E in some fixed set) find A 1 , . . ., A d , V of (III.7) (modulo gauge transformations (III.6)).

Concerning the results given in literature on Problem III.1 without the assumption that A 1 ≡ 0, . . ., A d ≡ 0, see, e.g., [START_REF] Nakamura | Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field[END_REF][START_REF] Panchenko | An inverse problem for the magnetic Schrödinger equation and quasi-exponential solutions of nonsmooth partial differential equations[END_REF][START_REF] Santos Ferreira | Determining a magnetic Schrödinger operator from partial Cauchy data[END_REF][START_REF] Krupchyk | Inverse problems with partial data for a magnetic Schrödinger operator in an infinite slab or on a bounded domain[END_REF][START_REF] Imanuvilov | Inverse problem by Cauchy data on an arbitrary sub-boundary for systems of elliptic equations[END_REF][START_REF] Krupchyk | Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential[END_REF] for n = 1 and [START_REF] Eskin | Global uniqueness in the inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials[END_REF] for n ≥ 1. Besides, see [START_REF] Novikov | Global uniqueness and reconstruction for the multi-channel Gelfand-Calderón inverse problem in two dimensions[END_REF][START_REF]Monochromatic recontruction algorithms for two-dimensional multichannel inverse problems[END_REF] for the case d = 2, A 1 ≡ 0, A 2 ≡ 0, n ≥ 1. Concerning the results for the case n = 1, A 1 ≡ 0, . . ., A d ≡ 0, see [START_REF]Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF][START_REF] Bukhgeim | Recovering a potential from Cauchy data in the twodimensional case[END_REF][START_REF]An effectivization of the global reconstruction in the Gelfand-Calderón inverse problem in three dimensions[END_REF][START_REF] Burov | Application of the functional-analytical Novikov algorithm for the purposes of ocean tomography[END_REF][START_REF]Effectivized Hölder-logarithmic stability estimates for the gelfand inverse problem[END_REF][START_REF] Santacesaria | A Hölder-logarithmic stability estimate for an inverse problem in two dimensions[END_REF] and references therein.

Concerning the results given in literature on Problem III.2 without the assumption A 1 ≡ 0, . . ., A d ≡ 0, see, e.g., [START_REF] Shiota | An inverse problem for the wave equation with first order perturbation[END_REF][START_REF] Henkin | A multidimensional inverse problem in quantum and acoustic scattering[END_REF][START_REF]The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF][START_REF] Eskin | Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy[END_REF][START_REF]Inverse scattering problems for Schrödinger operators with magnetic and electric potentials[END_REF][START_REF] Arians | Geometric approach to inverse scattering for the Schrödinger equation with magnetic and electric potentials[END_REF][START_REF] Nicoleau | A stationary approach to inverse scattering for Schrödinger operators with first order perturbation[END_REF][START_REF] Päivärinta | Inverse scattering for the magnetic Schrödinger operator[END_REF][START_REF] Agaltsov | Riemann-Hilbert problem approach for two-dimensional flow inverse scattering[END_REF] for n = 1 and [START_REF] Henkin | The Yang-Mills fields, the Radon-Penrose transform, and the Cauchy-Riemann equations[END_REF][START_REF]Inverse scattering problems for the Schrödinger operators with external Yang-Mills potentials[END_REF][START_REF] Eskin | Global uniqueness in the inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials[END_REF][START_REF] Xiaosheng | Inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials in two dimensions at fixed energy[END_REF] for n ≥ 1. The case A 1 ≡ 0, . . ., A d ≡ 0, n ≥ 1, was considered, e.g., in [START_REF]Monochromatic recontruction algorithms for two-dimensional multichannel inverse problems[END_REF]. Concerning the results for the case n = 1, A 1 ≡ 0, . . ., A d ≡ 0, see [START_REF]An iterative approach to non-overdetermined inverse scattering at fixed energy[END_REF] and references therein.

The main results of the present work consist of Theorems III.1 and III.2 of Section 2. In Theorem III.1 we give, in particular, formulas and equations for finding S E from Φ(E) -Φ 0 (E), where S E and Φ(E) correspond to coefficients A 1 , . . ., A d , V and Φ 0 (E) corresponds to zero coefficients A 0 1 ≡ 0, . . ., A 0 d ≡ 0, V 0 ≡ 0. In Theorem III.2 we give a result on the solvability of equations of Theorem III.1.

In fact, the formulas and equations of Theorem III.1 are also valid if either V 0 (x) is a diagonal matrix for all x ∈ D or V 0 is a product of a constant matrix by a scalar function, see Theorems III.1 and III.2 of Section 2. In this case, the potential V 0 is supposed to be known. This generalization to the case when V 0 (x) is diagonal for all x ∈ D is useful, in particular, in the framework of Problem III.1 for the case of mode wave equation, see, e.g., [START_REF] Baykov | Mode tomograpgy of moving ocean[END_REF] and Subsection 3.1 of [START_REF]Monochromatic recontruction algorithms for two-dimensional multichannel inverse problems[END_REF].

Thus, due to the results of Theorems III. For the case when n = 1, A 1 ≡ 0, . . ., A d ≡ 0, A 0 1 ≡ 0, . . ., A 0 d ≡ 0, V 0 ≡ 0, Theorems III.1 and III.2 were obtained for the first time in [START_REF]Multidimensional inverse spectral problem for the equation -∆ψ+(v(x)-Eu(x))ψ = 0[END_REF]. These theorems were generalized to the case when n = 1, A 1 ≡ 0, . . ., A d ≡ 0, A 0 1 ≡ 0, . . ., A 0 d ≡ 0, V 0 ≡ 0, in [START_REF]Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF]. In [START_REF]Monochromatic recontruction algorithms for two-dimensional multichannel inverse problems[END_REF] the authors give formulas and equations for the case when

d = 2, n ≥ 1, A 1 ≡ 0, A 2 ≡ 0, A 0 1 ≡ 0, A 0 2 ≡ 0, V 0 ≡ 0.
In the present paper we generalize these results to the case when n ≥ 1, A 1 ≡ 0, . . ., A d ≡ 0, A 0 1 ≡ 0, . . ., A 0 d ≡ 0, V 0 ≡ 0. To our knowledge, these results are new even for the general scalar case when n = 1 and V 0 ≡ 0.

The main results of the present work are presented in Section 2.

Main results

Consider equation (III.7) under assumption (III.15). We define the sets E, E γ , γ ∈ S d-1 , and E + , as follows :

E = ζ ∈ C d \ R d : equation (III.11) at k = ζ is not uniquely solvable for ψ = e ikx µ, where µ ∈ W 1,∞ (R d , M n (C)) ,
(III. [START_REF] Berezansky | On uniqueness theorem in inverse problem of spectral analysis for Schrödinger equation[END_REF])

E γ = ζ ∈ R d \ 0 : equation (III.11) at k = ζ + i0γ is not uniquely solvable for ψ ∈ W 1,∞ (R d , M n (C)) ,
(III.17)

E + = ζ ∈ R d \ 0 : equation (III.8) at k = ζ + i0ζ/|ζ| is not uniquely solvable for ψ ∈ W 1,∞ (R d , M n (C)) . (III.18)
The properties of sets E, E γ , E + are similar to the properties of the analogs of sets E, E γ , E + in the case when n = 1, A j ≡ 0, j = 1, . . ., d. For the properties of the latter sets see, e.g., [START_REF]Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF] and references therein. Restrictions in space and time prevent us from studying the properties of sets E, E γ , E + in the present paper.

Theorem III.1. Let D satisfy (III.2) and E be fixed. Suppose that E is not a Dirichlet eigenvalue for operators L and -∆ in D. Let A 1 , . . ., A d , V satisfy (III.15). Let Φ(E) correspond to coefficients A 1 , . . ., A d , V and Φ 0 (E) correspond to coefficients A 0 1 ≡ 0, . . ., A 0 d ≡ 0, V 0 ≡ 0. Denote by (Φ -Φ 0 )(x, y, E), x, y ∈ ∂D, the Schwartz kernel of operator Φ(E) -Φ 0 (E). Then the following formulas and equations hold :

h(k, l) = (2π) -d ∂D ∂D e -ilx (Φ -Φ 0 )(x, y, E)ψ(y, k) dy dx, (III.19)
where k, l

∈ C d \ R d , Im k = Im l, k 2 = l 2 = E, k ∈ E ; ψ(x, k) = e ikx Id n + ∂D A(x, y, k)ψ(y, k) dy, x ∈ ∂D, (III.20) A(x, y, k) = ∂D G(x -z, k)(Φ -Φ 0 )(z, y, E) dz, x, y ∈ ∂D, (III.21)
where

k ∈ C d \ (R d ∪ E), k 2 = E,
and G is defined in formula (III.12) ;

h γ (k, l) = (2π) -d ∂D ∂D e -ilx (Φ -Φ 0 )(x, y, E)ψ γ (y, k) dy dx, (III.22) where γ ∈ S d-1 , k, l ∈ R d \ 0, k 2 = l 2 = E, k ∈ E γ , ψ γ (x, k) = e ikx Id n + ∂D A γ (x, y, k)ψ γ (y, k) dy, x ∈ ∂D, (III.23) A γ (x, y, k) = ∂D G γ (x -z, k)(Φ -Φ 0 )(z, y, E) dz, x, y ∈ ∂D, (III.24) G γ (x, k) def == G(x, k + i0γ), x ∈ R d , (III.25)
where

γ ∈ S d-1 , k ∈ R d \ (0 ∪ E γ ), k 2 = E ; f (k, l) = (2π) -d ∂D ∂D e -ilx (Φ -Φ 0 )(x, y, E)ψ + (y, k) dy dx, (III.26)
where

k, l ∈ R d \ 0, k 2 = l 2 = E, k ∈ E + , ψ + (x, k) = e ikx Id n + ∂D A + (x, y, k)ψ + (y, k) dy, x ∈ ∂D, (III.27) A + (x, y, k) = ∂D G + (x -z, k)(Φ -Φ 0 )(z, y, E) dz, x, y ∈ ∂D, (III.28)
where k ∈ R d \ (0 ∪ E + ), k 2 = E, and G + is defined in formula (III.9).

Actually, we consider (III.20), (III.23), (III.27) as integral equations for finding ψ, ψ γ , ψ + , respectively, from Φ(E) -Φ 0 (E).

In addition, we consider (III. [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF]), (III.22), (III.26) as explicit formulas for finding h, h γ , f from Φ(E) -Φ 0 (E) and ψ, ψ γ , ψ + , respectively.

For fixed 0 < β ≤ 1 we denote by C 1,β (∂D, M n (C)) the Banach space of functions from C 1 (∂D, M n (C)) with component-wise Hölder-continuous derivatives.

Theorem III.2. Let the assumptions of Theorem III.1 be fulfilled. Let 0 < β < 1 be fixed.

1. Fix k ∈ C d \ R d , k 2 = E. Then equation (III.20) is a Fredholm integral equation of second kind for ψ ∈ C 1,β (∂D, M n (C)) which is uniquely solvable if and only if k ∈ E. 2. Fix γ ∈ S d-1 , k ∈ R d \ 0, k 2 = E. Then equation (III.23) is a Fredholm integral equation of second kind for ψ γ ∈ C 1,β (∂D, M n (C)) which is uniquely solvable if and only if k ∈ E γ . 3. Fix k ∈ R d \ 0, k 2 = E. Then equation (III.27) is a Fredholm integral equation of second kind for ψ + ∈ C 1,β (∂D, M n (C)) which is uniquely solvable if and only if k ∈ E + .
In fact, Theorems III.1 and III.2 are particular cases of more general Theorems III.1 and III.2 given below. To formulate these results we need to introduce some notations.

Let coefficients A 0 1 , . . .,

A 0 d , V 0 on R d satisfy A 0 1 ≡ 0, . . . , A 0 d ≡ 0, (III.29)
and either V 0 (x) be a diagonal matrix for all x, (III.30)

or V 0 = V 0 v 0 , where V 0 ∈ M n (C)
and v 0 is a slalar function of x.

(III.31)

We also suppose that V 0 is zero outide of D and coefficients A 0 1 , . . ., A 0 d , V 0 restricted to D satisfy (III.15).

Define

L V 0 , E V 0 , E V 0 ,γ , γ ∈ S d-1
, and E + V 0 by formulas (III.1), (III.16), (III.17), (III.18), respectively, using coefficients A 0 1 ≡ 0, . . ., A 0 d ≡ 0, V 0 in (III.1), (III.11), (III.8) instead of A 1 , . . ., A d , V .

Note that, in fact, in the definition of set

E V 0 (or sets E V 0 ,γ , E + V 0 ) it is suf- ficient to consider the solvability of corresponding equations for ψ = e ikx µ with µ ∈ L ∞ (R d , M n (C)) (for ψ ∈ L ∞ (R d , M n (C)), respectively).
We consider the functions R 0 , R 0 γ , γ ∈ S d-1 , and R +,0 defined as follows :

R 0 (x, y, k) = G(x -y, k)Id n + R d G(x -z, k)V 0 (z)R 0 (z, y, k) dz, (III.32)
where x, y ∈ R d , k ∈ C d \ R d and G is defined in formula (III.12) ;

R 0 γ (x, y, k) def == R 0 (x, y, k + i0γ), (III.33) R +,0 (x, y, k) def == R 0 k/|k| (x, y, k), (III.34) where x, y ∈ R d , k ∈ R d \ 0, γ ∈ S d-1 .
We consider (III.32) at fixed y, k as an integral equation for

R 0 (x, y, k) = G(x -y, k)Id n + e ik(x-y) r 0 (x, y, k), (III.35)
where

r 0 (•, y, k) ∈ L ∞ (R d , M n (C)).
It follows from (III.32), (III.35) that r 0 satisfies the following equation :

r 0 (x, y, k) = R d g(x -z, k)V 0 (z)g(z -y, k) dz + R d g(x -z, k)V 0 (z)r 0 (z, y, k) dy, (III.36)
where x, y ∈ R d and g is defined in formula (III.12). Note that under assumption (III.15) for coefficients A 0 1 ≡ 0, . . ., A 0 d ≡ 0, V 0 the following statements are true :

1. Fix k ∈ C d \ R d . Then equation (III.36) is uniquely solvable for r 0 (•, y, k) ∈ L ∞ (R d , M n (C)) for any y ∈ R d if and only if k ∈ E V 0 . 2. Fix ζ ∈ R d \ 0, γ ∈ S d-1 . Then equation (III.36) with k = ζ + i0γ is uniquely solvable for r 0 (•, y, k) ∈ L ∞ (R d , M n (C)) for any y ∈ R d if and only if ζ ∈ E V 0 ,γ . 3. Fix ζ ∈ R d \ 0. Then equation (III.36) with k = ζ + i0ζ/|ζ| is uniquely solvable for r 0 (•, y, k) ∈ L ∞ (R d , M n (C)) for any y ∈ R d if and only if ζ ∈ E + V 0 . Besides, if equation (III.36) at fixed k is uniquely solvable for r 0 (•, y, k) ∈ L ∞ (R d , M n (C))
for any y ∈ R d , then function r 0 has the following properties :

r 0 (•, •, k) ∈ C(R d × R d , M n (C)) ∩ L ∞ (R d × R d , M n (C)) at fixed k, (III.37) R d g(x -z, k)V 0 (z)r 0 (z, y, k) dz = R d r 0 (x, z, k)V 0 (z)g(z -y, k) dz, (III.38)
where x, y ∈ R d . We also consider the function ψ 0 γ defined as follows :

ψ 0 γ (x, k, l) = e ilx Id n + R d G γ (x -y, k)V 0 (y) ψ 0 γ (y, k, l) dy, (III.39)
where

x ∈ R d , k, l ∈ R d \ 0, k 2 = l 2 , γ ∈ S d-1
. We consider (III.39) at fixed k, l, γ as an integral equation for

ψ 0 γ (•, k, l) ∈ L ∞ (R d , M n (C)
). Theorem III.1 . Let D satisfy (III.2) and E be fixed. Suppose that E is not a Dirichlet eigenvalue for operators L, L V 0 and -∆ in D. Consider two sets of coefficients A 1 , . . ., A d , V and A 0 1 , . . ., A 0 d , V 0 , satisfying (III.15). Let A 0 1 , . . ., A 0 d , V 0 satisfy (III.29) and either (III. [START_REF] Deligne | Quantum fields and strings : A course for mathematicians[END_REF]

or (III.31). Let Φ, ψ, h, ψ γ , h γ , ψ + , f , E, E γ , E + correspond to A 1 , . . ., A d , V (as defined above) and Φ V 0 , ψ 0 , h 0 , ψ 0 γ , ψ 0 γ , h 0 γ , ψ +,0 , f 0 , R 0 , R 0 γ , R +,0 , E V 0 , E V 0 ,γ , E + V 0 correspond to A 0 1 , . . ., A 0 d , V 0 (as defined above with coefficients A 0 1 , . . ., A 0 d , V 0 instead of A 1 , . . ., A d , V ). Denote by (Φ -Φ V 0 )(x, y, E), x, y ∈ ∂D, the Schwartz kernel of operator Φ(E) -Φ V 0 (E).
Then the following formulas and equations hold :

h(k, l) = h 0 (k, l) + (2π) -d ∂D ∂D ψ 0 (x, -l)(Φ -Φ V 0 )(x, y, E)ψ(y, k) dy dx, (III.40) where k, l ∈ C d \ R d , Im k = Im l, k 2 = l 2 = E, k ∈ E ∪ E V 0 , ψ(x, k) = ψ 0 (x, k) + ∂D A(x, y, k)ψ(y, k) dy, x ∈ ∂D, (III.41) A(x, y, k) = ∂D R 0 (x, z, k)(Φ -Φ V 0 )(z, y, E) dz, x, y ∈ ∂D, (III.42)
where

k ∈ C d \ (R d ∪ E ∪ E V 0 ), k 2 = E ; h γ (k, l) = h 0 γ (k, l) +(2π) -d ∂D ∂D ψ 0 -γ (x, -k, -l)(Φ -Φ V 0 )(x, y, E)ψ γ (y, k) dy dx, (III.43) where γ ∈ S d-1 , k, l ∈ R d \ 0, k 2 = l 2 = E, k ∈ E γ ∪ E V 0 ,γ , ψ γ (x, k) = ψ 0 γ (x, k) + ∂D A γ (x, y, k)ψ γ (y, k) dy, x ∈ ∂D, (III.44) A γ (x, y, k) = ∂D R 0 γ (x, z, k)(Φ -Φ V 0 )(z, y, E) dz, x, y ∈ ∂D, (III.45)
where

γ ∈ S d-1 , k ∈ R d \ (0 ∪ E γ ∪ E V 0 ,γ ), k 2 = E ; f (k, l) = f 0 (k, l) + (2π) -d ∂D ∂D ψ +,0 (x, -l)(Φ -Φ V 0 )(x, y, E)ψ + (y, k) dy dx, (III.46) where k, l ∈ R d \ 0, k 2 = l 2 = E, k ∈ E + ∪ E + V 0 , ψ + (x, k) = ψ +,0 (x, k) + ∂D A + (x, y, k)ψ + (y, k) dy, x ∈ ∂D, (III.47)
A + (x, y, k) = ∂D R +,0 (x, z, k)(Φ -Φ V 0 )(z, y, E) dz, x, y ∈ ∂D, (III.48)
where

k ∈ R d \ (0 ∪ E + ∪ E + V 0 ), k 2 = E.
In a similar way with Theorem III.1, we consider (III.41), (III.44), (III.47) as integral equations for finding ψ, ψ γ , ψ + from Φ(E) -Φ V 0 (E) and ψ 0 , R 0 ; ψ 0 γ , R 0 γ ; ψ +,0 , R +,0 , respectively.

We also consider (III.40), (III.43), (III.46) as explicit formulas for finding h, h γ , f from Φ(E) -Φ V 0 (E) and h 0 , ψ 0 , ψ ; h 0 γ , ψ 0 -γ , ψ γ ; f 0 , ψ +,0 , ψ + , respectively. Theorem III.1 is proved in Section 3.

Theorem III.2 . Let the assumptions of Theorem III.1 be fulfilled. Let 0 < β < 1 be fixed.

1. Fix k ∈ C d \ (R d ∪ E V 0 ), k 2 = E. Then equation (III.41) is a Fredholm integral equation of second kind for ψ ∈ C 1,β (∂D, M n (C)) which is uniquely solvable if and only if k ∈ E. 2. Fix γ ∈ S d-1 , k ∈ R d \(0∪E V 0 ,γ ), k 2 = E. Then equation (III.44) is a Fredholm integral equation of second kind for ψ γ ∈ C 1,β (∂D, M n (C)) which is uniquely solvable if and only if k ∈ E γ . 3. Fix k ∈ R d \ (0 ∪ E + V 0 ), k 2 = E. Then equation (III.47) is a Fredholm integral equation of second kind for ψ + ∈ C 1,β (∂D, M n (C)) which is uniquely solvable if and only if k ∈ E + .
Theorem III.2 is proved in Section 4.

Remark 1. Note that the proofs of equations and formulas of Theorems III.1, III.1 remain valid without the assumption that coefficients A 1 , . . ., A d , V , V 0 have compact supports in D. The assumption that the coefficients have compact supports in D was introduced in order to simplify the choice of functional spaces for solving equations (III. [START_REF] Bukhgeim | Recovering a potential from Cauchy data in the twodimensional case[END_REF] On the other hand, it is known that the solution to equation (III.41) will be relatively stable if the norm of the integral operator involved in this equation is less then 1. If at fixed k coefficients A 1 , . . ., A d are sufficiently small whereas coefficient V is not small but is sufficiently close to coefficient V 0 , then the integral operator in equation (III.41) will have much smaller norm than the integral operator in equation (III.20) (e.g., as a norm of operator on C 1,β (∂D, M n (C)), 0 < β < 1). In particular, the norm will be less then 1 and we will be able to use the method of successive approximations to solve (III.41). Hence equation (III.41) and formula (III.40) will give much more stable way to find ψ and h than equation (III.20) and formula (III. [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF], respectively. For more details, see pp. 262-263 of [START_REF]Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF] and Section 3.2 of [START_REF]Monochromatic recontruction algorithms for two-dimensional multichannel inverse problems[END_REF] for related discussion.

3 Proof of Theorem III.1'

Integral identity

Note that we have the identity

∂D u 0 (x)(Φ(E) -Φ V 0 (E))(u| ∂D )(x) dx = D u 0 (x) -2i d j=1 A j (x)∂ x j + V (x) -V 0 (x) u(x) dx.
(III. [START_REF] Faddeev | Quantum scattering theory for several particle systems[END_REF] for any sufficiently regular M n (C)-valued functions u, u 0 on D (for example, for u,

u 0 ∈ C 2 (D, M n (C)) ∩ C 1 (D, M n (C))) satisfying -∆u -2i d j=1 A j (x)∂ x j u + V (x)u = Eu, x ∈ D,
(III.50)

-∆u 0 + V 0 (x)u 0 = Eu 0 , x ∈ D, (III.51)
where u 0 also satisfies

V 0 (x)u 0 (x) = u 0 (x)V 0 (x), x ∈ D. (III.52)
Identity (III.49) for the case when n = 1, A 1 ≡ 0, . . ., A d ≡ 0 first appeared in [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF]. It was generalized to the case when n ≥ 2, A 1 ≡ 0, . . ., A d ≡ 0 in [START_REF] Novikov | Global uniqueness and reconstruction for the multi-channel Gelfand-Calderón inverse problem in two dimensions[END_REF].

Identity (III.49) can be deduced from the second Green formula. More precisely, formula (III.49) follows from the following chain of equalities :

D u 0 (x) -2i d j=1 A j (x)∂ x j + V (x) -V 0 (x) u(x) dx (III.50),(III.52) ===== D u 0 (x)(∆ + E)u(x) -V 0 (x)u 0 (x)u(x) dx (III.51) == D u 0 (x)∆u(x) -∆u 0 (x)u(x) dx = ∂D u 0 (x)(Φ(E) -Φ V 0 (E))(u| ∂D )(x) dx + ∂D u 0 (x)Φ V 0 (E)(u| ∂D )(x) -Φ V 0 (E)(u 0 | ∂D )(x)u(x) dx = ∂D u 0 (x)(Φ(E) -Φ V 0 (E))(u| ∂D )(x) dx + D u 0 (x)V 0 (x) u(x) -V 0 (x)u 0 (x) u(x) dx (III.52) == ∂D u 0 (x)(Φ(E) -Φ V 0 (E))(u| ∂D )(x) dx,
where u satisfies (III.51) and u| ∂D = u| ∂D .

Symmetries of the scattering functions

We denote by L ∞ c (R d ) the set of compactly supported functions from L ∞ (R d ).

Lemma III.1. Let V 0 ∈ L ∞ c (R d ) satisfy either (III.30) or (III.31). Then the following identities hold :

V 0 (x)ψ 0 (x, k) = ψ 0 (x, k)V 0 (x), (III.53) V 0 (x)R 0 (x, y, k) = R 0 (x, y, k)V 0 (x), , (III.54) R 0 (x, y, k) = R 0 (y, x, -k),
(III.55)

where x, y ∈ R d , x = y, k ∈ C d \ (R d ∪ E V 0 ). Démonstration. Let k ∈ C d \ (R d ∪ E V 0 ) be fixed. Then equation (III.11) with A 1 ≡ 0, . . ., A d ≡ 0, V ≡ V 0 is uniquely solvable for ψ 0 = e ikx µ 0 with µ 0 ∈ L ∞ (R d , M n (C)).
Suppose, first, that (III.30) holds. In this case it follows from formula (III.11) that ψ 0 (x, k) is a diagonal matrix for all x ∈ R d . Hence (III.53) holds.

Suppose now that (III.31) holds, so that

V 0 (x) = V 0 v 0 (x), x ∈ R d . Let U be a non-degenerated n × n matrix such that U V 0 U -1 def == Λ =    Λ 1 • • • 0 . . . . . . . . . 0 • • • Λ s    , Λ j =        λ j 1 0 • • • 0 0 λ j 1 • • • 0 . . . . . . . . . . . . . . . 0 0 0 • • • 1 0 0 0 • • • λ j       
, where Λ j ∈ M n j (C), j = 1, . . ., s. Define ψ = U ψ 0 U -1 . Then ψ satisfies the equation

ψ (x, k) = e ikx Id n + R d G(x -y, k)Λv 0 (y)ψ (y, k) dy. (III.56)
Since the solution to (III.56) is unique it follows that ψ has the block-diagonal form :

ψ =    ψ 1 • • • 0 . . . . . . . . . 0 • • • ψ s    ,
where ψ j ∈ M n j (C), j = 1, . . ., s. Hence the following equations hold and have the unique solutions :

ψ j (x, k) = e ikx Id n j + R d G(x -y, k)Λ j v 0 (y)ψ j (y, k) dy, (III.57)
where j = 1, . . ., s.

We write ψ j,il for the element in position (i, l) in matrix ψ j . Fix j and consider the last row of matrix equation (III.57). We have the following equations :

ψ j,il (x, k) = λ j R d G(x -y, k)v 0 (y)ψ j,il (y, k) dy, i = n j , l < n j .
(III.58)

We claim that equation (III.58) has only the trivial solution. Suppose that, on the contrary, there is a nontrivial solution φ to equation (III.58). Then we can construct a solution ψ j to (III.57) different from ψ j , putting ψ j,11 = ψ j,11 + φ and ψ j,il = ψ j,il for all other i, l. This is a contradiction since we showed that equation (III.57) has the unique solution. This shows that equation (III.58) has only the trivial solution and ψ j,il ≡ 0 for i = n j and l < n j .

Writing the equation (III.57) componentwise for rows with numbers i = n j -1, . . ., 2 we show by induction that ψ j,il ≡ 0 for i > l.

Fix i, l with i = l. Subtracting equation (III.57) for the element in position (l, l) from equation (III.57) for the element in position (i, i) we get the equation

ψ j,ii (x, k) -ψ j,ll (x, k) = λ j R d G(x -y, k)v 0 (y) ψ j,ii (y, k) -ψ j,ll (y, k) dy.
Since equation (III.58) has only the trivial solution, it follows that ψ j,ii ≡ ψ j,ll . Now fix i, l with i = l, i > 1, l > 1. Write equation (III.57) for the elements in positions (i -1, i) and (l -1, l) and subtract one from another. This leads to equation

ψ j,i-1,i (x, k) -ψ j,l-1,l (x, k) = λ j R d G(x -y, k)× ×v 0 (y) ψ j,i-1,i (y, k) -ψ j,l-1,l (y, k) dy + ψ j,ii (x, k) -ψ j,ll (x, k).
But we showed that ψ j,ii ≡ ψ j,ll and that equation (III.58) has only the trivial solution. This implies that ψ j,i-1,i ≡ ψ j,l-1,l .

Proceeding inductively, we obtain that ψ j has the following upper triangular form :

ψ j =        ψ j,11 ψ j,12 ψ j,13 • • • ψ j,1,n-1 ψ j,1n 0 ψ j,11 ψ j,12 • • • ψ j,1,n-2 ψ j,1,n-1 . . . . . . . . . . . . . . . . . . 0 0 0 • • • ψ j,11 ψ j,12 0 0 0 • • • 0 ψ j,11        .
It follows that ψ j (x, k) commutes with Λ j for all x ∈ R d , j = 1, . . ., s. Hence ψ (x, k) commutes with Λ and ψ(x, k) commutes with V 0 for all x ∈ R d . Property (III.53) is proved.

Formula (III.54) can be proved in a similar way. The proof of (III.55) for the case when n = 1 was given in [No3]. This proof also works in the case when n ≥ 2 if either (III.30) or (III.31) holds.

Remark 2. In a similar way with Lemma III.1 it can be proved that if V 0 ∈ L ∞ c (R d ) satisfies either (III.30) or (III.31), then the following formulas hold :

V 0 (x)ψ 0 γ (x, k) = ψ 0 γ (x, k)V 0 (x), (III.59) V 0 (x)R 0 γ (x, y, k) = R 0 γ (x, y, k)V 0 (x), (III.60) R 0 γ (x, y, k) = R 0 -γ (y, x, -k), (III.61)
where

γ ∈ S d-1 , x, y ∈ R d , x = y, k ∈ R d \ (0 ∪ E V 0 ,γ ) ; V 0 (x)ψ +,0 (x, k) = ψ +,0 (x, k)V 0 (x), (III.62) V 0 (x)R +,0 (x, y, k) = R +,0 (x, y, k)V 0 (x), (III.63) R +,0 (x, y, k) = R +,0 (y, x, -k), (III.64) where x, y ∈ R d , x = y, k ∈ R d \ (0 ∪ E + V 0 ). Lemma III.2. Let V 0 ∈ L ∞ c (R d ). Then : 1. if k, l ∈ C d \ R d , k 2 = l 2 , Im k = Im l, k ∈ E V 0 , then l ∈ E V 0
and the following formula holds : R 0 (x, y, k) = R 0 (x, y, l), (III.65)

where x, y ∈ R d , x = y ; 2. if k, l ∈ R d \ 0, k 2 = l 2 , k ∈ E +
V 0 , then l ∈ E + V 0 and the following formula holds : R +,0 (x, y, k) = R +,0 (x, y, l), (III.66)

where x, y ∈ R d , x = y.

Démonstration. Part 1 follows from (III.32) and from the following formula of [START_REF] Novikov | The ∂-equation in the multidimensional inverse scattering problem[END_REF] :

G(x, k) = G(x, l), x ∈ R d \ 0, k, l ∈ C d \ R d , k 2 = l 2 , Im k = Im l.
Part 2 follows from formulas (III.32), (III.34) and from identity

G + (x, k) = G + (x, l), x ∈ R d \ 0, k, l ∈ R d \ 0, k 2 = l 2 ,
which is a consequence of the following well-known explicit formulas :

G + (x, k) = - i 4 H (1) 0 (|k||x|), d = 2, G + (x, k) = - e i|k||x| 4π|x| , d = 3,
where H

(1) 0 is the Hankel function of the first kind.

ψ(x, k) -ψ 0 (x, k) - R d G(x -y, k)V 0 (y)(ψ(y, k) -ψ 0 (y, k)) dy = R d G(x -y, k) -2i d j=1 A j (y)∂ y j + V (y) -V 0 (y) ψ(y, k) dy, where x ∈ R d , k ∈ C d \ (R d ∪ E ∪ E V 0 ). Comparing this equation with (III.32), we obtain ψ(x, k) = ψ 0 (x, k) + R d R 0 (x, y, k)× × -2i d j=1 A j (y)∂ y j + V (y) -V 0 (y) ψ(y, k) dy, (III.67) where x ∈ R d , k ∈ C d \ (R d ∪ E ∪ E V 0 ).
Similarly, we can obtain the following equations :

ψ γ (x, k) = ψ 0 γ (x, k) + R d R 0 γ (x, y, k)× × -2i d j=1 A j (y)∂ y j + V (y) -V 0 (y) ψ + γ (y, k) dy, (III.68) where γ ∈ S d-1 , x ∈ R d , k ∈ R d \ (0 ∪ E γ ∪ E V 0 ,γ ) ; ψ + (x, k) = ψ +,0 (x, k) + R d R +,0 (x, y, k)× × -2i d j=1 A j (y)∂ y j + V (y) -V 0 (y) ψ + (y, k) dy, (III.69) where x ∈ R d , k ∈ R d \ (0 ∪ E + ∪ E + V 0
). From formula (III.67) with A 1 ≡ 0, . . ., A d ≡ 0, V ≡ 0 and from formulas (III.54), (III.55), (III.65) we obtain

e -ilx Id n = ψ 0 (x, -l) - R d e -ily V 0 (y)R 0 (y, x, k) dy, (III.70) where x ∈ R d , k, l ∈ C d \ R d , k 2 = l 2 , Im k = Im l, k ∈ E V 0 .
Further, we have the following chain of equalities :

(2π) d h(k, l) (III.13) == R d e -ilx V 0 (x)ψ(x, k) dx + R d e -ilx -2i d j=1 A j (x)∂ x j + V (x) -V 0 (x) ψ(x, k) dx (III.70) == R d e -ilx V 0 (x)ψ(x, k) dx + R d ψ 0 (x, -l)× × -2i d j=1 A j (x)∂ x j + V (x) -V 0 (x) ψ(x, k) dx - R d e -ily V 0 (y) R d R 0 (y, x, k)× × -2i d j=1 A j (x)∂ x j + V (x) -V 0 (x) ψ(x, k) dx dy, where k, l ∈ C d \ R d , k 2 = l 2 , Im k = Im l, k ∈ E ∪ E V 0 .
From this formula and formula (III.67) it follows that

h(k, l) = h 0 (k, l) + (2π) -d R d ψ 0 (x, -l)× × -2i d j=1 A j (x)∂ x j + V (x) -V 0 (x) ψ(x, k) dx, (III.71)
where

k, l ∈ C d \ R d , k 2 = l 2 , Im k = Im l, k ∈ E ∪ E V 0 .
In a similar way with formula (III.71), using formulas (III.69), (III.63), (III.64), (III.66) instead of (III.67), (III.54), (III.55), (III.65), we can obtain formula We recall that function G(x, k) satisfies the following equation at fixed k ∈ C d \R d , see, e.g., [START_REF]Growing solutions of the Schrödinger equation[END_REF] :

f (k, l) = f 0 (k, l) + (2π) -d R d ψ +,0 (x, -l)× × -2i d j=1 A j (x)∂ x j + V (x) -V 0 (x) ψ + (x, k) dx, (III.72) where k, l ∈ R d \ 0, k 2 = l 2 , k ∈ E + ∪ E + V 0 .

The final part of the proof of Theorem

∆ x G(x, k) + k 2 G(x, k) = δ(x
) that R 0 (x, y, k) at fixed k ∈ C d \ (R d ∪ E V 0 )
, satisfies the equations :

(∆ x -V 0 (x) + k 2 )R(x, y, k) = δ y (x)Id n , at fixed y ∈ R d , (∆ y -V 0 (y) + k 2 )R(x, y, k) = δ x (y)Id n , at fixed x ∈ R d . (III.74)
Taking this into account, using formula (III.54), applying identity (III.49) to (III.67) with x ∈ D and passing to the limit in the resulting formula as x tends to a point at ∂D, we obtain formula (III.41). Formulas (III.44) and (III.47) can be obtained in a similar way, if we use equations (III.68) and (III.69) instead of (III.67). We will prove formula (III.43) using the ideas of [START_REF]Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF]. Note that formula (III.39) can be written in the form

e ilx Id n -ψ 0 γ (x, k, l) - R d G γ (x -y, k)V 0 (y) e ily -ψ 0 γ (y, k, l) dy = - R d
G(x -y, k)V 0 (y)e ily dy.

(III.75)

From formulas (III.25), (III.32), (III.33) it follows that function

R 0 γ (•, y, k) is well- defined at fixed y ∈ R d , k ∈ R d \ (0 ∪ E V 0 ,γ ) and satisfies R 0 γ (x, y, k) = G γ (x -y, k) + R d G γ (x -z, k)V 0 (z)R 0 γ (z, y, k) dz, (III.76) where x, y ∈ R d , γ ∈ S d-1 , k ∈ R d \ (0 ∪ E V 0 ,γ ).
Comparing (III.75) with (III.76), we obtain the following formula :

e ilx Id n = ψ 0 γ (x, k, l) - R d R 0 γ (x, y, k)V 0 (y)e ily dy, (III.77) where x ∈ R d , k ∈ R d \ (0 ∪ E V 0 ,γ ), l ∈ R d , k 2 = l 2 , γ ∈ S d-1 .
Replacing k, l, γ by -k, -l, -γ in formula (III.77) and using formulas (III.60), (III.61), we obtain the equality

e -ilx Id n = ψ 0 -γ (x, -k, -l) - R d e -ily V 0 (y)R 0 γ (y, x, k) dy, (III.78) where x ∈ R d , k, l ∈ R d \ 0, k 2 = l 2 , k ∈ E V 0 ,γ , γ ∈ S d-1 .
Formula (III.78) is an analog of formula (III.70). The remaining part of the proof of (III.43) is similar to the proof of formula (III.40).

4 Proof of Theorem III.2'

Auxilary results

Consider the following Dirichlet problem for function ψ :

   Lψ ≡ -∆ψ -2i d j=1 A j (x)∂ x j ψ + V (x)ψ = Eψ, x ∈ D, ψ| ∂D = ϕ, (III.79)
where ϕ is some given function on ∂D.

Our goal in this subsection is to prove the following lemma. We need two other lemmas to prove Lemma III.3.

Lemma III.4. Let A 1 , . . ., A d , V ∈ C 0,α c (D, M n (C)) for some 0 < α ≤ 1. Suppose that E is not a Dirichlet eigenvalue for operator L in D. Then : 3. there exists a constant C > 0, not depending on ψ 0 , such that

1. for any ψ 0 ∈ C 1 (D, M n (C)) ∩ C 2 (D, M n (C)) satisfying (∆ + E)ψ 0 =
ψ C 1 (D) ≤ C ψ 0 C 1 (D) . (III.82)
Proof of Lemma III.4. 1. Reduction to a system of integral equations. We introduce the following notations :

ψ 0 (x) = ψ(x), ψ j (x) = ∂ x j ψ(x), j = 1, . . . , d, ψ 0 0 (x) = ψ 0 (x), ψ 0 j (x) = ∂ x j ψ 0 (x), j = 1, . . . , d, a 0 (x) = V (x), a j (x) = -2iA j (x), j = 1, . . . , d, Γ 0 (x, y, k) = Γ(x, y, k), Γ j (x, y, k) = ∂ x j Γ(x, y, k), j = 1, . . . , d.
Differentiating equation (III.80) we obtain the following system of coupled linear integral equations for ψ j ∈ C(D), j = 0, . . ., d : One can see that solutions ψ of class C 1 (D, M n (C)) to (III.80) are in bijective correspondence with solutions ψ j ∈ C(D, M n (C)), j = 0, . . ., d, to system (III.83).

ψ j (x) = ψ 0 j (x) +
2. Smoothness of solution to (III.80). It follows from properties of fundamental solution Γ that any solution ψ ∈ C 1 (D, M n (C)) to (III.80) belongs to C 1,γ loc (D, M n (C)) (the space of continuously differentiable M n (C)-valued functions on D with locally Hölder continuous derivatives) for any 0 < γ < 1.

In a similar way with Lemma 4.2, p. 55 of [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF] it can be shown then that ψ ∈ C 2 (D, M n (C)) and that ψ satisfies (III.79) with ϕ = ψ 0 | ∂D .

3. Existence and uniqueness. As we noted in part 1 of the proof of this lemma, system of equations (III.83) can be considered as a Fredholm equation of second kind in Banach space (C(D, M n (C))) d+1 . To show that this equation has the unique solution it is sufficient to show that the corresponding homogeneous equation has only the trivial solution. But if follows from parts 1, 2 of the proof that any solution to the homogeneous equation gives rise to a solution to (III.79) with zero boundary condition. Since E is not a Dirichlet eigenvalue for L in D, it follows that the homogeneous equation corresponding to (III.83) can have only the trivial solution.

Hence, using part 1 of the proof, we obtain that system (III.83) has the unique solution and this solution gives rise to the unique solution ψ ∈ C 1 (D, M n (C)) to (III.80). By part 2 of the proof this function ψ belongs to C 2 (D, M n (C)) and satisfies (III.79) with ϕ = ψ 0 | ∂D .

Property 3 follows from Fredholm alternative for system (III.83) which asserts the existence of continuous inverse for Fredholm operator of (III.83).

Lemma III.5. Suppose that E is not a Dirichlet eigenvalue for operator -∆ in D. Then for any ϕ ∈ C 1,β (∂D, M n (C)), 0 < β < 1, there exists the unique

ψ 0 ∈ C 2 (D, M n (C)) ∩ C 1 (D, M n (C))
solving Dirichlet problem (III.81). Furthermore, there exists such C β > 0, not depending on ϕ, that In this subsection we show that the integral operator in equation (III.41) is a compact linear operator on C 1,β (∂D, M n (C)) for any fixed 0 < β < 1. The case of operators in equations (III.44), (III.47) can be considered in a similar way.

ψ 0 C 1 (D) ≤ C β ϕ C 1,β (
Throughout this subsection we assume that k

∈ C d \ (R d ∪ E V 0 ) is fixed.
Rewrite equation (III.41) in the following form :

ψ = ψ 0 + R 0 (k) Φ(E) -Φ V 0 (E) ψ, (III.85)
where R 0 (k) is the integral operator with Schwartz kernel R 0 (x, y, k), x, y ∈ ∂D.

We are going to show that R 0 (k) is a linear continuous operator on the space C 1,β (∂D, M n (C)). To show this we use the representation (III.35).

It follows from Theorems 2.12, 2.17 of [START_REF] Colton | Integral equation methods in scattering theory[END_REF] that the integral operator on C 1,β (∂D, M n (C)) with Schwartz kernel G(x -y, k), x, y ∈ ∂D, is continuous.

To see that the integral operator with Schwartz kernel e ik(x-y) r 0 (x, y, k), x, y ∈ ∂D, is continuous on C 1,β (∂D, M n (C)), we will use formula (III.36).

It follows from (III.36), (III.37) (if we take into account that V 0 has a compact support in D) that derivatives ∂ x i r 0 (x, y, k), ∂ x i ∂ x j r 0 (x, y, k), i, j = 1, . . ., d, exist and are continuous for x, y belonging to some neighborhood of set ∂D. Hence the operator with Schwartz kernel e ik(x-y) r 0 (x, y, k), x, y ∈ ∂D, is continuous on C 1,β (∂D, M n (C)). Hence R 0 (k) is also a linear continuous operator on C 1,β (∂D, M n (C)).

If we show that Φ(E)-Φ V 0 (E) is a compact operator on C 1,β (∂D, M n (C)), formula (III.85) will imply that integral operator in formula (III.41) is compact.

Let S be the operator defined in part 3 of Lemma III.3. Define S V 0 by the same formula using coefficients A 0 1 ≡ 0, . . ., A 0 d ≡ 0, V 0 in (III.80). It follows from Lemma III.3, part 3, that S, S V 0 are linear continuous operators from

C 1,β (∂D, M n (C)) to C 1 (D, M n (C)).
Taking into account equation (III.80), we obtain the following formula :

Φ(E) -Φ V 0 (E) = N S -N V 0 S V 0 , (III.86)
where N , N V 0 are the linear continuous operators acting from C 1 (D, M n (C)) to C 2 (∂D, M n (C)) defined by the following formulas :

(N ψ)(x) = D ∂Γ ∂ν x (x, y, k) -2i d j=1
A j (y)∂ y j + V (y) ψ(y) dy,

(N V 0 ψ)(x) = D ∂Γ ∂ν x (x, y, k)V 0 (y)ψ(y) dy,
where Γ is the Green function for Dirichlet problem for operator ∆ + E in D and ν x denotes the unit exterior normal to ∂D at x ∈ ∂D.

Taking into accout that inclusion

C 2 (∂D, M n (C)) → C 1,β (∂D, M n (C)) is compact, we obtain that N , N V 0 are compact operators from C 1 (D, M n (C)) to C 1,β (∂D, M n (C)).
It follows from continuity of S, S V 0 , from compactness of N , N V 0 and from formula (III.86) that Φ(E) -Φ V 0 (E) is a compact operator on C 1,β (∂D, M n (C)). Now formula (III.85) implies that the integral operator in equation (III.41) is compact on C 1,β (∂D, M n (C)).

In a similar way it can be shown that the integral operators in equations (III.44), (III.47) are compact.

Unique solvability of equations (III.41), (III.44), (III.47)

In this subsection we will finish the proof of the part 1 of Theorem III.2 . The proof of parts 2, 3 of Theorem III.2 can be finished in a similar way.

In fact, in this subsection we will prove the following lemma.

Lemma III.6.

Let k ∈ C d \ (R d ∪ E V 0 ), k 2 = E,
ψ(x, k) = e ikx µ(x, k) with µ ∈ W 1,∞ (R d , M n (C)).
Lemma III.6 implies the statement of part 1 of Theorem III.2 since equation (III.67

) at fixed k ∈ C d \ (R d ∪ E V 0 ) is uniquely solvable for ψ(x, k) = e ikx µ(x, k) with µ ∈ W 1,∞ (R d , M n (C)) if and only if k ∈ E.
Before passing to the proof of Lemma III.6, note that function ψ 0 (x, k) of (III.41) is defined as the solution to (III.11) with coefficients A 0 1 ≡ 0, . . ., A 0 d ≡ 0, V 0 , such that

ψ 0 (x, k) = e ikx µ 0 (x, k), µ 0 ∈ W 1,∞ (R d , M n (C)). Note that function ψ 0 (x, k) belongs to C 2 (R d , M n (C)
) and satisfies the equation

-∆ψ 0 + V 0 (x)ψ 0 = Eψ 0 , x ∈ R d .
(III.87)

The proof of this fact is similar to the proof of parts 1, 2 of Lemma III.4. We will prove Lemma III.6 in two steps.

== D ∆ y + E -V 0 (y) R 0 (x, y, k)ϕ + (y) dy + ∂D R 0 (x, y, k) ∂ϕ + ∂ν y (y) - ∂R 0 ∂ν y (x, y, k)ϕ(y) dy (III.74) == D δ x (y)ϕ + (y) dy + ∂D R 0 (x, y, k) ∂ϕ + ∂ν y (y) - ∂R 0 ∂ν y (x, y, k)ϕ(y) dy.
Using formulas (III.90), (III.91), we obtain formula where ν x is the unit exterior normal to ∂D at x ∈ ∂D, and the argument x + 0ν x (or x -0ν x ) means that we evaluate function at x + εν x (or x -εν x ), ε > 0, and then pass ε → +0. Besides, the following equality follows from (III.74), (III.91) and from definition of function ϕ + :

ϕ -(x) = ψ 0 (x, k) + ∂D R 0 (x,
∆ x -V 0 (x) + E ∂D R 0 (x, y, k) ∂ϕ + ∂ν y (y) dy - ∂D ∂R 0 ∂ν y (x, y, k)ϕ(y) dy = 0, (III.94)
where x ∈ D. Formulas (III.87), (III.94) imply that the function on the right hand side of equation (III.92) is annihilated by operator ∆ -V 0 + E in D and formula (III.93) together with property ϕ -| ∂D = ϕ imply that this function has limit zero as x approaches ∂D from D. Since E is not a Dirichlet eigenvalue for operator L V 0 ≡ -∆ + V 0 in D the following formula is valid :

ψ 0 (x, k) + ∂D R 0 (x, y, k) ∂ϕ + ∂ν y (y) - ∂R 0
∂ν y (x, y, k)ϕ(y) dy = 0, (III. [START_REF] Norton | Reconstruction of stratified fluid flow from reciprocal scattering measurements[END_REF] where x ∈ D. Define From formulas (III.36), (III.37) it follows that functions r 0 (x, y, k), ∂ x j r 0 (x, y, k), j = 1, . . ., d, are uniformly bounded for y ∈ D, x ∈ D, dist(x, ∂D) > 1. This fact together with the property that ψ 0 (x, k) = e ikx µ 0 (x, k), µ 0 ∈ W 1,∞ (R d , M n (C)) and with formula (III.67) imply that ψ(x, k) = e ikx µ(x, k) with µ ∈ W 1,∞ (R d , M n (C)).

ψ(x) =      ϕ -(x), x ∈ R d \ D, ϕ(x), x ∈ ∂D, ϕ + (x), x ∈ D.
Thus

(x, k) = e ikx µ(x, k), µ ∈ W 1,∞ (R d , M n (C)), be a solution to (III.67). It follows from inclusion W 1,∞ (R d , M n (C)) ⊂ C(R d , M n (C)) that ϕ = ψ| ∂D is a continuous function on ∂D.
Repeating the proof of Theorem III.1 , we can see that ϕ is a solution to (III.41). We are going to show that ϕ ∈ C 2 (∂D, M n (C)).

It follows from formula (III.67 We have shown that solutions ϕ to (III.41) of class C 1,β (∂D, M n (C)) are in bijective correspondence with solutions ψ to (III.67) such that ψ(x, k) = e ikx µ(x, k), µ ∈ W 1,∞ (R d , M n (C)). This finishes the proof of Lemma III.6.

) that ψ ∈ C 1 (R d , M n (C)). Since A 1 , . . . , A d , V have compact

Introduction

We consider the equation

-∆ψ -2iA(x)∇ψ + V (x)ψ = Eψ, x = (x 1 , x 2 ) ∈ R 2 , E > 0, (IV.1)
where ∆ = ∂ 2

x 1 + ∂ 2 x 2 , ∇ = (∂ x 1 , ∂ x 2 ), ∂ x k = ∂/∂x k , k = 1, 2
, and A = (A 1 , A 2 ) and V are vector and scalar potentials on R 2 , respectively. In addition we assume that A 1 , A 2 and V are sufficiently regular functions on R 2 with sufficient decay at infinity. (IV.2)

1. CMAP, Ecole polytechnique, CNRS, Université Paris-Saclay, 91128, Palaiseau, France ; IEPT RAS, 117997, Moscow, Russia ; Moscow Institute of Physics and Technology, Dolgoprudny, Russia ; email : novikov@cmap.polytechnique.fr Equation (IV.1) can be considered as a model equation for the time-harmonic exp(-iωt) acoustic pressure ψ in a two-dimensional moving fluid, see, e.g., [START_REF] Roussef | Two-dimensional vector flow inversion by diffraction tomography[END_REF][START_REF] Rychagov | Reconstruction of fluid motion in acoustic diffraction tomography[END_REF]. In this setting

E = ω c 0 2 , A(x) = ω c 0 u(x), V (x) = 1 -n 2 (x) ω c 0 2 , (IV.3)
where c 0 is a reference sound speed, n(x) is a scalar index of refraction, u(x) is a normalized fluid velocity vector. Equation (IV.1) can be also considered as the two-dimensional Schrödinger equation at fixed energy E with magnetic potential A and electric potential v, where

V (x) = A 2 (x) -i∇ • A(x) + v(x) = A 2 1 (x) + A 2 2 (x) -i ∂A 1 (x) ∂x 1 + ∂A 2 (x) ∂x 2 + v(x),
see, e.g., [START_REF] Novikov | The ∂-equation in the multidimensional inverse scattering problem[END_REF][START_REF]Inverse scattering problems for Schrödinger operators with magnetic and electric potentials[END_REF].

For equation (IV.1) we consider the classical scattering solutions ψ + continuous and bounded on R 2 with their first derivatives and specified by the following asymptotics as |x| → ∞ :

ψ + (x, k) = e ikx + C(|k|) e i|k||x| |x| 1/2 f k, |k| x |x| + o 1 |x| 1/2 , (IV.4) x ∈ R 2 , k ∈ R 2 , k 2 = E, C(|k|) = -πi √ 2πe -iπ/4 |k| 1/2 ,
where a priori unknown function f = f (k, l), k, l ∈ R 2 , k 2 = l 2 = E, arising in (IV.4) is the classical scattering amplitude for (IV.1). Given potentials A, V , to determine ψ + and f one can use the following Lippmann-Schwinger integral equation (IV.5) and formula (IV.6) ; see, e.g., [START_REF] Novikov | The ∂-equation in the multidimensional inverse scattering problem[END_REF] :

ψ + (x, k) = e ikx + R 2 G + (x -y, k)× × -2iA(y)∇ y ψ + (y, k) + V (y)ψ + (y, k) dy, (IV.5) G + (x, k) = -(2π) -2 R 2 e iξx dξ ξ 2 -k 2 -i0 = - i 4 H 1 0 (|x||k|), where x ∈ R 2 , k ∈ R 2 , k 2 = E, H 1 0 is the Hankel function of the first kind ; f (k, l) = (2π) -2 R 2 e -ily -2iA(y)∇ y ψ + (y, k) + V (y)ψ + (y, k) dy, (IV.6)
where k ∈ R 2 , l ∈ R 2 , k 2 = l 2 = E. Actually, we consider (IV.5) and its differentiated version, where ∇ is applied to both sided of (IV.5), as a system of linear integral equations for ψ + and ∇ψ + .

One can see that the scattering amplitude f for equation (IV.1) at fixed E is defined on

M E = k ∈ R 2 , l ∈ R 2 : k 2 = l 2 = E , E > 0. (IV.7)
Note that f on M E is invariant with respect to the gauge transformations

A → A + ∇ϕ, V → V -i∆ϕ + (∇ϕ) 2 + 2A∇ϕ, (IV.8)
where ϕ is a sufficiently regular function on R 2 with sufficient decay at infinity, see e.g. [START_REF] Henkin | A multidimensional inverse problem in quantum and acoustic scattering[END_REF][START_REF]Inverse scattering problems for Schrödinger operators with magnetic and electric potentials[END_REF]. In addition, ψ + is transformed as

ψ + → e -iϕ ψ + (IV.9)
with respect to (IV.8).

In this work we consider the following inverse scattering problem for equation (IV.1) under assumptions (IV.2) :

Problem IV.1. Given scattering amplitude f on M E at fixed E > 0, find potentials A and V on R 2 (at least approximately).

Problem IV.1 for the case when A ≡ 0 was studied, in particular, in [START_REF] Novikov | Construction of two-dimensional Schrödinger operator with given scattering amplitude at fixed energy[END_REF][START_REF] Grinevich | Inverse scattering problem for the twodimensional Schrödinger operator, the ∂-method and nonlinear equations[END_REF][START_REF]The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF][START_REF]Transparent potentials at fixed energy in dimension two. Fixed-energy dispersion relations for the fast decaying potentials[END_REF][START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF][START_REF] Bogatyrev | Numerical realization of algorithm for exact solution of two-dimensional monochromatic inverse problem of acoustical scattering[END_REF][START_REF] Burov | Multifrequency generalization of the Novikov algorithm for the two-dimensional inverse scattering problem[END_REF] and in [START_REF]Multidimensional inverse spectral problem for the equation -∆ψ+(v(x)-Eu(x))ψ = 0[END_REF][START_REF] Bukhgeim | Recovering a potential from Cauchy data in the twodimensional case[END_REF].

Problem IV.1 for the general case was considered, in particular, on page 457 of [START_REF]The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF] and in [START_REF] Xiaosheng | Inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials in two dimensions at fixed energy[END_REF].

Problem IV.1 is also related with several other inverse problems for the Schrödinger equation in magnetic field (and for the time-harmonic wave equation with first-order perturbation) in dimension d ≥ 2. Concerning these other inverse problems see [START_REF] Dubrovin | The Schrödinger equation in a periodic field and Riemann surfaces[END_REF][START_REF] Shiota | An inverse problem for the wave equation with first order perturbation[END_REF][START_REF] Henkin | A multidimensional inverse problem in quantum and acoustic scattering[END_REF][START_REF] Norton | Reconstruction of stratified fluid flow from reciprocal scattering measurements[END_REF][START_REF] Roussef | Two-dimensional vector flow inversion by diffraction tomography[END_REF][START_REF] Rychagov | Reconstruction of fluid motion in acoustic diffraction tomography[END_REF][START_REF] Baykov | Mode tomograpgy of moving ocean[END_REF][START_REF] Eskin | Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy[END_REF][START_REF] Nicoleau | A stationary approach to inverse scattering for Schrödinger operators with first order perturbation[END_REF][START_REF] Weder | On inverse scattering at a fixed energy for potentials with a regular behaviour at infinity[END_REF][START_REF] Guillarmou | Identification of a connection from Cauchy data on a Riemann surface with boundary[END_REF][START_REF] Imanuvilov | Inverse problem by Cauchy data on an arbitrary sub-boundary for systems of elliptic equations[END_REF] and references therein.

Note that approximate finding A and V in Problem IV.1 means, in particular, finding modulo transformations (IV.8). However, for real-valued A and V there is no gauge nonuniqueness (IV.8) in Problem IV.1 ! In addition, A and V of formulas (IV.3) (of moving fluid acoustics) are real if n is real.

In this work we are mainly motivated by applications to the acoustic tomography of moving fluid discribed in [START_REF] Norton | Reconstruction of stratified fluid flow from reciprocal scattering measurements[END_REF][START_REF] Roussef | Two-dimensional vector flow inversion by diffraction tomography[END_REF][START_REF] Rychagov | Reconstruction of fluid motion in acoustic diffraction tomography[END_REF][START_REF] Baykov | Mode tomograpgy of moving ocean[END_REF]. Note that in their reconstruction results the works [START_REF] Norton | Reconstruction of stratified fluid flow from reciprocal scattering measurements[END_REF][START_REF] Roussef | Two-dimensional vector flow inversion by diffraction tomography[END_REF][START_REF] Rychagov | Reconstruction of fluid motion in acoustic diffraction tomography[END_REF][START_REF] Baykov | Mode tomograpgy of moving ocean[END_REF] proceed from the near-field scattering data (e.g., from some near field information on ψ + ) instead of the scattering amplitude f . But it is also known that near-field scattering data can be transformed into values of f , see, e.g., [START_REF] Berezansky | On uniqueness theorem in inverse problem of spectral analysis for Schrödinger equation[END_REF][START_REF] Baykov | Mode tomograpgy of moving ocean[END_REF].

Results of the present work can be described as follows :

In Section 2 we give formulas for solving Problem IV.1 in the Born approximation. To our knowledge these formulas were not yet given explicitely in the literature for the case when A ≡ 0. These formulas are proved in Section 5.

In Section 3 we give a nonlinearized reconstruction algorithm for Problem IV.1. For the case when A ≡ 0 this algorithm is reduced to the algorithm of [START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF] and was implemented numerically in [START_REF] Burov | Multifrequency generalization of the Novikov algorithm for the two-dimensional inverse scattering problem[END_REF]. For the general case this algorithm can be also regarded as simplication and development of the algorithm mentioned on page 457 of [START_REF]The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF] and based on the Riemann-Hilbert-Manakov problem approach of [START_REF] Grinevich | Analogs of multisoliton potentials for the two-dimensional Schrödinger operator[END_REF][START_REF] Novikov | Construction of two-dimensional Schrödinger operator with given scattering amplitude at fixed energy[END_REF]. A derivation of the reconstruction algorithm of Subsection 3.2 is given in Section 6.

In Section 4 we show that in the Born approximation the algorithm of Section 3 is reduced to formulas of Section 2. Related proofs are given in Section 7.

In a similar way with results of [START_REF]Monochromatic recontruction algorithms for two-dimensional multichannel inverse problems[END_REF] the reconstruction algorithm of Subsection 3.2 can be generalized to the multi-channel case. This generalization will be given in a subsequent work.

In the present work we are focused on approximate reconstructions for Problem IV.1, admitting stable numerical implementation. Issues related with theorems of uniqueness and examples of nonuniqueness for Problem IV.1 will be considered in a subsequent work.

Inverse scattering in the Born approximation

If A = (A 1 , A 2 ) and V satisfy (IV.2) and are sufficiently small, then proceeding from (IV.5), (IV.6) we have the following Born approximation formulas for direct scattering :

ψ + (x, k) ≈ e ikx , ∇ψ + (x, k) ≈ e ikx ik, f (k, l) ≈ f lin (k, l),
(IV.10)

f lin (k, l) def == (2π) -2 R 2 e i(k-l)x 2kA(x) + V (x) dx, (IV.11)
where x, k, l ∈ R 2 , k 2 = l 2 = E. In particular, formulas (IV.10) can be specified as (IV.21). Note that f lin on M E is invariant with respect to the gauge transformations

A → A + ∇ϕ, V → V -i∆ϕ, (IV.12)
where ϕ is a sufficiently regular function on R 2 with sufficient decay at infinity. This invariance follows from (IV.11), (IV.12), integrating by parts and using that k 2 -l 2 = 0. We consider (IV.12) as a linearization of (IV.8) for small A, V and ϕ.

In this section we consider the following linearized inverse scattering problem for equation (IV.1) under assumptions (IV.2) :

Problem IV.2. Given linearized scattering amplitude f lin on M E at fixed E > 0, find potentials A and V on R 2 (at least approximately).

Note that approximate finding A and V in Problem IV.2 means, in particular, finding modulo transformations (IV.12). However, in a similar way with Problem IV.1, there is no gauge nonuniqueness (IV.12) in Problem IV.2 for the case of real-valued A and V .

Problem IV.2 is a linearization of Problem IV.1.

To study Problem IV.2 it is convenient to introduce ϕ div , A div,0 , V div,0 and ϕ ± , A ±,0 , V ±,0 such that :

∆ϕ div (x) = -∇ • A(x), ϕ div (x) → 0 as |x| → ∞, A div,0 (x) = A(x) + ∇ϕ div (x), V div,0 (x) = V (x) -i∆ϕ div (x), (IV.13)
where x ∈ R 2 ;

∂ z ϕ -(x) = - 1 2 (A 1 (x) -iA 2 (x)), ϕ -(x) → 0 as |x| → ∞, A -,0 (x) = A(x) + ∇ϕ -(x), V -,0 (x) = V (x) -i∆ϕ -(x),
(IV.14)

∂ z ϕ + (x) = - 1 2 (A 1 (x) + iA 2 (x)), ϕ + (x) → 0 as |x| → ∞, A +,0 (x) = A(x) + ∇ϕ + (x), V +,0 (x) = V (x) -i∆ϕ + (x), (IV.15)
where

∂ z = 1 2 (∂ x 1 -i∂ x 2 ), ∂ z = 1 2 (∂ x 1 + i∂ x 2 ), x = (x 1 , x 2 ) ∈ R 2 .
One can see that

∇ • A div,0 (x) = 0, A -,0 1 (x) -iA -,0 2 (x) = 0, A +,0 1 (x) + iA +,0 2 (x) = 0, where x ∈ R 2 , A ±,0 = (A ±,0 1 , A ±,0
2 ). It is also convenient to transform formula (IV.11) to the form

f lin (k, l) -f lin (-l, -k) = 2(k + l) A(k -l), f lin (k, l) + f lin (-l, -k) = 2(k -l) A(k -l) + 2 V (k -l), (IV.16) A(p) = (2π) -2 R 2 e ipx A(x) dx, V (p) = (2π) -2 R 2 e ipx V (x) dx, (IV.17) where (k, l) ∈ M E , p ∈ R 2 . Note that (k, l) ∈ M E =⇒ k -l ∈ B 2 √ E , p ∈ B 2 √ E =⇒ p = k -l for some (k, l) ∈ M E , (IV.18) where B r = {p ∈ R 2 : |p| ≤ r}, r > 0. (IV.19)
We define

C N,σ (R 2 ) = u ∈ C N (R 2 ) : u N,σ < +∞ , u N,σ = max |n| N sup x∈R 2 (1 + |x| 2 ) σ/2 |∂ n u(x)|, N ∈ N ∪ {0}, σ 0, (IV.20)
where C N (R 2 ) is the space of N -times continuously differentiable complex-valued functions on R 2 ,

∂ n = ∂ n 1 x 1 ∂ n 2 x 2 , n = (n 1 , n 2 ) ∈ N ∪ {0} 2 , |n| = n 1 + n 2 . Note that if A 1 , A 2 , V ∈ C 0,σ (R 2 ) for some σ > 2 and A j 0,σ ≤ q, j = 1, 2, V 0,σ ≤ q, then ψ + (x, k) = e ikx + O(q), ∇ψ + (x, k) = e ikx ik + O(q), f (k, l) = f lin (k, l) + O(q 2 ) as q → +0, (IV.21) uniformly with repsect to x, k, l ∈ R 2 , k 2 = l 2 = E, at fixed E > 0. Theorem IV.1. Suppose that A 1 , A 2 , V are real-valued and A 1 , A 2 , V ∈ C N,σ (R 2 ) for some N ≥ 3, σ > 2.
Then the following formulas for solving Problem IV.2 are valid :

A(k -l) = f lin (k, l) -f lin (l, k) 2 k -l |k -l| 2 + f lin (k, l) -f lin (-l, -k) 2 k + l |k + l| 2 , V (k -l) = f lin (l, k) + f lin (-l, -k) 2 , (IV.22)
where A, V are defined by (IV.17) and (k, l) ∈ M E ;

A(x) = A appr (x, E) + A err (x, E), x ∈ R 2 , E > 0, A appr (x, E) = |p|≤2 √ E e -ipx A(p) dp, A err (x, E) = |p|≥2 √ E e -ipx A(p) dp, (IV.23) V (x) = V appr (x, E) + V err (x, E), x ∈ R 2 , E > 0, V appr (x, E) = |p|≤2 √ E e -ipx V (p) dp, V err (x, E) = |p|≥2 √ E e -ipx V (p) dp, (IV.24)
where A(p) and V (p) for |p| ≤ 2 √ E are given in terms of f lin on M E according to (IV.18), (IV.22) and

|A err,j (x, E)| ≤ c 1 (N, σ) A j N,σ E -N -2
2 , (IV.25)

|V err (x, E)| ≤ c 1 (N, σ) V N,σ E -N -2 2 , (IV.26) where x ∈ R 2 , j = 1, 2, A err = (A err,1 , A err,2 ), E ≥ 1 4 and c 1 (N, σ) = 4 (N -2)(σ -2) . (IV.27) Theorem IV.2. Suppose that A 1 , A 2 , V ∈ C N,σ (R 2 )
for some N ≥ 4 and σ > 2. Let A div,0 , V div,0 be defined according to (IV.13). Then the following formulas for solving Problem IV.2 are valid :

A div,0 (k -l) = f lin (k, l) -f lin (-l, -k) 2 k + l |k + l| 2 , V div,0 (k -l) = f lin (k, l) + f lin (-l, -k) 2 ,
(IV.28)

where A div,0 , V div,0 are the Fourier transforms of A div,0 , V div,0 (see (IV.17)) and (k, l) ∈ M E ;

A div,0 (x) = A div,0 appr (x, E) + A div,0 err (x, E), x ∈ R 2 , E > 0, (IV.29)

A div,0 appr (x, E) = |p|≤2 √ E e -ipx A div,0 (p) dp, A div,0 err (x, E) = |p|≥2 √ E e -ipx A div,0 (p) dp, V div,0 (x) = V div,0 appr (x, E) + V div,0 err (x, E), x ∈ R 2 , E > 0, (IV.30) V div,0 appr (x, E) = |p|≤2 √ E e -ipx V div,0 (p) dp, V div,0 err (x, E) = |p|≥2 √ E e -ipx V div,0 (p) dp,
where A div,0 (p) and V div,0 (p) for |p| ≤ 2 √ E are given in terms of f lin on M E according to (IV.18), (IV.28) and

|A div,0 err,j (x, E)| ≤ (1 + √ 2)c 1 (N, σ) A N,σ E -N -2 2 , (IV.31) |V div,0 err (x, E)| ≤ c 1 (N, σ) V N,σ E -N -2 2 + √ 2 A N,σ E -N -3 2 , (IV.32) A N,σ = max A 1 N,σ , A 2 N,σ , (IV.33) where x ∈ R 2 , j = 1, 2, E ≥ 1 4 , A div,0 err = (A div,0 err,1 , A div,0 err,2 ) and c 1 (N, σ) is defined by (IV.27). Furthermore, if ∇ • A = 0 then A div,0 = A, V div,0 = V . Theorem IV.3. Suppose that A 1 , A 2 , V ∈ C N,σ (R 2 )
for some N ≥ 4 and σ > 2. Let A ±,0 , V ±,0 be defined according to (IV.14)-(IV.15). Then the following formulas for solving Problem IV.2 are valid :

A ±,0 1 (k -l) = 1 2 f (k, l) -f (-l, -k) k 1 + l 1 ± i(k 2 + l 2 ) , A ±,0 2 (k -l) = ±i A ±,0 1 (k -l), V ±,0 (k -l) = (l 1 ± il 2 )f (k, l) + (k 1 ± ik 2 )f (-l, -k) k 1 + l 1 ± i(k 2 + l 2 ) , (IV.34)
where A ±,0 , V ±,0 are the Fourier transforms of A ±,0 , V ±,0 (see (IV.17)) and (k, l) ∈ M E ;

A ±,0 (x) = A ±,0 appr (x, E) + A ±,0 err (x, E), x ∈ R 2 , E > 0, (IV.35)

A ±,0 appr (x, E) = |p|≤2 √ E e -ipx A ±,0 (p) dp, A ±,0 err (x, E) = |p|≥2 √ E e -ipx A ±,0 (p) dp, V ±,0 (x) = V ±,0 appr (x, E) + V ±,0 err (x, E), x ∈ R 2 , E > 0, (IV.36) V ±,0 appr (x, E) = |p|≤2 √ E e -ipx V ±,0 (p) dp, V ±,0 err (x, E) = |p|≥2 √ E e -ipx V ±,0 (p) dp,
where A ±,0 (p) and V ±,0 (p) for |p| ≤ 2 √ E are given in terms of f lin on M E according to (IV.18), (IV.34) and

|A ±,0 err,j (x, E)| ≤ (1 + √ 2)c 1 (N, σ) A N,σ E -N -2 2 , (IV.37) |V ±,0 err | ≤ c 1 (N, σ) V N,σ E -N -2 2 + √ 2 A N,σ E -N -3 2 , (IV.38)
where x ∈ R 2 , j = 1, 2, A ±,0 err = (A ±,0 err,1 , A ±,0 err,2 ), A N,σ is defined by (2.23) and c 1 (N, σ) is given by (IV.27). Furthermore, if

A 1 ± iA 2 = 0 then A = A ±,0 , V = V ±,0 .
Theorems IV.1-IV.3 are proved in Section 5.

Nonlinearized inverse scattering 3.1 Some notations

To study Problem IV.1 it is convenient to introduce ϕ div , A div , V div and ϕ ± , A ± , V ± , where ϕ div and ϕ ± are defined according to (IV.13)-(IV.15) and

A div = A + ∇ϕ div , V div = V -i∆ϕ div + (∇ϕ div ) 2 + 2A∇ϕ div , A ± = A + ∇ϕ ± , V ± = V -i∆ϕ ± + (∇ϕ ± ) 2 + 2A∇ϕ ± .
(IV.39)

In this section we give a nonlinearized algorithm for approximate finding A ± , V ± and A div , V div on R 2 from f on M E . This algorithm takes into account multiple scattering effects and can be regarded as a nonlinear version of formulas for A ±,0 appr , V ±,0 appr , A div,0 appr , V div,0 appr of (IV.35), (IV.36), (IV.29), (IV.30).

It is convenient to use the following notations :

z = x 1 + ix 2 , z = x 1 -ix 2 , (IV.40) λ = E -1/2 (k 1 + ik 2 ), λ = E -1/2 (l 1 + il 2 ), (IV.41) where x = (x 1 , x 2 ) ∈ R 2 , k = (k 1 , k 2 ) ∈ Σ E , l = (l 1 , l 2 ) ∈ Σ E , Σ E = m = (m 1 , m 2 ) ∈ C 2 : m 2 1 + m 2 2 = E , E > 0. (IV.42)
In these notations

k 1 = 1 2 E 1/2 (λ + λ -1 ), k 2 = i 2 E 1/2 (λ -1 -λ), (IV.43) l 1 = 1 2 E 1/2 (λ + λ -1 ), l 2 = i 2 E 1/2 (λ -1 -λ ), (IV.44) exp(ikx) = exp i 2 E 1/2 (λz + λ -1 z) , where λ, λ ∈ C \ {0}, z ∈ C 2 , k, l ∈ Σ E .
In addition, using formulas (IV.7), (IV.41)-(IV.44) one can see that

Σ E ∼ = C \ {0}, Σ E ∩ R 2 = S 1 √ E ∼ = T, M E ∼ = T × T, where S 1 r = m ∈ R 2 : |m| = r , r > 0, T = λ ∈ C : |λ| = 1 .
In addition, the functions ψ + , f of (IV.4)-(IV.6) can be written as

ψ + = ψ + (z, λ, E), f = f (λ, λ , E),
where λ, λ ∈ T , z ∈ C, E > 0.

Reconstruction algorithm

Our nonlinearized algorithm for approximate finding A ± , V ± and A div , V div on R 2 from f on M E has the following scheme

f -→ h ± -→ µ + -→ µ ± -→ A ± appr , V ± appr -→ A div appr , V div appr
and consists of the following steps :

Step 1. Find functions h ± (λ, λ , E), λ, λ ∈ T , from the following linear integral equations : Step 2. Solve the following linear integral equation for µ + (z, λ, E), z ∈ C, λ ∈ T , E > 0 :

h ± (λ, λ , E) -πi T h ± (λ, λ , E)χ ±i λ λ - λ λ × ×f (λ , λ , E) |dλ | = f (λ, λ , E),
µ + (z, λ, E) + T B(λ, λ , z, E)µ + (z, λ , E) |dλ | = 1, (IV.47)
where

B(λ, λ , z, E) = 1 2 T h -(ζ, λ , z, E)χ -i ζ λ - λ ζ dζ ζ -λ(1 -0) - - 1 2 T h + (ζ, λ , z, E)χ i ζ λ - λ ζ dζ ζ -λ(1 + 0) , (IV.48) h ± (λ, λ , z, E) def == h ± (λ, λ , E)× × exp -i √ E 2 (λ -λ )z + (λ -1 -λ -1 )z , (IV.49)
and λ, λ ∈ T , z ∈ C, E > 0.

Step 3. Define functions µ ± (z, λ, E), z ∈ C, λ ∈ T , E > 0, by formulas

µ ± (z, λ, E) = µ + (z, λ, E) + πi T h ± (λ, λ , z, E)× ×χ ±i λ λ - λ λ µ + (z, λ , E) |dλ |, (IV.50)
where functions h ± (λ, λ , z, E) are given by (IV.49) and χ is defined by (IV.46).

Step 4. Functions A ± appr,j (x, E), V ± appr (x, E), x ∈ R 2 , j = 1, 2, E > 0, are defined by formulas

A - appr,1 (x, E) = i 4 a - z (z, E), A - appr,2 (x, E) = 1 4 a - z (z, E), a - z (z, E) = 4∂ z ln T µ + (z, ζ, E) |dζ|, V - appr (x, E) = √ E π T ∂ z µ -(z, ζ, E) dζ,
(IV.51) 91 and

A + appr,1 (x, E) = i 4 a + z (z, E), A + appr,2 (x, E) = - 1 4 a + z (z, E), a + z (z, E) = -4∂ z ln T µ + (z, ζ, E) |dζ|, V + appr (z, E) = 2i √ E∂ z T µ + (z, ζ, E) dζ ζ 2 T µ + (z, ζ, E) dζ ζ , (IV.52)
where z is given by (IV.40).

Step 5. Find A div appr,j (x, E), V div appr (x, E), x ∈ R 2 , j = 1, 2, E > 0, from formulas

A div appr,1 (x, E) = i 8 (a - z (z, E) + a + z (z, E)), A div appr,2 (x, E) = 1 8 (a - z (z, E) -a + z (z, E)), (IV.53) V div appr (x, E) = 1 2 V - appr (x, E) + V + appr (z, E) - 1 8 a - z (z, E)a + z (z, E),
where z is defined by (IV.40) and functions a - z , a + z , V ± appr are defined in (IV.51), (IV.52). A derivation of this reconstruction algorithm is based on the method of the Riemann-Hilbert problem and on the ∂-method. This derivation is given in Section 6.

For the case when A ≡ 0 this algorithm is reduced to the algorithm of [START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF] for approximatie finding V on R 2 from f on M E . The algorithm of [START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF] consists of the same aforementioned steps 1, 2, 3 and the formula V appr = V - appr , where V - appr is defined in (IV.51). This algorithm of [START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF] was implemented numerically in [START_REF] Burov | Multifrequency generalization of the Novikov algorithm for the two-dimensional inverse scattering problem[END_REF].

For the general case this algorithm can be also regarded as simplication and development of the algorithm mentioned (in few lines) on page 457 of [START_REF]The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF]. Actually, in [START_REF]The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF] the part of the algorithm consisting in finding µ ± from h ± is realized in a more complicated way. In addition, in [START_REF]The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF] the algorithm is mentioned for the case when

A 1 = A 1 , A 2 = A 2 , -2i∇ • A + V = V,
i.e. for the self-adjoint case, whereas this assumption is not necessary for the algorithm.

Properties of the algorithm

Let u 1 L 2 (T ) = T |u 1 (λ)| 2 |dλ| 1/2 , u 2 L 2 (T 2 ) = T 2 |u 2 (λ, λ )| 2 |dλ| |dλ | 1/2 , T 2 = T × T,
where u 1 and u 2 are test functions on T and T 2 , respectively.

Proposition IV.1. Let E > 0 be fixed. Suppose that

f ∈ L 2 (T 2 ), f L 2 (T 2 ) < 1 π , (IV.54)
where f = f (λ, λ , E). Then equation (IV.45) is uniquely solvable for h ± ∈ L 2 (T 2 ) and

h ± L 2 (T 2 ) < f L 2 (T 2 ) 1 -π f L 2 (T 2 ) , B L 2 (T 2 ) < 2π f L 2 (T 2 ) 1 -π f L 2 (T 2 )
,

where B is defined by (IV.48), (IV.49) (at fixed z, E). In addition, if

f L 2 (T 2 ) < 1 3π , (IV.55) then B L 2 (T 2 ) < 1, equation (IV.47) at fixed z, E is uniquely solvable for µ + ∈ L 2 (T ) and µ + L 2 (T ) < (2π) 1/2 1 -B L 2 (T 2 ) , µ + -1 L 2 (T ) < (2π) 1/2 B L 2 (T 2 ) 1 -B L 2 (T 2 ) , µ ± -1 L 2 (T ) < 3π(2π) 1/2 f L 2 (T 2 ) 1 -3π f L 2 (T 2 ) ,
where µ ± are defined by (IV.50). In addition, at least, if

f L 2 (T 2 ) < 1 6π , (IV.56) then T µ + (z, λ, E) |dλ| = 0 for all z ∈ C, (IV.57)
and A ± appr,j , A div appr,j , j = 1, 2, as well as V ± appr , V div appr are bounded on R 2 . Proposition IV.1 is based on solving the linear integral equations (IV.45) and (IV.47) by the method of successive approximations in L 2 (T 2 ) and L 2 (T ), respectively, and on standard estimates of L 2 -analysis for B, h ± , µ ± of (IV.48)-(IV.50) and for the integral of (IV.57).

Note that assumptions (IV.54), (IV.55), (IV.56) of Proposition IV.1 are only some surplus sufficient conditions on f for unique solvability of integral equations (IV.45), (IV.47), fulfilment of (IV.57) and for boundedness of A div appr , V div appr .

Theorem IV.4. Let f ∈ L 2 (T 2 ) at fixed E > 0. Suppose that f satisfies (IV.56) and is a smooth function on T 2 and A div appr , V div appr are constructed from f via the algorithm of Subsection 3.2. Then A div appr,1 , A div appr,2 , V div appr are bounded functions on R 2 , decaying at infinity. In addition, f is the scattering amplitude for equation (IV.1) with A = A div appr (x, E), V = V div appr (x, E).

For simplicity one can assume that f ∈ C ∞ (T 2 ) in Theorem IV.4. However, very limited smoothness of f is already sufficient. As regards to smoothness of A div appr,1 , A div appr,2 , V div appr of Theorem IV.4 (which are complex-valued, in general), these functions are real-analytic functions of x ∈ R 2 . In addition, it is just for simplicity that we assume IV.56 in Theorem IV. [START_REF] Agaltsov | Riemann-Hilbert problem approach for two-dimensional flow inverse scattering[END_REF].

The proof of Theorem IV.4 is similar to the proof of Theorem 9.2 of [START_REF]The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF] for the case when A ≡ 0. Results of this type go back to [START_REF] Novikov | Construction of two-dimensional Schrödinger operator with given scattering amplitude at fixed energy[END_REF]. In the present work restrictions in time prevent us from proving Theorem IV.4 in details.

Finally, suppose that f = f (λ, λ , E) is the scattering amplitude for equation (IV.1) under assumptions (IV.2) and that A div appr = A div appr (x, E), V div appr = V div appr (x, E) are constructed from f via the algorithm of Subsection 3.2 at fixed E. In the present work restrictions on time prevent us from obtaining estimates for A div -A div appr (•, E) and V div -V div appr (•, E) for sufficiently large E. For the linearized case such error estimates are given by formulas (IV.31), (IV.32). For the nonlinearized case with A ≡ 0 such error estimates were given in [START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF].

Reduction of the nonlinearized reconstruction algorithm to inversion formulas of the Born approximation

Suppose that we are given f on M E ∼ = T × T = T 2 at fixed E, where

f ∈ L 2 (T 2 ), f L 2 (T 2 ) ≤ ε. (IV.58)
Proposition IV.2. Suppose that f satisfies (IV.58) at fixed E > 0. Then, for ε → +0, the nonlinearized reconstruction algorithm of Subsection 3.2 is reduced to the following formulas at fixed E > 0 :

A ± appr,j (x, E) = A ± appr,j (x, E) + O(ε 2 ), j = 1, 2, V ± appr (x, E) = V ± appr (x, E) + O(ε 2 ), (IV.59) A div appr (x, E) = A div appr,j (x, E) + O(ε 2 ), j = 1, 2, V div appr (x, E) = V div appr (x, E) + O(ε 2 ), (IV.60)
where O(ε 2 ) is considered in the uniform sense with respect to x ∈ R 2 and where functions A ± appr,j , V ± appr , j = 1, 2, and A div appr,j , V div appr , j = 1, 2, are defined by the following linear formulas with respect to f :

A - appr,1 (x, E) = - i 4 √ E T 2 sgn 1 i λ λ - λ λ (λ -λ )f (λ, λ , z, E) |dλ| |dλ |, A - appr,2 (x, E) = -iA - appr,1 (x, E), (IV.61) V - appr (x, E) = i E 2 T 2 (1 -λλ ) sgn 1 i λ λ - λ λ f (λ, λ , z, E) |dλ| |dλ |, A + appr,1 (x, E) = i 4 √ E T 2 sgn 1 i λ λ - λ λ (λ -λ )f (λ, λ , z, E) |dλ| |dλ |, A + appr,2 (x, E) = iA + appr,1 (x, E), (IV.62) V + appr (x, E) = -i E 2 T 2 (1 -λλ ) sgn 1 i λ λ - λ λ f (λ, λ , z, E) |dλ| |dλ |, A div appr,j (x, E) = 1 2 A + appr,j (x, E) + A - appr,j (x, E) , j = 1, 2, V div appr (x, E) = E 2 T 2 1 2i λ λ - λ λ f (λ, λ , z, E) |dλ| |dλ |, (IV.63) where f (λ, λ , z, E) def == f (λ, λ , E)× × exp -i √ E 2 (λ -λ )z + (λ -1 -λ -1 )z , (IV.64)
λ ∈ T , λ ∈ T and z, z are given by (IV.40).

Proposition IV.3. Suppose that A 1 , A 2 , V ∈ C 2,σ (R 2 ) for some σ > 2, where C N,σ (R 2
) is defined by (IV.20). Let f lin be defined by (IV.11) and let A div,0 appr,j , V div,0 appr , A ±,0 appr,j , V ±,0 appr , j = 1, 2, be defined by (IV.28), (IV.29), (IV.30), (IV.34), (IV.35), (IV.36) in terms of f lin . Suppose also that functions A div appr,j , V div appr , A ± appr,j , V ± appr , j = 1, 2, are given by (IV.63), (IV.61), (IV.62) with f = f lin . Then the following equalities are valid :

A div,0 appr,j (x, E) = A div appr,j (x, E), V div,0 appr (x, E) = V div appr (x, E),
(IV.65)

A ±,0 appr,j (x, E) = A ± appr,j (x, E), V ±,0 appr (x, E) = V ± appr (x, E),
(IV.66)

where x ∈ R 2 , j = 1, 2, E > 0.
5 Proofs of Theorems IV.1, IV.2, IV.3

Let us use the notations

u(p) = (2π) -2 R 2 e ipx u(x) dx, u err (x, E) = |p|≥2 √ E e -ipx u(p) dp, (IV.67) where p ∈ R 2 , x ∈ R 2 , E > 0. Lemma IV.1. Let u ∈ C N,σ (R 2 ), where N ≥ 3, σ > 2.
Then the following formula holds :

|u err (x, E)| ≤ c 1 (N, σ) u N,σ E -N -2 2 , (IV.68)
where x ∈ R 2 , E ≥ 1/2 and c 1 (N, σ) is defined by (IV.27).

Proof of Lemma IV.1. We have that

∂ n u(p) = (-ip 1 ) n 1 (-ip 2 ) n 2 u(p),
where

∂ n is defined in (2.13), p = (p 1 , p 2 ) ∈ R 2 , n = (n 1 , n 2 ) ∈ (N ∪ {0}) 2 , |n| ≤ N .
Using this equality we obtain that

| u(p)| ≤ 2 N -1 π(σ -2) u N,σ 1 + |p| 2 -N 2 for each p ∈ R 2 , |p| ≥ 1.
Using the latter inequality we obtain (IV.68).

Proof of Theorem IV.1. Since potentials A and V are real-valued it follows from (IV.16) that the following formula holds :

f lin (k, l) -f lin (l, k) = 2(k -l) A(k -l), (IV.69)
where (k, l) ∈ M E . We consider (IV.16), (IV.69) as a system of linear equations for finding A(k -l) and V (k -l). In addition, we use that (k -l)(k +l) = 0 for (k, l) ∈ M E , i.e. that vectors (k -l) and (k + l) are orthogonal. As a result we obtain formulas (IV.22). Formulas (IV.23), (IV.24) can be regarded as definitions of A appr , A err , V appr , V err . Estimates (IV.25), (IV.26) follow from Lemma IV.1.

Lemma IV.2. Suppose that A 1 , A 2 ∈ C N,σ (R 2 )
, where N ≥ 4, σ > 2. Let ϕ div be the solution of (IV.13). Let ∇ϕ div , (∇ϕ div ) err , ∆ϕ div , (∆ϕ div ) err be defined according to (IV.67). Then the following estimates hold :

(∂ j ϕ div ) err (x, E) ≤ √ 2 c 1 (N, σ) A N,σ E -N -2 2 , (∆ϕ div ) err (x, E) ≤ √ 2 c 1 (N, σ) A N,σ E -N -3 2 , (IV.70)
where x ∈ R 2 , j = 1, 2, E ≥ 1/4, A N,σ is defined by (IV.33) and c 1 (N, σ) is defined by (IV.27).

Proof of Lemma IV.2. The solution ϕ div of (IV.13) is given by :

ϕ div (x) = -i R 2 e -ipx p A(p) |p| -2 dp, x ∈ R 2 . (IV.71)
Using (IV.67), (IV.71) we obtain that

∇ϕ div (p) = -p p A(p) |p| -2 , ∆ϕ div (p) = ip A(p), (IV.72)
where p ∈ R 2 \ {0}. Formulas (IV.72) imply the following inequalities :

∂ j ϕ div (p) ≤ √ 2 max k=1,2 | A k (p)|, ∆ϕ div (p) ≤ √ 2 |p| max k=1,2 | A k (p)|, (IV.73)
where p ∈ R 2 \ {0}, j = 1, 2.

We have that

∂ n A j (p) = (-ip 1 ) n 1 (-ip 2 ) n 2 A j (p), where p = (p 1 , p 2 ) ∈ R 2 , n = (n 1 , n 2 ) ∈ (N ∪ {0}) 2 , |n| ≤ N .
Using this equality we obtain that

| A j (p)| ≤ 2 N -1 π(σ -2) A j N,σ (1 + |p| 2 ) -N 2 (IV.74) for p ∈ R 2 , |p| ≥ 1, j = 1, 2.
Formulas (IV.73), (IV.74) imply the estimates

∂ j ϕ div (p) ≤ √ 2 2 N -1 π(σ -2) A N,σ (1 + |p| 2 ) -N 2 , ∆ϕ div (p) ≤ √ 2 2 N -1 π(σ -2) A N,σ (1 + |p| 2 ) -N -1 2 , (IV.75)
where p ∈ R 2 , |p| ≥ 1, j = 1, 2. Using (IV.75) we obtain (IV.70).

Proof of Theorem IV.2. Taking into account invariance of f lin with respect to transformations (IV.12) and using (IV.13), (IV.16) we obtain the following equalities :

(k -l) A div,0 (k -l) = 0,

f lin (k, l) -f lin (-l, -k) = 2(k + l) A div,0 (k -l), f lin (k, l) + f lin (-l, -k) = 2 V div,0 (k -l), (IV.76)
where (k, l) ∈ M E . Using (IV.76) and orthogonality of vectors (k -l) and (k + l) we obtain (IV.28). Formulas (IV.29), (IV.30) can be regarded as definitions of A div,0 appr , A div,0 err , V div,0 appr , V div,0 err . From (IV.13), (IV.29), (IV.30) we derive

A div,0 err,j (x, E) = (A j ) err (x, E) + (∂ j ϕ div ) err (x, E), j = 1, 2, V div,0 err (x, E) = V err (x, E) -i(∆ϕ div ) err (x, E), (IV.77)
where x ∈ R 2 , E > 0, and (A 1 ) err , (A 2 ) err , V err , (∇ϕ div ) err , (∆ϕ div ) err are defined according to (IV.67). From (IV.77) using inequalities (IV.68) for (A 1 ) err , (A 2 ) err , V err and using inequalities (IV.70) we obtain formulas (IV.31), (IV.32).

Lemma IV.3. Suppose that A 1 , A 2 ∈ C N,σ (R 2 )
, where N ≥ 4, σ > 2. Let ϕ -, ϕ + be the solutions of (IV.14), (IV.15), respectively. Let ∇ϕ ± , (∇ϕ ± ) err , ∆ϕ ± , (∆ϕ ± ) err be defined according to (IV.67). Then the following estimates hold :

(∂ j ϕ ± ) err (x, E) ≤ √ 2 c 1 (N, σ) A N,σ E -N -2 2 , (∆ϕ ± ) err (x, E) ≤ √ 2 c 1 (N, σ) A N,σ E -N -3 2 , (IV.78)
where x ∈ R 2 , j = 1, 2, E ≥ 1/4, A N,σ is defined by (IV.33) and c 1 (N, σ) is defined by (IV.27).

Proof of Lemma IV.3. The solutions ϕ ± of (IV.14), (IV.15) are given by :

ϕ ± (x) = -i R 2 e -ipx A 1 (p) ± i A 2 (p) p 1 ± ip 2 dp, x ∈ R 2 . (IV.79)
Using (IV.67), (IV.79) we obtain that

∇ϕ ± (p) = - A 1 (p) ± i A 2 (p) p 1 ± ip 2 p, ∆ϕ ± (p) = i A 1 (p) ± i A 2 (p) p 1 ± ip 2 |p| 2 , (IV.80)
where p ∈ R 2 \ {0}. Formulas (IV.80) imply the following inequalities :

∂ j ϕ ± (p) ≤ √ 2 max k=1,2 | A k (p)|, ∆ϕ ± (p) ≤ √ 2 |p| max k=1,2 | A k (p)|, (IV.81) where p ∈ R 2 \ {0}, j = 1, 2.
As in the proof of Lemma IV.2 we have estimates (IV.74). Formulas (IV.74), (IV.81) imply the following estimates :

∂ j ϕ ± (p) ≤ √ 2 2 N -1 π(σ -2) A N,σ (1 + |p| 2 ) -N 2 , ∆ϕ ± (p) ≤ √ 2 2 N -1 π(σ -2) A N,σ (1 + |p| 2 ) -N -1 2 , (IV.82)
where p ∈ R 2 , |p| ≥ 1, j = 1, 2. Using (IV.82) we obtain (IV.78).

Proof of Theorem IV.3. Taking into account invariance of f lin with respect to transformations (IV.12) and using (IV.14), (IV.15), (IV.16) we obtain the following equalities :

A ±,0 2 (k -l) = ±iA ±,0 1 (k -l), f lin (k, l) -f lin (-l, -k) = 2 k 1 + l 1 ± i(k 2 + l 2 ) A ±,0 1 (k -l), (IV.83)
f lin (k, l) + f lin (-l, -k) = 2 k 1 -l 1 ± i(k 2 -l 2 ) A ±,0 1 (k -l) + 2 V ±,0 (k -l),
where (k, l) ∈ M E . Using (IV.83) and orthogonality of vectors (k -l) and (k + l) we obtain (IV.34). Formulas (IV.35), (IV.36) can be regarded as definitions of A ±,0 appr , A ±,0 err , V ±,0 appr , V ±,0 err . From (IV.14), (IV.15), (IV.35), (IV.36) we derive formulas

A ±,0 err,j (x, E) = (A j ) err (x, E) + (∇ϕ ± ) err (x, E), j = 1, 2, V ±,0 err (x, E) = V err (x, E) -i(∆ϕ ± ) err (x, E), (IV.84)
where x ∈ R 2 , E > 0, and (A 1 ) err , (A 2 ) err , V err , (∇ϕ ± ) err , (∆ϕ ± ) err are defined by (IV.67). From (IV.84) using inequalities (IV.68) for (A 1 ) err , (A 2 ) err , V err and using inequalities (IV.78) we obtain formulas (IV.37), (IV.38).

6 Derivation of the reconstruction algorithm of Section 3

Faddeev functions

For equation (IV.1), under assumptions (IV.2), we consider the Faddeev functions ψ, h (see, e.g., [START_REF]Growing solutions of the Schrödinger equation[END_REF][START_REF]Inverse problem of quantum scattering theory. II[END_REF] and Subsection 5.1 of [START_REF] Novikov | The ∂-equation in the multidimensional inverse scattering problem[END_REF]) :

ψ(x, k) = e ikx µ(x, k), µ(x, k) = 1 + R 2 g(x -y, k)× × -2iA(y)∇ y µ(y, k) + 2A(y)k + V (y) µ(y, k) dy, (IV.85) g(x, k) = -(2π) -2 R 2 e iξx ξ 2 + 2kξ dξ, where x ∈ R 2 , k ∈ Σ E \ R 2 ; h(k, l) = (2π) -2 R 2 e i(k-l)y × × -2iA(y)∇ y µ(y, k) + 2A(y)k + V (y) µ(y, k) dy, (IV.86)
where k, l ∈ Σ E \ R 2 , Im k = Im l. Here (IV.85) and its differentiated version, where ∇ is applied to both sides of (IV.85), are considered as a system of linear integral equations for bounded µ and ∇µ, Σ E is defined by (IV.42). We recall that ψ are "growing" solutions of (IV.1), in the sense of [START_REF]Growing solutions of the Schrödinger equation[END_REF], parametrized by k ∈ Σ E \R 2 , and G = e ikx g is the Faddeev's Green function for the operator ∆+k 2 . Equation (IV.85) for µ and formula (IV.86) for h are analogs in the complex domain in k of equation (IV.5) for ψ + and formula (IV.6) for f . Note that

k, l ∈ Σ E \ R 2 , Im k = Im l =⇒ l = k or l = -k.
Therefore, the function h of (IV.86) splits to the functions

a(k) = h(k, k), b(k) = h(k, -k), k ∈ Σ E \ R 2 . (IV.87)
Note also that a and b are invariant with respect to transformations (IV.8) and ψ, µ are transformed as ψ → e -iϕ ψ, µ → e -iϕ µ (IV.88)

with respect to (IV.8).

Analytic properties of ψ

Using notations of Section 3 coefficients A 1 , A 2 , V of equation (IV.1), functions ψ + , f of (IV.5), (IV.6) and functions ψ, µ, b of (IV.85), (IV.87) can be written as

A 1 = A 1 (z), A 2 = A 2 (z), V = V (z), ψ + = ψ + (z, λ, E), f = f (λ, λ , E), λ, λ ∈ T, ψ = ψ(z, λ, E), µ = µ(z, λ, E), b = b(λ, E), λ ∈ C \ (T ∪ 0), (IV.89) where z ∈ C, E > 0.
It is known that the function ψ (or µ) has the following properties at fixed z ∈ C and E > 0 (see page 448 of [START_REF]The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF]) :

∂ ∂λ µ(z, λ, E) = r(λ, z, E)µ z, - 1 λ , E , (IV.90) for λ ∈ C \ (T ∪ 0), where r(λ, z, E) = exp -i √ E 2 λz + z λ + λz + z λ × × π λ sgn λλ -1 b(λ, E); µ(z, λ, E) = µ - 0 (z) + o(1) for λ → ∞, µ(z, λ, E) = µ + 0 (z) + o(1) for λ → 0, (IV.91)
where

∂ z µ + 0 (z) = 1 2i A 1 (z) + iA 2 (z) µ + 0 (z), ∂ z µ - 0 (z) = 1 2i A 1 (z) -iA 2 (z) µ - 0 (z), µ ± 0 (z) → 1 as z → ∞;
(IV.92)

ψ ± (z, λ, E) = ψ + (z, λ, E) + πi T h ± (λ, λ , E)χ ±i λ λ - λ λ × ×ψ + (z, λ , E) |dλ | (IV.93)
for λ ∈ T , where

ψ ± (z, λ, E) = ψ(z, λ(1 ∓ 0), E) = exp i √ E 2 λz + z λ × ×µ ± (z, λ, E), µ ± (z, λ, E) = µ(z, λ(1 ± 0), E), λ ∈ T, (IV.94)
ψ + is the function of (IV.5), (IV.89), h ± are the functions related with the scattering amplitude f by equations (IV.45), χ is defined by (IV.46).

More precisely, equation (IV.90) is fulfied if the system of linear equations for µ and ∇µ related with (IV.85) is uniquely solvable for k = k 1 (λ, E), k 2 (λ, E) for fixed λ ∈ C \ (T ∪ 0), where k 1 , k 2 are given by (IV.43), and relation (IV.93) is fulfiled if the aforementioned system is uniquely solvable for k = k 1 (λ(1 ∓ 0), E), k 2 (λ(1 ∓ 0), E) for fixed λ ∈ T . In particular, all these conditions are fulfiled if coefficients A 1 , A 2 , V of (IV.1) are sufficiently small for fixed E.

Inverse scattering from f and b

Using the definitions of ϕ ± , A ± , V ± of (IV.14), (IV.15), (IV.39), the invariance of f and b with respect to (IV.8) and formulas (IV.9), (IV.88) one can see that for A, V transformed to A ± , V ± (respectively) formulas (IV.90)-(IV.94) are fulfiled with µ ± 0 ≡ 1 (respectively).

(IV.95)

Properties (IV.90)-(IV.95) of ψ, µ yield the following approach to inverse scattering for equation (IV.1) from f and b :

1. find ψ, µ satisfying (IV.90)-(IV.91), (IV.93), (IV.94) with a priori unknown ψ + in (IV. [START_REF] Nicoleau | A stationary approach to inverse scattering for Schrödinger operators with first order perturbation[END_REF], where

µ - 0 ≡ 1, µ(z, •, E) ∈ C(C \ T ), h ± are related with f by (IV.45) ; 2. find A -, V -using that A - 1 (z) -iA - 2 (z) = 0, A - 1 (z) + iA - 2 (z) = 2i∂ z ln µ + 0 (z), (IV.96) V -(z)ψ(z, λ, E) = 4∂ z ∂ z + +2i A - 1 (z) + iA - 2 (z) ∂ z + E ψ(z, λ, E), (IV.97) where z ∈ C, λ ∈ C \ (T ∪ 0).
Or alternatively :

1'. find ψ, µ satisfying (IV.90)-(IV.91), (IV.93), (IV.94) with a priori unknown ψ + in (IV. [START_REF] Nicoleau | A stationary approach to inverse scattering for Schrödinger operators with first order perturbation[END_REF], where

µ + 0 ≡ 1, µ(z, •, E) ∈ C (C ∪ ∞) \ T , h ± are related with f by (IV.45) ; 2'. find A + , V + using that A + 1 (z) -iA + 2 (z) = 2i∂ z ln µ - 0 (z), A + 1 (z) + iA + 2 (z) = 0, (IV.98) V + (z)ψ(z, λ, E) = 4∂ z ∂ z + +2i A + 1 (z) -iA + 2 (z) ∂ z + E ψ(z, λ, E), (IV.99) where z ∈ C, λ ∈ C \ (T ∪ 0).
Note that (IV.96) arises from (IV.92) with µ - 0 ≡ 1, (IV.98) arises from (IV.92) with µ + 0 ≡ 1, (IV.97) and (IV.99) arise from equation (IV.1) for the Faddeev functions ψ of Subsections 6.1, 6.2 in the gauge setting related with A -, V -and A + , V + , respectively. In addition, ψ, µ, µ + 0 of steps 1, 2 and ψ = ψ , µ = µ , µ - 0 of steps 1', 2' are related by the formulas

ψ (z, λ, E) = µ + 0 (z) -1 ψ(z, λ, E), µ (z, λ, E) = µ + 0 (z) -1 µ(z, λ, E), µ - 0 (z) = µ + 0 (z) -1 , (IV.100) where z ∈ C, λ ∈ C \ (T ∪ 0).
As soon as A, V are recovered as A -, V -or A + , V + , then these coefficients can be transformed from A -, V -or from A + , V + to other possible gauge setting via (IV.8) and, in particular, to A div , V div of (IV.39).

Note that different ideas of the aforementioned approach to inverse scattering go back to [START_REF] Manakov | The inverse scattering transform for the time dependent Schrödinger equation and Kadomtsev-Petviashvili equation[END_REF][START_REF] Ablowitz | On the inverse scattering transform for the Kadomtsev-Petviashvili equation[END_REF][START_REF] Fokas | On the inverse scattering of the time-dependent Schrödinger equation and the associated Kadomtsev-Petviashvili equation[END_REF][START_REF] Grinevich | Analogs of multisoliton potentials for the two-dimensional Schrödinger operator[END_REF][START_REF] Grinevich | Inverse scattering problem for the twodimensional Schrödinger operator, the ∂-method and nonlinear equations[END_REF][START_REF]The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF][START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF]. In particular, finding ψ, µ of the aforementioned steps 1 or 1' for the case when b ≡ 0 at fixed E is reduced to solving a non-local Riemann-Hilbert problem for holomorphic functions, see Subsection 6.4. Such nonlocal Riemann-Hilbert problems go back to [START_REF] Manakov | The inverse scattering transform for the time dependent Schrödinger equation and Kadomtsev-Petviashvili equation[END_REF][START_REF] Fokas | On the inverse scattering of the time-dependent Schrödinger equation and the associated Kadomtsev-Petviashvili equation[END_REF] ; see also [START_REF] Zhou | Inverse scattering transform for the time dependent Schrödinger equation with applications to the KPI equation[END_REF][START_REF] Fokas | Kadomtsev-Petviashvili equation revisited and integrability in 4+2 and 3 + 1[END_REF][START_REF] Grinevich | Analogs of multisoliton potentials for the two-dimensional Schrödinger operator[END_REF][START_REF] Grinevich | Inverse scattering problem for the twodimensional Schrödinger operator, the ∂-method and nonlinear equations[END_REF][START_REF]The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF][START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF].

Inverse scattering with b ≡ 0 at fixed E

In the Born approximation at fixed E we have that

f (k, l) ≈ f lin (k, l) = 2k A(k -l) + V (k -l), (k, l) ∈ M E , (IV.101) b(k) ≈ b lin (k) = 2k A(2 Re k) + V (2 Re k), k ∈ Σ E \ R 2 , (IV.102)
where A, V are defined by (IV.17). Here formula (IV.101) is equivalent to the formulas for f of (IV.10), (IV.11) and formula (IV.102) follows from (IV.85), (IV.86), (IV.87) in a similar way that (IV.101) follows from (IV.5), (IV.6). Note also that

k ∈ Σ E \ R 2 =⇒ 2 Re k ∈ R 2 \ B 2 √ E , E > 0, (IV.103)
where B r is defined by (IV. [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF]). Using (IV.18), (IV.101) and (IV.102), (IV.103) one can see that the expression for f lin involves A, V on B 2 √ E , only, and the expression for b lin involves A, V on R 2 \B 2 √ E , only, at fixed E. Further, using also (IV.28)-(IV.30), (IV.34)-(IV.36) one can see that the expressions for A ±,0 appr , V ±,0 appr , A div,0 appr , V div,0 appr involve f lin , only, and are independent of b lin at fixed E.

In a similar way, in Section 3 in order to construct nonlinear analogs of A ±,0 appr , V ±,0 appr , A div,0 appr , V div,0 appr we use inverse scattering of Subsection 6.3 without b or, in other words, with b ≡ 0 at fixed E.

In this case steps 1 and 1' of Subsection 6.3 consist in solving the following nonlocal Riemann-Hilbert problems for holomorphic functions : [START_REF] Nicoleau | A stationary approach to inverse scattering for Schrödinger operators with first order perturbation[END_REF]), (IV.94) with a priori unknown ψ + in (IV. [START_REF] Nicoleau | A stationary approach to inverse scattering for Schrödinger operators with first order perturbation[END_REF], where

1. find ψ = exp (i/2) √ E(λz + z/λ) µ(z, λ, E), z ∈ C, λ ∈ C \ (T ∪ 0), satisfying (IV.
∂ ∂λ µ(z, λ, E) = 0, λ ∈ C \ (T ∪ 0), µ(z, λ, E) → 1, as λ → ∞, µ(z, •, E) ∈ C(C \ T ),
or, alternatively : [START_REF] Nicoleau | A stationary approach to inverse scattering for Schrödinger operators with first order perturbation[END_REF]), (IV.94) with a propri unknown ψ + in (IV. [START_REF] Nicoleau | A stationary approach to inverse scattering for Schrödinger operators with first order perturbation[END_REF], where

1'. find ψ = exp (i/2) √ E(λz + z/λ) µ(z, λ, E), z ∈ C, λ ∈ C \ (T ∪ 0), satisfying (IV.
∂ ∂λ µ(z, λ, E) = 0, λ ∈ C \ (T ∪ 0), µ(z, λ, E) → 1, as λ → 0, µ(z, •, E) ∈ C((C ∪ ∞) \ T ).
We recall that h ± of (IV.45) are related with f by (IV.45). Actually, it is also assumed that µ(z, •, E) admits continuous extension on T from its both sides. Now due to considerations of Section 2 of [START_REF]Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension 2[END_REF] we have that finding µ of step 1 is reduced to : (a) solving the linear integral equation (IV.47) for µ + (z, •, E) on T , where

ψ + = exp (i/2) √ E(λz + z/λ) µ + (z, λ, E), z ∈ C, λ ∈ T , (b) finding µ ± (z,
•, E) on T by formulas (IV.50) (or, in other words, by formulas (IV.93) rewritten in terms of µ ± and µ + ), (c) finding µ(z, •, E) on C \ T by the Cauchy formulas :

µ(z, λ, E) = 1 2πi T µ + (z, ζ, E) ζ -λ dζ, |λ| < 1, µ(z, λ, E) = 1 - 1 2πi T µ -(z, ζ, E) ζ -λ dζ, |λ| > 1, (IV.104) z ∈ C, λ ∈ C \ T .
In addition, due to (IV.100), (IV.104) finding ψ = ψ , µ = µ of step 1' is reduced to finding ψ, µ of step 1 and to the formulas

ψ (z, λ, E) = µ + 0 (z) -1 ψ(z, λ, E), µ (z, λ, E) = µ + 0 (z) -1 µ(z, λ, E), µ + 0 (z) = 1 2πi T µ + (z, ζ, E) ζ dζ, (IV.105) where z ∈ C, λ ∈ C \ (T ∪ 0).
In addition, due to (IV.104) we have that

µ(z, λ, E) = 1 + µ - 1 (z)λ -1 + O(λ -2 ) as λ → ∞, µ(z, λ, E) = µ + 0 (z) + µ + 1 (z)λ + O(λ 2 ) as λ → 0, (IV.106) where z ∈ C, λ ∈ C \ (T ∪ 0), µ + 0 (z) is given in (IV.105), µ + 1 (z) = 1 2πi T µ + (z, ζ, E) ζ 2 dζ, µ - 1 (z) = 1 2πi T µ -(z, ζ, E) dζ.
(IV.107)

Note that the non-local Riemann-Hilbert problems of steps 1 and 1' are better known in the literature (going back to [START_REF] Manakov | The inverse scattering transform for the time dependent Schrödinger equation and Kadomtsev-Petviashvili equation[END_REF]) for the case when relation (IV.93) between ψ + and ψ -on T with a priori unknown ψ + is given in the form

ψ + (λ) = ψ -(λ) + T ρ(λ, λ )ψ -(λ ) |dλ |, λ ∈ T, (IV.108)
see, e.g., [START_REF] Manakov | The inverse scattering transform for the time dependent Schrödinger equation and Kadomtsev-Petviashvili equation[END_REF][START_REF] Grinevich | Analogs of multisoliton potentials for the two-dimensional Schrödinger operator[END_REF][START_REF] Grinevich | Inverse scattering problem for the twodimensional Schrödinger operator, the ∂-method and nonlinear equations[END_REF][START_REF]The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF]. Note also that in our case h ± of (IV.45) are related with ρ of (IV.108) by the following formulas and equations :

h 1 (λ, λ , E) = χ i λ λ - λ λ h + (λ, λ , E)- -χ -i λ λ - λ λ h -(λ, λ , E), h 2 (λ, λ , E) = χ i λ λ - λ λ h -(λ, λ , E)- -χ -i λ λ - λ λ h + (λ, λ , E), ρ(λ, λ , E) + πi T ρ(λ, λ , E)χ -i λ λ - λ λ × × h 1 (λ , λ , E) |dλ | = -πih 1 (λ, λ , E), ρ(λ, λ , E) + πi T χ i λ λ - λ λ × × h 2 (λ , λ , E) |dλ | = -πih 2 (λ, λ , E),
where λ, λ ∈ T , see [START_REF]The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF].

Futher, due to results of [START_REF] Grinevich | Analogs of multisoliton potentials for the two-dimensional Schrödinger operator[END_REF][START_REF]The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF] and of Proposition IV.1 of the present work we have that, at least under assumption (IV.55), the non-local Riemann-Hilbert problems for ψ and for ψ = ψ of steps 1 and 1' are uniquely solvable and

-4∂ z ∂ z + a - z (z)∂ z + V -(z) ψ(z, λ, E) = Eψ(z, λ, E), (IV.109) a - z (z) = 4∂ z ln µ + 0 (z), V -(z) = 2i √ E∂ z µ - 1 (z), (IV.110) -4∂ z ∂ z + a + z (z)∂ z + V + (z) ψ (z, λ, E) = Eψ (z, λ, E), (IV.111) a + z (z) = 4∂ z ln 1 µ + 0 (z) , V + (z) = 2i √ E∂ z µ + 1 (z) µ + 0 (z) , (IV.112)
where z ∈ C, λ ∈ C\(T ∪0), µ + 0 , µ - 1 , µ + 1 are the coefficients of (IV.106). Here (IV.109), (IV.110) correspond to (IV.96), (IV.97) and (IV.111), (IV.112) correspond to (IV.98), (IV.99).

Formulas (IV.51), (IV.52) follow from (IV.109)-(IV.111) and the integral expression for µ + 0 , µ - 1 , µ + 1 of (IV.105), (IV.107). Finally, formulas (IV.53) arise from considerations of the gauge transformations (IV.8) between A -, V -and A + , V + and A div , V div of (IV.39). In particular, in these considerations we use that for equation (IV.1) written as

-4∂ z ∂ z + a z ∂ z + a z ∂ z + V ψ = Eψ, a z = -2i(A 1 + iA 2 ), a z = -2i(A 1 -iA 2 ),
the gauge transformations (IV.8), (IV.9) can be written as

a z → a z -4i∂ z ϕ, a z → a z -4i∂ z ϕ, V → V -4i∂ z ∂ z ϕ + 4∂ z ϕ∂ z ϕ + ia z ∂ z ϕ + ia z ∂ z ϕ, ψ → e -iϕ ψ,
and that the equations for ϕ div , ϕ -, ϕ + of (IV.13)-(IV.15), (IV.39) can be written as

8i∂ z ∂ z ϕ div = ∂ z a z + ∂ z a z , ϕ div (z) → 0 as |z| → ∞, 4i∂ z ϕ -= a z , ϕ -(z) → 0 as |z| → ∞, 4i∂ z ϕ + = a z , ϕ + (z) → 0 as |z| → ∞.
7 Proofs of Propositions IV.2 and IV.3

Proof of Proposition IV.2. The method of successive approximations for solving (IV.45) with repsect to h ± ∈ L 2 (T 2 ) and assumptions (IV.58) imply that

h ± = f + O(ε 2 ), ε → +0, (IV.113)
where O(ε 2 ) is considered in the sense of • L 2 (T 2 ) . Consider the following operators acting in L 2 (T )

(C ± u)(λ) = 1 2πi T u(ζ) ζ -λ(1 ∓ 0) dζ, λ ∈ T, (IV.114)
where u is a test function. We recall that

C ± u L 2 (T ) ≤ u L 2 (T ) . (IV.115)
Using (IV.48), (IV.49), (IV.113), (IV.115) and the equality

exp -i √ E 2 (λ -λ )z + (λ -1 -λ -1 )z = 1, λ, λ ∈ T, z ∈ C, E > 0, (IV.116)
we obtain that

B(λ, λ , z, E) = 1 2 T f (ζ, λ , z, E)χ -i ζ λ - λ ζ dζ ζ -λ(1 -0) - - 1 2 T f (ζ, λ , z, E)χ i ζ λ - λ ζ dζ ζ -λ(1 + 0) + O(ε 2 ), (IV.117)
where λ, λ ∈ T , z ∈ C, f (ζ, λ , z, E) is given by (IV.64), O(ε 2 ) is considered in the sense of • L 2 (T 2 ) and is uniform with respect to z ∈ C. From (IV.47) we derive the following equalities :

∂ z µ + (z, λ, E) + T B(λ, λ , z, E)∂ z µ + (z, λ , E) |dλ | = = - T ∂ z B(λ, λ , z, E)µ + (z, λ , E) |dλ |, ∂ z µ + (z, λ, E) + T B(λ, λ , z, E)∂ z µ + (z, λ , E) |dλ | = = - T ∂ z B(λ, λ , z, E)µ + (z, λ , E) |dλ |, (IV.118) where λ ∈ T , z ∈ C, E > 0.
The method of successive approximations for solving (IV.47) and (IV.118) with respect to µ + ∈ L 2 (T ) and ∂ z µ + , ∂ z µ + ∈ L 2 (T ) and estimates (IV.58), (IV.117) imply that :

µ + (z, λ, E) = 1 + O(ε), ∂ z µ + (z, λ, E) = - T ∂ z B(λ, λ , z, E) |dλ | + O(ε 2 ), ∂ z µ + (z, λ, E) = - T ∂ z B(λ, λ , z, E) |dλ | + O(ε 2 ), (IV.119) where z ∈ C, λ ∈ T , O(ε), O(ε 2 ) are considered in the sense of • L 2 (T )
and are uniform with respect to z ∈ C. Using (IV.50), (IV.113), (IV.119) we obtain that

µ ± (z, λ, E) = 1 + O(ε), ∂ z µ ± (z, λ, E) = - T ∂ z B(λ, λ , z, E) |dλ |+ + πi T ∂ z f (λ, λ , z, E)χ ±i λ λ - λ λ |dλ | + O(ε 2 ), ∂ z µ ± (z, λ, E) = - T ∂ z B(λ, λ , z, E) |dλ |+ + πi T ∂ z f (λ, λ , z, E)χ ±i λ λ - λ λ |dλ | + O(ε 2 ), (IV.120)
where λ ∈ T , f (λ, λ , z, E) is given by (IV.64) and O(ε), O(ε 2 ) are considered in the sense of • L 2 (T ) and are uniform with repsect to z ∈ C.

Using (IV.51), (IV.52), (IV.120) we obtain that :

V - appr (x, E) = - √ E π T 2 ∂ z B(λ, λ , z, E) dλ |dλ |+ + i √ E T 2 ∂ z f (λ, λ , z, E)χ -i λ λ - λ λ dλ |dλ | + O(ε 2 ), V + appr (x, E) = √ E π T 2 ∂ z B(λ, λ , z, E) λ -2 dλ |dλ |- -i √ E T 2 ∂ z f (λ, λ , z, E)χ i λ λ - λ λ λ -2 dλ |dλ | + O(ε 2 ), a - z (z, E) = 2i π T 2 ∂ z B(λ, λ , z, E) λ -1 dλ |dλ |+ (IV.121) + 2 T 2 ∂ z f (λ, λ , z, E)χ i λ λ - λ λ λ -1 dλ |dλ | + O(ε 2 ), a + z (z, E) = - 2i π T 2 ∂ z B(λ, λ , z, E) λ -1 dλ |dλ |- -2 T 2 ∂ z f (λ, λ , z, E)χ i λ λ - λ λ λ -1 dλ |dλ | + O(ε 2 ),
where x ∈ R 2 , z is given by (IV.40), and O(ε 2 ) is considered in the uniform sense with respect to x ∈ R 2 . Note that the following formulas hold for each u ∈ L 2 (T ) :

T (C + u)(λ) dλ = 0, T (C -u)(λ) dλ = - T u(λ) dλ, T (C + u)(λ) dλ λ = T u(λ) dλ λ , T (C -u)(λ) dλ λ = 0, T (C + u)(λ) dλ λ 2 = T u(λ) dλ λ 2 , T (C -u)(λ) dλ λ 2 = 0, (IV.122)
where C ± are defined by (IV.114).

Formulas (IV.117), (IV.121), (IV.122) imply estimates

V - appr (x, E) = i √ E T 2 s(λ, λ )∂ z f (λ, λ , z, E) dλ |dλ | + O(ε 2 ), V + appr (x, E) = i √ E T 2 s(λ, λ )∂ z f (λ, λ , z, E) λ -2 dλ |dλ | + O(ε 2 ), a - z (z, E) = -2 T 2 s(λ, λ )∂ z f (λ, λ , z, E) λ -1 dλ |dλ | + O(ε 2 ), a - z (z, E) = 2 T 2 s(λ, λ )∂ z f (λ, λ , z, E) λ -1 dλ |dλ | + O(ε 2 ), s(λ, λ ) def == sgn 1 i λ λ - λ λ , (IV.123)
where x ∈ R 2 , z is given by (IV.40), and O(ε 2 ) is considered in the uniform sense with respect to x ∈ R 2 .

In addition, due to (IV.64) we have that

∂ z f (λ, λ , z, E) = -i √ E 2 λ -1 -λ -1 f (λ, λ , z, E), ∂ z f (λ, λ , z, E) = -i √ E 2 λ -λ f (λ, λ , z, E), (IV.124)
for each λ, λ ∈ T , z ∈ C, E > 0. Formulas (IV.59), (IV.60), (IV.61), (IV.62), (IV.63) follow immediately from (IV.51), (IV.52), (IV.53), (IV.123), (IV.124).

Proposition IV.2 is proved.

Lemma IV. [START_REF] Agaltsov | Riemann-Hilbert problem approach for two-dimensional flow inverse scattering[END_REF]. Let E > 0 be fixed. Let u(λ, λ , E), (λ, λ ) ∈ T 2 , be a complex valued function such that u ∈ L 1 (T 2 ) and

u(λ, λ , E) = u(-λ , -λ, E), (λ, λ ) ∈ T 2 . (IV.125) Suppose that g(p, E), p = (p 1 , p 2 ) ∈ R 2 , |p| ≤ 2 √ E, is the function defined by the formula g √ E Re(λ -λ ), √ E Im(λ -λ ), E = u(λ, λ , E), (IV.126)
where (λ, λ ) ∈ T 2 . Then

|p|≤2 √ E e -ipx g(p, E) dp = E 2 T 2 u(λ, λ , z, E) 1 2i λ λ - λ λ |dλ| |dλ |, u(λ, λ , z, E) def == u(λ, λ , E)× × exp -i √ E 2 (λ -λ )z + (λ -1 -λ -1 )z , (IV.127)
where x ∈ R 2 , z is given by (IV.40).

Actually, (IV.127) arises from the following change of variables in the integration with respect to p :

p 1 = √ E Re(λ -λ ) = √ E(cos φ -cos φ ), p 2 = √ E Im(λ -λ ) = √ E(sin φ -sin φ ), (IV.128)
where λ = e iφ , λ = e iφ .

Proof of Proposition IV.3. Let λ ∈ T , λ ∈ T be defined by (IV.41). It follows from (IV.43), (IV.44) that the following formulas are valid :

2(k 1 + l 1 ) = √ E λ + λ -1 + λ + λ -1 , 2(k 2 + l 2 ) = -i √ E λ -λ -1 + λ -λ -1 , k 1 ± ik 2 = √ Eλ ±1 , l 1 ± il 2 = √ Eλ ±1 , k 1 + l 2 ± i(k 2 + l 2 ) = √ E λ ±1 + λ ±1 , |k + l| 2 = E|λ + λ | 2 , (IV.129) where (k, l) ∈ M E .
Using Lemma IV.4 and formulas (IV.129) we derive from (IV.28), (IV.29), (IV.30), (IV.34), (IV.35), (IV.36) the following formulas :

A div,0 appr,j (x, E) = E 2 T 2
A div,0 appr,j (λ, λ , z, E)

1 2i λ λ - λ λ |dλ| |dλ |, V div,0 appr (x, E) = E 2 T 2
V div,0 appr (λ, λ , z, E)

1 2i λ λ - λ λ |dλ| |dλ |, A ±,0 appr,1 (x, E) = E 2 T 2
A ±,0 appr,1 (λ, λ , z, E)

1 2i λ λ - λ λ |dλ| |dλ |, V ±,0 appr (x, E) = E 2 T 2
V ±,0 appr (λ, λ , z, E)

1 2i λ λ - λ λ |dλ| |dλ |, (IV.130)
where j = 1, 2, x ∈ R 2 , 2, z is given by (IV.40) and where

A div,0 appr,1 (λ, λ , z, E) = f lin (λ, λ , z, E) -f lin (-λ , -λ, z, E) 4 √ E × × 1 λ -1 + λ -1 + 1 λ + λ , A div,0 appr,2 (λ, λ , z, E) = f lin (λ, λ , z, E) -f lin (-λ , -λ, z, E) 4i √ E × × 1 λ -1 + λ -1 - 1 λ + λ , V div,0 appr (λ, λ , z, E) = f lin (λ, λ , z, E) + f lin (-λ , -λ, z, E) 2 , A ±,0 appr,1 (λ, λ , z, E) = 1 2 √ E f (λ, λ , z, E) -f (-λ , -λ, z, E) λ ±1 + λ ±1 , V ±,0 appr (λ, λ , z, E) = λ ±1 f lin (λ, λ , z, E) + λ ±1 f lin (-λ , -λ, z, E) λ ±1 + λ ±1 , (IV.131)
where λ, λ ∈ T , z ∈ C, f lin (λ, λ , z, E) is defined according to (IV.64). Now using (IV.131) we represent each integral of (IV.130) as a sum of the integral containing f lin (λ, λ , z, E) and the integral containing f lin (-λ , -λ, z, E) within the integrands.

Making the change of variables (λ, λ ) → (-λ , -λ) in each integral containing f lin (-λ , -λ, z, E) and taking into account (IV.34) for A ± 2 we obtain formulas (IV.65), (IV.66).

where

D is an open bounded domain in R d , d ∈ {2, 3}, with connected C ∞ boundary ∂D, (V.3) A ω (x) = ωv(x) c 2 (x) + i∇ ln ρ 1 2 (x), U ω (x) = ω 2 c 2 (x) + 2iω 1+ζ(x) α 0 (x) c(x) . (V.4)
Note that the operator L ω is a special case of the so-called magnetic Schroedinger operator.

The model equation (V.1) was studied in different particular cases in [START_REF] Henkin | A multidimensional inverse problem in quantum and acoustic scattering[END_REF][START_REF] Roussef | Two-dimensional vector flow inversion by diffraction tomography[END_REF][START_REF] Rychagov | Reconstruction of fluid motion in acoustic diffraction tomography[END_REF][START_REF] Rumyantseva | Increased resolution of two-dimensional tomography imaging along the transverse coordinate and separate reconstruction of elastic and viscous scatterer characteristics[END_REF][START_REF] Burov | Simulation of a functional solution to the acoustic tomography problem for data from quasipoint transducers[END_REF][START_REF]A global uniqueness result for acoustic tomography of moving fluid[END_REF][START_REF]Uniqueness and non-uniqueness in acoustic tomography of moving fluid[END_REF].

In the present article we suppose that In what follows we always assume that 0 is not a Dirichlet eigenvalue for operator L ω in D. (V.7)

c ∈ C 2 (D), c > 0 in D, (V.5a) v ∈ C 2 (D, R d ), (V.5b) ρ ∈ C 2,β (D), β ∈ (0, 1], ρ > 0 in D, (V.5c) ζ ∈ C(D), α 0 ∈ C(D), ζ > 0, α 0 is real in D, (V.
Note that the set of ω's for which (V.7) does not hold is locally finite. Besides, (V.7) always holds for ω = 0. For equation (V.1) under the assumption (V.7) we consider the Dirichlet-to-Neumann type operator Λ ω which maps a sufficiently regular function f on ∂D to the function

Λ ω f = ∂ψ ∂ν + i(ν • A ω )ψ ∂D ,
where ψ is the solution of equation (V.1) in D with Dirichlet boundary condition ψ| ∂D = f and ν is the unit exterior normal field to ∂D. Note that it is possible to get rid of the assumption (V.7) by considering a general Robin-to-Robin map instead of the Dirichlet-to-Neumann map, see [START_REF] Isayev | Reconstruction of a potential from the impedance boundary map[END_REF].

We consider the following problem :

Problem V.1. Find c, v, ρ and α in D from Λ ω given for ω in some fixed set Ω and from c, v and ρ.

This problem was studied, in particular, in [START_REF]A global uniqueness result for acoustic tomography of moving fluid[END_REF][START_REF]Uniqueness and non-uniqueness in acoustic tomography of moving fluid[END_REF]. In these works it was shown that (a) if ρ ≡ const, α 0 ≡ 0 and Ω = {ω 1 } then the Problem V.1 is uniquely solvable ;

(b) if α 0 ≡ 0, Ω = {ω 1 , ω 2 }, ω 1 < ω 2 , then the Problem V.1 is uniquely solvable ;

(c) if Ω = {ω 1 , ω 2 , ω 3 }, ω 1 < ω 2 < ω 3 , and ζ = 0 in D then the Problem V.1 is uniquely solvable.

Reconstruction results for Problem V.1 at fixed ω can be summarized as follows :

1. Uniqueness modulo gauge transformations in the most general case follows from the results of [START_REF] Guillarmou | Identification of a connection from Cauchy data on a Riemann surface with boundary[END_REF] in dimension d = 2 and of [START_REF] Krupchyk | Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential[END_REF] in dimension d = 3.

2. An approximate reconstruction algorithm modulo gauge transformations in dimension d = 2 was developed in [START_REF] Agaltsov | Riemann-Hilbert problem approach for two-dimensional flow inverse scattering[END_REF][START_REF] Agaltsov | Finding scattering data for a time-harmonic wave equation with first order perturbation from the Dirichlet-to-Neumann map[END_REF].

3. A possible reconstruction approach modulo gauge transformations in dimension d = 3 was outlined in [START_REF] Henkin | A multidimensional inverse problem in quantum and acoustic scattering[END_REF].

In the present article we assume that the coefficients A ω and U ω of equation (V.1) are already recovered from Λ ω up to a gauge transformation using some appropriate method at fixed ω. The goal of the present work is to show how to get rid of the gauge non-uniqueness and recover c, v, ρ and α using boundary measurements at several frequencies. In this respect, the present article can be considered as a development of the article [START_REF]A global uniqueness result for acoustic tomography of moving fluid[END_REF] where the particular case ρ ≡ const, α 0 ≡ 0, Ω = {ω 1 } was considered, and of the article [START_REF]Uniqueness and non-uniqueness in acoustic tomography of moving fluid[END_REF] where the corresponding uniqueness theorems for equation (V.1) with general operator of the form (V.2) were obtained.

For a vector field V = (V 1 , . . . , V d ) and a function f in D we set by definition

curl V = ∂ 1 V 2 -∂ 2 V 1 , d = 2, ∂ 2 V 3 -∂ 3 V 2 , ∂ 3 V 1 -∂ 1 V 3 , ∂ 1 V 2 -∂ 2 V 1 , d = 3, curl f = ∂ 2 f, -∂ 1 f , d = 2, (V.8)
where

∂ j = ∂/∂ x j .
In the present article it is assumed that the following functions are already recovered from the operator Λ ω at fixed ω :

F = curl v c 2
in D, (V.9)

q ω = f 1 -ω 2 f 2 + iωf 3 -2iω 1+ζ α 0 c in D, (V.10)
where

f 1 = ρ 1 2 ∆ρ -1 2 , f 2 = 1 c 2 + v c 2 • v c 2 , f 3 = ∇ • v c 2 -v•∇ ln ρ c 2 . (V.11)
For the corresponding identifiability results see [START_REF] Guillarmou | Identification of a connection from Cauchy data on a Riemann surface with boundary[END_REF] (for d = 2) and [START_REF] Krupchyk | Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential[END_REF] (for d = 3) ; for an approximate reconstruction algorithm see [START_REF] Agaltsov | Riemann-Hilbert problem approach for two-dimensional flow inverse scattering[END_REF][START_REF] Agaltsov | Finding scattering data for a time-harmonic wave equation with first order perturbation from the Dirichlet-to-Neumann map[END_REF]. Thereby, in the present article we study the following problems.

Problem V.2. Find c, v and ρ in D from q ω given for ω ∈ Ω = {ω 1 , ω 2 }, ω 1 < ω 2 , and from F , c, v, ρ.

Problem V.3. Find c, v, ρ, ζ and α 0 in D from q ω given for ω ∈ Ω = {ω 1 , ω 2 , ω 3 }, ω 1 < ω 2 < ω 3 , and from F , c, v, ρ.

2 Solution of Problem V.2

We are going to derive the explicit formulas for solving the Problem V.2. We consider (V.10) with ω ∈ Ω as a system of linear equations for f 1 , f 2 and f 3 . Solving this system we obtain

f 1 = ω 2 2 Re q ω 2 -ω 2 1 Re q ω 1 ω 2 2 -ω 2 1 , f 2 = Re q ω 1 -Re q ω 2 ω 2 2 -ω 2 1 , f 3 = ω -1 1 Im q ω 1 . (V.12)
Set g = ρ -1 2 . It follows from formula (V.11) that g satisfies the following equation

g(x) = g 0 (x) + D G(x, y)f 1 (y)g(y) dy, x ∈ D, (V.13) g 0 (x) = ∂D ∂G(x, y) ∂ν y ρ-1 2 (y) dy, (V.14)
where G(x, y) is the (non-positive) Dirichlet Green's function for operator ∆ in D and ν y is the unit exterior normal to ∂D at point y. Note that g 0 is just the the harmonic extension of ρ -1 2 to D. The existence of function G follows from assumption (V.3) and from [START_REF] Aubin | Nonlinear analysis on manifolds. Monge-Ampère equations[END_REF]Theorem 4.17,p. 112].

Lemma V.1. Equation (V.13) is uniquely solvable for g ∈ C(D).

Démonstration. Suppose that g 1 , g 2 ∈ C(D) are two solutions of equation (V.13). Then their difference h = g 1 -g 2 satisfies

h(x) = D G(x, y)f 1 (y)h(y) dy, x ∈ D. (V.15)
Using formulas (V.5c) and (V.11) we obtain that f 1 ∈ C 0,β (D). Taking this into account and using formula (V.15) and Lemma [54, Lemma 4.2], we obtain that h ∈ C 2 (D) ∩ C(D) and that

-∆h + f 1 (x)h = 0 in D, (V.16a) h| ∂D = 0. (V.16b)
Using formulas (V.12) and (V.16a) we can rewrite equation (V.16a) as -∆(ρ

1 2 h) + ∇ρ • ∇(ρ 1 2 h) = 0, in D. (V.17)
It follows from [54, Lemma 4.2 and Corollary 8.2] and from formulas (V.16b) and (V.17) that h ≡ 0. Hence, ρ -

a × b = a 1 b 2 -a 2 b 1 , d = 2, a 2 b 3 -a 3 b 2 , a 3 b 1 -a 1 b 3 , a 1 b 2 -a 2 b 1 , d = 3. (V.18)
Recall that the Helmholtz decomposition of the vector field v c 2 is given by the following formula :

v c 2 = ∇Φ -curl V in D, (V.19)
where

Φ(x) = D G 0 (x -y)∇ y • v(y) c 2 (y) dy - ∂D G 0 (x -y) ν y • v(y) c 2 (y) dy, (V.20) V (x) = D G 0 (x -y)F (y) dy - ∂D G 0 (x -y) ν y × v(y) c 2 (y) dy, (V.21) G 0 (x) = - 1 2π ln |x|, d = 2, (V.22) G 0 (x) = 1 4π 1 |x| , d = 3, (V.23)
where ν y is the unit exterior normal to ∂D at point y. Note that the vector field V is known since F , v and c are given. Using formula (V. [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF]) we can recover the function Φ| ∂D modulo an additive constant (which does not matter). Fix a point x 0 ∈ ∂D. Let x : [0, 1] → ∂D be a smooth curve linking x 0 to some given point x ∈ ∂D. Then

Φ(x) -Φ(x 0 ) = 1 0 v c 2 + curl V | x(t) • ẋ(t) dt, ẋ = dx dt . (V.24)
It follows from formulas (V.11) and (V. [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF]) that Φ satisfies the equation

-∆Φ + ∇ ln ρ • ∇Φ = -f 3 + curl V • ∇ ln ρ in D, or -∆η + f 1 (y)η = ρ -1 2 (-f 3 + curl V • ∇ ln ρ) in D, 117
where η = ρ -1 2 (Φ -Φ(x 0 )). The function η can be found from the following integral equation :

η(x) = η 0 (x) + η 1 (x) + D G(x, y)f 1 (y)η(y) dy, x ∈ D, (V.25) η 0 (x) = D G(x, y)ρ -1 2 (y) f 3 (y) -curl V (y) • ∇ ln ρ(y) dy, η 1 (x) = ∂D ∂G(x, y) ∂ν y ρ-1 2 (Φ(y) -Φ(x 0 )) dy,
where G(x, y) is the (non-positive) Dirichlet Green's function for ∆ in D. Note that η 1 is just the harmonic extension of ρ-1 2 (Φ| ∂D -Φ(x 0 )) to D. Also note that equation (V.25) has the same kernel as equation (V.13). Therefore, it is uniquely solvable for η ∈ C(D).

After recovering Φ -Φ(x 0 ) in D we can find v c 2 using formula (V.19). Finally, using formulas (V.11) and (V. [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF]) we obtain

1 c 2 = f 2 -(∇Φ -curl V ) 2 , v = c 2 (∇Φ -curl V ).
(V.26)

The described algorithm for solving Problem V.2 is summarized in the following theorem.

Theorem V.1. Suppose that D satisfies (V.3), Ω = {ω 1 , ω 2 }, ω 1 = ω 2 and (V.7) holds for all ω ∈ Ω, and suppose that c, v, ρ satisfy (V.5a)-(V.5c). Then Problem V.2 can be solved as follows :

1. Define f 1 , f 2 , f 3 and V using formulas (V.12) and (V.21).

2. Find g as the unique solution of class C(D) to equation (V.13). Set ρ = g -2 .

3. Fix x 0 ∈ ∂D and find Φ| ∂D -Φ(x 0 ) using formula (V.24).

4. Find η as the unique solution of class C(D) to equation (V.25). Set Φ-Φ(x 0 ) = ρ 1 2 η. 5. Find c and v using the explicit formulas (V.26).

Remark 3. Suppose that d = 2 and D = x ∈ R 2 | |x| ≤ 1}. Then the function g 0 from formula (V.13) and the function G(x, y) from formulas (V.13) and (V.25) can be found explicitly :

g 0 (x) = S 1 ρ -1 2 (ϑ) 1 -|x| 2 |ϑ -x| 2 dϑ 2π , G(x, y) = 1 2π ln |x||y -x| y|x| 2 -x ,
where S 1 = ∂D. Furthermore, if f 1 C(D) < 4 then equations (V.13) and (V.25) can be solved using the method of successive approximations in C(D).

3 Solution of Problem V.3

Define the sets D 0 and D 1 by the formulas

D 0 = x ∈ D | Im qω 1 (x) ω 1 = Im qω 2 (x) ω 2 , D 1 = D \ D 0 . (V.27)
It follows from formulas (V.5d) and (V.10) that D 0 = {x ∈ D | α 0 (x) = 0} and that the functions f 1 , f 2 and f 3 defined in (V.10) can be found in D 0 using formulas (V.12). Using formula (V.10) for x ∈ D 1 and ω ∈ Ω we obtain that

ω -1 2 Im q ω 2 (x) -ω -1 1 Im q ω 1 (x) ω -1 3 Im q ω 3 (x) -ω -1 1 Im q ω 1 (x) = ω 2 ω 1 ζ(x) -1 ω 3 ω 1 ζ(x) -1 . (V.28)
We consider (V.28) as an equation for finding ζ(x) at fixed x ∈ D 1 . This equation is uniquely solvable for ζ(x) at fixed x ∈ D 1 as the following lemma shows.

Lemma V.2. The right side of equation (V.28) at fixed x ∈ D 1 is a strictly decreasing function of ζ(x) ∈ (0, +∞).

Démonstration. It is sufficient to show that for any p, q such that 1 < p < q and for any t > 1 the equality

p t -1 q t -1 = p -1 q -1 . (V.29)
can not hold. Assuming the equality (V.29), we define

λ 1 = p -1 q -1 , λ 2 = q -p q -1 . (V.30)
Using formulas (V.29) and (V.30) we obtain the formulas

λ 1 + λ 2 = 1, λ 1 > 0, λ 2 > 0, (V.31a) λ 1 q + λ 2 = p, λ 1 q t + λ 2 = p t . (V.31b)
Since the function f (s) = s t is strictly convex, relations (V.31a)-(V.31b) can not hold. Therefore the initial assumption that (V.29) holds must be false.

Next we find functions f 1 , f 2 , f 3 and α 0 /c in the domain D 1 . It follows from formula (V.10) that f 1 and f 2 in D 1 can be found using formulas (V.12). It also follows from (V.10) that f 3 and α 0 /c in D 1 can be found from the following formulas :

f 3 = ω ζ 1 ω 2 Im q ω 2 - ω ζ 2 ω 1 Im q ω 1 ω ζ 1 -ω ζ 2 , α 0 c = 1 2 ω -1 2 Im q ω 2 -ω -1 1 Im q ω 1 ω ζ 1 -ω ζ 2 . (V.32)
Using the values of f 1 , f 2 and f 3 in D we can find c, ρ and v in D using the steps 2-5 mentioned in Theorem V.1. Finally, we find α 0 using α 0 /c and c. The algorithm for solving Problem V.3 is summarized in the following proposition.

Theorem V.2. Suppose that D satisfies (V.3), Ω = {ω 1 , ω 2 , ω 3 }, ω 1 < ω 2 < ω 3 , and (V.7) holds for all ω ∈ Ω. Suppose also that c, v, ρ, ζ and α 0 satisfy (V.5a)-(V.5d).

Then Problem V.3 can be solved as follows :

1. Define D 0 and D 1 using formula (V.27). Find f 1 and f 2 in D using formulas (V.12). Find f 3 using formula (V.12) in D 0 and formula (V.32) in D 1 .

2. Find ζ(x) at fixed x ∈ D 1 as the unique positive solution to equation (V.28).

3. Find α 0 /c in D 1 using formula (V.32).

4. Find g as the unique solution of class C(D) to equation (V.13). Set ρ = g -2 .

5. Fix x 0 ∈ ∂D and find Φ| ∂D -Φ(x 0 ) using formula (V.24).

6. Find η as the unique solution of class C(D) to equation (V.25). Set Φ-Φ(x 0 ) = ρ 1 2 η. 7. Find c and v using the explicit formulas (V.26). Set α 0 to zero in D 0 and find α 0 from α 0 /c and c in D 1 .

Remark 4. Note that the formulas and equations presented in Theorem V.1 (resp. V.2) require the knowledge of the function q ω at two (resp. three) frequencies ω. These formulas are exact but they can be not very stable with respect to the noise in the initial data. However, if q ω is known for a bigger number of frequencies it is possible to increase the stability of reconstruction by replacing formulas (V.12) and (V.32) with their least squares analogues. A numerical study of reconstruction stability will be carried out in a subsequent paper.

Introduction

We consider the time-independent Schrödinger equation

-∆ψ + v(x)ψ = Eψ, x ∈ R d , d ≥ 2, E > 0, (VI.1) where v ∈ L ∞ (R d ), supp v ⊂ D, (VI.2)
where D is some fixed open bounded domain in R d .

In quantum mechanics equation (VI.1) describes an elementary particle interacting with a macroscopic object contained in D at fixed energy E.

For equation (VI.1) we consider the classical scattering solutions

ψ + = ψ + (x, k), x ∈ R d , k ∈ R d , k 2 =
E. These solutions ψ + can be specified by the following asymptotics as |x| → ∞ :

ψ + (x, k) = e ikx + c(d, |k|) e i|k||x| |x| (d-1)/2 f (k, |k| x |x| ) + O(|x| -(d+1)/2 ), x ∈ R d , k ∈ R d , k 2 = E, c(d, |k|) = -πi(-2πi) (d-1)/2 |k| (d-3)/2 , (VI.3)
for some a priori unknown f . The function f arising in (VI.3) is defined on

M E = (k, l) ∈ R d × R d : k 2 = l 2 = E , (VI.4)
and is known as the classical scattering amplitude for equation (VI.1).

In quantum mechanics |f (k, l)| 2 describes the probability density of scattering of particle with initial impulse k into direction l/|l| = k/|k|, and is known as differential scattering cross section for equation (VI.1) ; see, e.g., [START_REF] Faddeev | Quantum scattering theory for several particle systems[END_REF]Chapter 1,Section 6].

The problem of finding ψ + and f from v is known as the direct scattering problem for equation (VI.1). For solving this problem, one can use, in particular, the Lippmann-Schwinger integral equation for ψ + and an explicit integral formula for f , see, e.g., [START_REF] Berezin | The Schrödinger equation, ser. Mathematics and its applications[END_REF][START_REF]Inverse problem of quantum scattering theory. II[END_REF][START_REF]Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions[END_REF].

In turn, the problem of finding v from f is known as the inverse scattering problem (with phase information) and the problem of finding v from |f | 2 is known as the phaseless inverse scattering problem for equation (VI.1).

There is a vast literature on the former inverse scattering problem with phase information ; see [START_REF] Alexeenko | Solution of the threedimensional acoustical inverse scattering problem. The modified Novikov algorithm[END_REF][START_REF] Barceló | Numerical approximation of the potential in the two-dimesional inverse scattering problem[END_REF][START_REF] Burov | Multifrequency generalization of the Novikov algorithm for the two-dimensional inverse scattering problem[END_REF][START_REF] Chadan | Inverse problems in quantum scattering theory[END_REF][START_REF]Lectures on linear partial differential equations[END_REF][START_REF] Faddeev | Uniqueness of the solution of the inverse scattering problem[END_REF][START_REF] Grinevich | The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed nonzero energy[END_REF][START_REF]New stability estimates for the inverse acoustic inhomogeneous medium problem and applications[END_REF][START_REF] Isaev | Exponential instability in the inverse scattering problem on the energy interval[END_REF][START_REF] Isaev | New global stability estimates for monochromatic inverse acoustic scattering[END_REF][START_REF]Multidimensional inverse spectral problem for the equation -∆ψ+(v(x)-Eu(x))ψ = 0[END_REF][START_REF]Rapidly converging approximation in inverse quantum scattering in dimension 2[END_REF][START_REF]The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF][START_REF]Approximate Lipschitz stability for non-overdetermined inverse scattering at fixed energy[END_REF][START_REF]An iterative approach to non-overdetermined inverse scattering at fixed energy[END_REF] and references therein. In particular, it is well known that the scattering amplitude f uniquely determines v via the Born approximation formulas at high energies :

v(k -l) = f (k, l) + O(E -1 2 ), E → +∞, (k, l) ∈ M E , (VI.5) v(p) = (2π) -d R d e ipx v(x) dx, p ∈ R d , (VI.6)
and the inverse Fourier transform ; see, e.g., [START_REF] Faddeev | Uniqueness of the solution of the inverse scattering problem[END_REF][START_REF]An iterative approach to non-overdetermined inverse scattering at fixed energy[END_REF].

On the other hand, the literature for the phaseless case is much more limited ; see [START_REF] Chadan | Inverse problems in quantum scattering theory[END_REF][START_REF]Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions[END_REF] and references therein for the case of the aforementioned phaseless inverse problem and see [START_REF] Klibanov | Phaseless inverse scattering problems in three dimensions[END_REF][START_REF] Klibanov | Two reconstruction procedures for a 3D phaseless inverse scattering problem for the generalized Helmholtz equation[END_REF][START_REF]Inverse scattering without phase information[END_REF][START_REF]Formulas for phase recovering from phaseless scattering data at fixed frequency[END_REF][START_REF]Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions[END_REF] and references therein for the case of some similar inverse problems without phase information. In addition, it is well known that the phaseless scattering data |f | 2 does not determine v uniquely, even if |f | 2 is given completely for all positive energies. In particular, it is known that

f y (k, l) = e i(k-l)y f (k, l), |f y (k, l)| 2 = |f (k, l)| 2 , k, l ∈ R d , k 2 = l 2 > 0, (VI.7)
where f is the scattering amplitude for v and f y is the scattering amplitude for v y = v(• -y), where y ∈ R d ; see [START_REF]Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions[END_REF] and references therein.

In the present work, in view of the aforementioned non-uniqueness for the problem of finding v from |f | 2 , we consider the modified phaseless inverse scattering problem formulated below as Problem VI.1. Let

S = {|f | 2 , |f 1 | 2 , . . . , |f m | 2 }, (VI.8)
where f is the scattering amplitude for v and f 1 , . . . , f m are the scattering amplitudes for v 1 , . . ., v m , where v j = v + w j , j = 1, . . . , m, (VI.9)

where w 1 , . . ., w m are additional a priori known background scatterers such that Studies of Problem VI.1 in dimension d ≥ 2 were started in [START_REF]Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions[END_REF]. In dimension d = 1 for m = 1 studies of Problem VI.1 were started earlier in [START_REF] Aktosun | Inverse problems on the line without phase information[END_REF], where phaseless scattering data was considered for all E > 0.

w j ∈ L ∞ (R d ), supp w j ⊂ Ω j , Ω j is an open bounded domain in R d , Ω j ∩ D = ∅, w j = 0, w j 1 = w j 2 if j 1 = j 2 (in L ∞ (R d )), (
Actually, the key result of [START_REF]Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions[END_REF] consists in a proper extension of formula (VI.5) for the Fourier transform v of v to the phaseless case of Problem VI.1 ; see Section 5.

In the present work we proceed from the aforementioned result of [START_REF]Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions[END_REF] and study related approximate reconstruction of v in the configuration space. In this connection our results consist in obtaining related error estimates in the configuration space at high energies E ; see Section 4.

In addition, results of the present work are necessary for extending the iterative algorithm of [START_REF]An iterative approach to non-overdetermined inverse scattering at fixed energy[END_REF] to the phaseless case of Problem VI.1. The latter extension will be given in a subsequent work.

Extension of formula (VI.5) to the phaseless case

Actually, the key result of [START_REF]Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions[END_REF] consists in the following formulas for solving Problem VI.1 in dimension d ≥ 2 for m = 2 at high energies E :

123 Re v Im v = 1 2 Re w 1 Im w 1 Re w 2 Im w 2 -1 | v 1 | 2 -| v| 2 -| w 1 | 2 | v 2 | 2 -| v| 2 -| w 2 | 2 , (VI.11) | v j (p)| 2 = |f j (k, l)| 2 + O(E -1 2 ), E → +∞, p ∈ R d , (k, l) ∈ M E , k -l = p, j = 0, 1, 2,
(VI.12)

where :

-v 0 = v, v j is defined by (VI.9), j = 1, 2, and The point is that using formulas (VI.12) for d ≥ 2 with where

f 0 = f , f 1 , f 2 are the scattering amplitudes for v 0 , v 1 , v 2 , respectively ; -v = v(p), v j = v j (p), w j = w j (p), p ∈ R d ,
k = k E (p) = p 2 + E -p 2 4 1/2 γ(p), l = l E (p) = -p 2 + E -p 2
p ∈ R d , |p| ≤ 2 √ E, one can reconstruct | v| 2 , | v 1 | 2 , | v 2 |
2 from S at high energies for any p ∈ R d . And then using formula (VI.11) one can reconstruct v completely, provided that condition (VI.13) is fulfilled for almost all p ∈ R d . Remark 5. Formulas (VI.12) can be precised as formula (2.15) of [START_REF]Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions[END_REF] :

| v j (p)| 2 -|f j (k, l)| 2 ≤ c(D j )N 3 j E -1 2 , p = k -l, (k, l) ∈ M E , E 1 2 ≥ ρ(D j , N j ), j = 0, 1 , 2, (VI.15)
where v j L ∞ (D j ) ≤ N j , j = 0, 1, 2, and D 0 = D, D j = D ∪ Ω j , j = 1, 2, and formulas for the constants c, ρ are given in [START_REF]Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions[END_REF].

In addition, from the experimental point of view it seems to be, in particular, convenient to consider Problem VI.1 with m = 2 for the case when w 2 is just a translation of w 1 :

w 2 (x) = w 1 (x -y), x ∈ R d , y ∈ R d . (VI.16)
In this case

w 2 (p) = e ipy w 1 (p), ζ w 1 , w 2 (p) = sin(py)| w 1 (p)| 2 , p ∈ R d . (VI.17)
On the level of analysis, the principal complification of (VI.11), (VI.12) in comparison with (VI.5) consists in possible zeros of the determinant ζ w 1 , w 2 of (VI.13). For some simplest cases, we study these zeros in the next section.

Zeros of the determinant

Let Z w 1 , w 2 = p ∈ R d : ζ w 1 , w 2 (p) = 0 , Z w j = p ∈ R d : w j (p) = 0 , j = 1, 2, (VI.18)
where ζ is defined by (VI.13). From (VI.13), (VI.18) it follows that

Z w 1 ∪ Z w 2 ⊆ Z w 1 , w 2 . (VI.19)
In view of (VI. [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF], in order to construct examples of w 1 , w 2 such that the set Z w 1 , w 2 is as simple as possible, we use the following lemma :

Lemma VI.1. Let w(x) = |x| ν K ν (|x|) R d q(x -y)q(y) dy, x ∈ R d , ν > 0, (VI.20) K ν (s) = Γ( 1 2 + ν) √ π 2 s ν ∞ 0 cos(st) dt (1 + t 2 ) 1 2 +ν , s > 0, (VI.21) q ∈ L ∞ (R d ), q = q, q = 0 in L ∞ (R d ), q(x) = 0 if |x| > r, q(x) = q(-x), x ∈ R d . (VI.22) Then w ∈ C(R d ), w = w, w(x) = 0 if |x| > 2r, x ∈ R d , w(p) = w(p) ≥ c 1 (1 + |p|) -β , p ∈ R d , (VI.23)
for β = d + 2ν and some positive constant c 1 = c 1 (q, ν), where w is the Fourier transform of w. In addition, if q ≥ 0, then w ≥ 0.

We recall that K ν defined by (VI.21) is the modified Bessel function of the second kind and order ν. In addition, Γ denotes the gamma function.

Lemma VI.1 is proved in Subsection 5.4. As a corollary of Lemma VI.1, functions

w j (x) = w(x -T j ), x ∈ R d , T j ∈ R d , (VI.24)
where w is constructed in Lemma VI.1, give us examples of w j satisfying (VI.10) for fixed D, Ω j and for appropriate radius r of Lemma VI.1 and translations T j of (VI.24), and such that

Z w j = ∅, | w j (p)| = w(p) ≥ c 1 (1 + |p|) -β , p ∈ R d , (VI.25)
where c 1 , β are the same as in (VI.23). In addition,

ζ w 1 , w 2 (p) = sin(py)| w(p)| 2 , y = T 2 -T 1 = 0, p ∈ R d , Z w 1 , w 2 = p ∈ R d : sin(py) = 0 = p ∈ R d : py ∈ πZ , (VI.26)
for w 1 , w 2 of (VI.24). As another corollary of Lemma VI.1, we have that

ζ w 1 , w 2 (p) = | w(p)| 2 ≥ c 2 1 (1 + |p|) -2β , p ∈ R d , Z w 1 , w 2 = ∅, if w 1 is
defined as in (VI.24) and w 2 = iw 1 .

(VI.27) Note that complex-valued v and w j naturally arise if we interpret equation (VI.1) as a time-harmonic Helmholtz equation of acoustics or electrodynamics.

Finally, note that

Z w 1 ,..., w d+1 = π s Z d , where Z w 1 ,..., w d+1 = Z w 1 , w 2 ∩ Z w 1 , w 3 ∩ • • • ∩ Z w 1 ,

Error estimates in the configuration space

We recall that for inverse scattering with phase information the scattering amplitude f on M E processed by (VI.5) and the inverse Fourier transform yield the approximate reconstruction

u(•, E) = v + O(E -α ) in L ∞ (R d ) as E → +∞, α = n -d 2n , (VI.30) if v ∈ W n,1 (R d )
, n > d (in addition to the initial assumption (VI.2)), where W n,1 (R d ) denotes the standard Sobolev space of n-times differentiable functions in L 1 (R d ) :

W n,1 (R d ) = u ∈ L 1 (R d ) : u n,1 < ∞ , u n,1 = max |J|≤n ∂ |J| u ∂x J L 1 (R d ) , n ∈ N ∪ {0}. (VI.31)
More precisely, the approximation u(•, E) in (VI.30) is defined by

u(x, E) = B r(E) e -ipx f (k E (p), l E (p)) dp, x ∈ R d , r(E) = 2τ E α n-d
for some fixed τ ∈ (0, 1],

(VI.32)

where

B r = p ∈ R d : |p| ≤ r , (VI.33)
α is defined in (VI.30), and k E (p), l E (p) are defined as in (VI.14) with some piecewise continuous vector-function γ on R d ; see, e.g., [START_REF]An iterative approach to non-overdetermined inverse scattering at fixed energy[END_REF]. Analogs of u(•, E) for the phaseless case are given below in this section. In particular, related formulas depend on the zeros of determinant ζ w 1 , w 2 of (VI.13).

We consider

U w 1 , w 2 = Re U w 1 , w 2 + i Im U w 1 , w 2 , Re U w 1 , w 2 Im U w 1 , w 2 = 1 2 M -1 w 1 , w 2 (p)b w 1 , w 2 (p, E), (VI.34) M w 1 , w 2 (p) = Re w 1 (p) Im w 1 (p) Re w 2 (p) Im w 2 (p) , (VI.35) M -1 w 1 , w 2 (p) = 1 ζ w 1 , w 2 (p) Im w 2 (p) -Im w 1 (p) -Re w 2 (p) Re w 1 (p) , (VI.36) b w 1 , w 2 (p, E) = |f 1 (p, E)| 2 -|f (p, E)| 2 -| w 1 (p)| 2 |f 2 (p, E)| 2 -|f (p, E)| 2 -| w 2 (p)| 2 , (VI.37) f (p, E) = f (k E (p), l E (p)), f j (p, E) = f j (k E (p), l E (p)), j = 1,
(x, E) = B r 1 (E) e -ipx U w 1 , w 2 (p, E) dp, x ∈ R d , r 1 (E) = 2τ E α 1 n-d , α 1 = n -d 2(n + β)
, for some fixed τ ∈ (0, 1],

(VI.39)

where U w 1 , w 2 is defined by (VI.34), B r is defined by (VI.33), β is the number of (VI.23), (VI.27). Then

u(•, E) = v + O(E -α 1 ) in L ∞ (R d ), E → +∞. (VI.40)
Theorem VI.1 is proved in Subsection 5.1. Next, we set (VI.42)

Z ε w 1 , w 2 = p ∈ R d : py ∈ (-ε, ε) + πZ , y ∈ R d \ 0, 0 < ε <
In addition to U w 1 , w 2 of (VI.34), we define

U ε w 1 , w 2 (p, E) = 1 2 U w 1 , w 2 (p ε -, E) + U w 1 , w 2 (p ε + , E) , p ε ± = p ⊥ + πz(p) y |y| 2 ± ε y |y| 2 , p ⊥ = p -(py) y |y| 2 , p ∈ B 2 √ E ∩ Z ε w 1 , w
(x, E) = u 1 (x, E) + u 2 (x, E), x ∈ R d , u 1 (x, E) = B r 2 (E) \Z ε 2 (E) w 1 , w 2 e -ipx U w 1 , w 2 (p, E) dp, u 2 (x, E) = B r 2 (E) ∩Z ε 2 (E) w 1 , w 2 e -ipx U ε w 1 , w 2 (p, E) dp, r 2 (E) = 2τ E α 2 n-d , ε 2 (E) = E -α 2 2 , α 2 = n-d 2 n+β+ n-d 2
, for some fixed τ ∈ (0, 1],

(VI.44)

where U w 1 , w 2 and U ε w 1 , w 2 are defined by (VI.34), (VI.43), B r and Z ε w 1 , w 2 are defined by (VI.33), (VI.41), and β is the number of (VI.23). Then

u(•, E) = v + O(E -α 2 ) in L ∞ (R d ), E → +∞. (VI.45) Theorem VI.2 is proved in Subsection 5.2. Next, we set Z ε w 1 ,..., w d+1 = B ε/s + π s Z d , 0 < ε < 1, B r = B r \ ∂B r , r > 0, (VI.46)
where w 1 , . . ., w d+1 are the same as in (VI.28), (VI.29), and B r is defined by (VI.33). One can see that Z ε w 1 ,..., w d+1 is the open ε s -neighborhood of Z w 1 ,..., w d+1 defined in (VI.28).

Note that for any p ∈ Z ε w 1 ,..., w d+1 there exists the unique z(p) ∈ Z d such that |sp -πz(p)| < ε.

(VI.47)

In addition, we consider i such that

i = i (p, s), p = (p 1 , . . . , p d ) ∈ R d \ π s Z d , s > 0, i take values in {2, . . . , d + 1}, | sin(sp i -1 )| ≥ | sin(sp i-1 )| for all i ∈ {2, . . . , d + 1}. (VI.48) Let U w 1 ,..., w d+1 (p, E) = U w 1 , w i (p, E), p ∈ R d \ π s Z d , (VI.49) U ε w 1 ,..., w d+1 (p, E) = 1 |S d-1 | S d-1 U w 1 ,..., w d+1 ( ε s ϑ + π s z(p), E) dϑ, p ∈ Z ε w 1 ,.
(x, E) = u 1 (x, E) + u 2 (x, E), x ∈ R d , u 1 (x, E) = B r 3 (E) \Z ε 3 (E) w 1 ,..., w d+1 e -ipx U w 1 ,..., w d+1 (p, E) dp, u 2 (x, E) = B r 3 (E) ∩Z ε 3 (E) w 1 ,..., w d+1 e -ipx U ε w 1 ,..., w d+1 (p, E) dp, r 3 (E) = 2τ E α 3 n-d , ε 2 (E) = E -α 3 d+1 , α 3 = n-d 2 n+β+ n-d d+1
, for some fixed τ ∈ (0, 1],

(VI.52)

where U w 1 ,..., w d+1 and U ε w 1 ,..., w d+1 are defined by (VI.49), (VI.50), B r and Z ε w 1 ,..., w d+1 are defined by (VI.33), (VI.46), and β is the number of (VI.23). Then

u(•, E) = v + O(E -α 3 ) in L ∞ (R d ), E → +∞. (VI.53)
Theorem VI.3 is proved in Subsection 5.3.

5 Proofs of the main results Then :

v(p) -U w 1 , w 2 (p, E) ≤ c 2 | w(p)| -1 E -1 2 for p ∈ B 2 √ E , E ≥ ρ 1 , c 2 = 2c(D 0 )N 3 0 + 2c(D 1 )N 3 1 , ρ 1 = max j=0,1 ρ(D j , N j ), (VI.54)
where w is the function of (VI.23), (VI.24), and c, ρ, N j , D j , j = 0, 1, 2, are the same as in estimates (VI.15

Ω 2 = Ω 1 , D 2 = D 1 , N 2 = N 1 .
(VI.55)

In turn, properties (VI.55) follow from (VI.9) for j = 1, 2, (VI.10) for j = 1, and from the equality w 2 = iw 1 assumed in (VI.27).

Next, we represent v as follows :

v(x) = v + (x, r) + v -(x, r), x ∈ R d , r > 0, v + (x, r) = Br e -ipx v(p) dp, v -(x, r) = R d \Br e -ipx v(p) dp. (VI.56) Since v ∈ W n,1 (R d ), n > d, we have |v -(x, r)| ≤ c 3 v n,1 r d-n , x ∈ R d , r > 0, c 3 = |S d-1 | (2π) -d d n n-d , (VI.57)
where • n,1 is defined in (VI.31), and |S d-1 | is the standard Euclidean volume of S d-1 . Indeed,

|p k 1 1 • • • p k d d v(p)| ≤ (2π) -d v n,1 , p = (p 1 , . . . , p d ) ∈ R d , for any k 1 , . . .k d ∈ N ∪ {0}, k 1 + • • • + k d ≤ n, (VI.58)
assuming also that p 0 j = 1. Taking an appropriate sum in (VI.58) over all such k 1 , . . .,

k d with k 1 + • • • + k d = m ≤ n, we get |p| m | v(p)| ≤ (|p 1 | + • • • + |p d |) m | v(p)| ≤ (2π) -d d m v n,1 , p ∈ R d . (VI.59)
The definition of v -of (VI.56) and inequalities (VI.59) for m = n imply (VI.57). In addition, using Proposition VI.1 and the estimate on w of (VI.23), we obtain :

v + (x, r) - Br e -ipx U w 1 , w 2 (p, E) dp ≤ c 2 E -1 2 Br | w(p)| -1 dp ≤ c -1 1 c 2 E -1 2 Br (1 + |p|) β dp ≤ c 4 E -1 2 r d+β , c 4 = |S d-1 | 2 d+β d+β c -1 1 c 2 , x ∈ R d , 1 ≤ r ≤ 2 √ E, E ≥ ρ 1 .
(VI.60)

As a corollary of (VI.56), (VI.57), (VI.60), we have

v(x) - Br e -ipx U w 1 , w 2 (p, E) dp ≤ c 3 v n,1 r d-n + c 4 E -1 2 r d+β , (VI.61) where x ∈ R d , 1 ≤ r ≤ 2 √ E, E ≥ ρ 1 . In addition, if r = r 1 (E), where r 1 (E) is defined in (VI.39), then r d-n = (2τ ) d-n E -α 1 , E -1 2 r d+β = (2τ ) d+β E -α
v(p) -U w 1 , w 2 (p, E) ≤ c 5 ε -1 (1 + |p|) β E -1 2 , p ∈ B 2 √ E \ Z ε w 1 , w 2 , E ≥ ρ 2 , 0 < ε < 1, (VI.63) c 5 = π 2 (2c(D 0 )N 3 0 + c(D 1 )N 3 1 + c(D 2 )N 3 2 )c 1 , ρ 2 = max j=0,1,2 ρ(D j , N j ), in addition, if v ∈ W n,1 (R d ), n ≥ 0, then : | v(p) -U ε w 1 , w 2 (p, E)| ≤ 2 β c 5 ε -1 (1 + |p|) β E -1 2 + c 6 ε(1 + π |y| |z(p)| + |p ⊥ |) -n , p ∈ B 2 √ E ∩ Z ε w 1 , w 2 , E ≥ ρ 2 , 0 < ε < min{1, 1 2 |y|};
(VI.64)

c 6 = 2 n (d+1) n+1 (2π) d |y| max j=1,...,d x j v n,1 ,
where c, ρ, N j , D j , j = 0, 1, 2, are the same as in estimates (VI.

(p) = w(p) cos(T 1 p) sin(T 1 p) cos(T 2 p) sin(T 2 p) , p ∈ R d , M -1 w 1 , w 2 (p) = 1 sin(py) w(p) sin(T 2 p) -sin(T 1 p) -cos(T 2 p) cos(T 1 p) , p ∈ R d \ Z ε w 1 , w 2 . (VI.65) Also note that | sin(py)| ≥ 2ε π , p ∈ R d \ Z ε w 1 , w 2 , 0 < ε < 1. ( VI
U ε w 1 , w 2 (p, E) -v(p) = ϕ ε 1 (p, E) + ϕ ε 2 (p), p ∈ B 2 √ E ∩ Z ε w 1 , w 2 , ϕ ε 1 (p, E) = 1 2 U ε w 1 , w 2 (p ε -, E) -v(p ε -) + 1 2 U ε w 1 , w 2 (p ε + , E) -v(p ε + ) , p ∈ B 2 √ E ∩ Z ε w 1 , w 2 , ϕ ε 2 (p) = 1 2 v(p ε -) + v(p ε + ) -v(p), p ∈ Z ε w 1 , w 2 .
(VI.67)

Using estimate (VI.63), formula (VI.67) and the definitions of p ε ± in (VI.43), we get

|ϕ ε 1 (p, E)| ≤ 1 2 c 5 ε -1 E -1 2 (1 + |p ε -|) β + (1 + |p ε + |) β ≤ c 5 ε -1 (1 + |p| + 2 ε |y| ) β E -1 2 ≤ 2 β c 5 ε -1 (1 + |p|) β E -1 2 , for ε as in (VI.64), p ∈ B 2 √ E ∩ Z ε w 1 , w 2 .
(VI.68)

Next, using the definition of ϕ ε 2 in (VI.67) and the mean value theorem, we obtain

|ϕ ε 2 (p)| ≤ ε |y| max{| y |y| ∇ v(ξ)| : ξ ∈ [p ε -, p ε + ]}, p ∈ Z ε w 1 , w 2 , (VI.69)
where [p ε -, p ε + ] denotes the segment joining p ε -to p ε + . Here, the mean value theorem was used for v(ξ) on [p ε -, p] and on [p, p

ε + ]. Note also that |∇ v(ξ)| ≤ d max j=1,...,d ∂ v ∂ξ j (ξ) , ξ = (ξ 1 , . . . , ξ d ) ∈ [p ε -, p ε + ]. (VI.70)
In addition, the following estimates hold :

∂ v ∂ξ j (ξ) ≤ (1 + d) n (2π) d (1 + |ξ|) n x j v n,1 , ξ ∈ [p ε -, p ε + ], j = 1, . . . , d. (VI.71)
Indeed, taking the sum in (VI.59) over all m = 0, . . ., n with the binomial coefficients, we get

(1 + |p|) n | v(p)| ≤ (1 + |p 1 | + • • • + |p d |) n | v(p)| ≤ (2π) -d (1 + d) n v n,
(p)| ≤ 2 -n c 6 ε max 1 + |ξ| -n : ξ ∈ [p ε -, p ε + ] , p ∈ Z ε w 1 , w 2 . (VI.73)
Using also that

ξ = τ y |y| + p ⊥ , where |τ -π |y| z(p)| ≤ ε |y| , if ξ ∈ [p ε -, p ε + ], (VI.74)
and that ε < |y|, we obtain

|ϕ ε 2 (p)| ≤ 2 -n c 6 ε 1 + 1 2 ( π |y| |z(p)| -ε |y| + |p ⊥ |) -n ≤ c 6 1 + π |y| |z(p)| + |p ⊥ | -n , p ∈ Z ε w 1 , w 2 .
(VI.75) Estimate (VI.64) follows from (VI.68) and (VI.75). Proposition VI.2 is proved.

The final part of the proof of Theorem VI.2 is as follows. In a similar way with (VI.56), we represent v as follows :

v(x) = v + 1 (x, r) + v + 2 (x, r) + v -(x, r), x ∈ R d , r > 0, v + 1 (x, r) = Br\Z ε w 1 , w 2 e -ipx v(p) dp, v + 2 (x, r) = Br∩Z ε w 1 , w 2 e -ipx v(p) dp, v -(x, r) = R d \Br e -ipx v(p) dp. (VI.76) Since v ∈ W n,1 (R d ), estimate (VI.57) holds.
Using estimates (VI.63), (VI.64), we get :

v + 1 (x, r) - Br\Z ε w 1 , w 2 e -ipx U w 1 , w 2 (p, E) dp +v 2 + (x, r) - Br∩Z ε w 1 , w 2 e -ipx U ε w 1 , w 2 (p, E) dp ≤ I 1 + I 2 ,
(VI.77)

I 1 = 2 β c 5 ε -1 E -1 2

Br

(1 + |p|) β dp, (VI.78)

I 2 = c 6 ε Br∩Z ε w 1 , w 2 1 + π |y| |z(p)| + |p ⊥ | -n dp, (VI.79) x ∈ R d , 1 ≤ r ≤ 2 √ E, E ≥ ρ 2 ,
where ρ 2 is the same as in Proposition VI.2. In addition :

I 1 ≤ c 7 ε -1 E 1 2 r d+β , c 7 = |S d-1 | 2 d+2β d+β c 5 ;
(VI.80)

I 2 = c 6 ε z∈Z τ 2 +p 2 ⊥ ≤r 2 |τ - π |y| z|≤ ε |y| 1 + π |y| |z| + |p ⊥ | -n dτ dp ⊥ , where τ ∈ R, p ⊥ ∈ R d , p ⊥ y = 0, I 2 ≤ 2c 6 ε 2 |y| z∈Z ξ∈R d-1 ,|ξ|≤r 1 + π |y| |z| + |ξ| -n dξ ≤ c 6 c 8 ε 2 , c 8 = 2 |y| |S d-2 | n -d + 1 z∈Z (1 + π |y| |z|) d-n-1 .
(VI.81)

In addition, if r = r 2 (E), ε = ε 2 (E), where r 2 (E), ε 2 (E) are defined in (VI.44), then

r d-n = (2τ ) d-n E -α 2 , ε -1 E -1 2 r d+β = (2τ ) d+β E -α 2 , ε 2 = E -α 2 .
(VI.

| v(p) -U w 1 ,..., w d+1 (p, E)| ≤ c 9 ε -1 (1 + |p|) β E -1 2 , p ∈ B 2 √ E \ Z ε w 1 ,..., w d+1 , E ≥ ρ 3 , 0 < ε < 1, (VI.83) c 9 = π √ d 2 2c(D 0 )N 3 0 + c(D 1 )N 3 1 + max j=2,...,d+1 c(D j )N 3 j c 1 , ρ 3 = max j=0,...,d+1 ρ(D j , N j ), in addition, if v ∈ W n,1 (R d ), n ≥ 0, then : | v(p) -U ε w 1 ,..., w d+1 (p, E)| ≤ 2 β c 9 ε -1 (1 + |p|) β E -1 2 + 2c 6 ε 1 + 2 π s z(p) 2 -n , p ∈ B 2 √ E ∩ Z ε w 1 ,..., w d+1 , E ≥ ρ 3 , 0 < ε < min{1,
M w 1 , w i (p) = w 1 (p) 1 0 cos(sp i -1 ) sin(sp i -1 ) , p ∈ R d \ π s Z d , M -1 w 1 , w i (p) = 1 sin(sp i -1 ) w 1 (p) sin(sp i -1 ) 0 -cos(sp i -1 ) 1 , p ∈ R d \ Z ε w 1 ,.
U ε w 1 ,..., w d+1 (p, E) -v(p) = ϕ ε 1 (p, E) + ϕ ε 2 (p), p ∈ B 2 √ E ∩ Z ε w 1 ,..., w d+1 ϕ ε 1 (p, E) = 1 |S d-1 | S d-1 U w 1 ,..., w d+1 (η, E) -v(η) η= ε s ϑ+ π s z(p) dϑ, ϕ ε 2 (p) = 1 |S d-1 | S d-1 v ε s ϑ + π s z(p) -v(p) dϑ, (VI.87)
where z(p) is defined in (VI.47).

Using formulas (VI.83), (VI.87), we obtain

|ϕ ε 1 (p, E)| ≤ c 9 ε -1 E -1 2 1 |S d-1 | S d-1 1 + ε s ϑ + π s z(p) β dϑ ≤ c 9 ε -1 E -1 2 1 |S d-1 | S d-1 1 + |p| + 2 ε s β dϑ ≤ 2 β c 9 ε -1 (1 + |p|) β E -1 2 ,
for ε as in (VI.84).

(VI.88)

Next, using the definition of ϕ ε 2 in formula (VI.87) and the mean value theorem, we get the following estimate : The final part of the proof of Theorem VI.3 is as follows. In a similar way with (VI.76), we represent v as follows : e -ipx U ε w 1 ,..., w d+1 (p, E) dp

|ϕ ε 2 (p)| ≤ 2 ε s max |∇ v(ξ)| : ξ ∈ R d , |ξ -π s z(p)| ≤ ε s , p ∈ Z ε w 1 ,.
|ϕ ε 2 (p)| ≤ 2 1-n c 6 ε max (1 + |ξ|) -n : |ξ -π s z(p)| ≤ ε s ≤ 2 1-n c 6 ε 1 + π s z(p) 2 -ε s -n ≤ 2c 6 ε 1 + 2 π s z(p) 2 -n , p ∈ Z ε w 1 ,.
v(x) = v + 1 (x, r) + v + 2 (x, r) + v -(x, r), x ∈ R d , r > 0, v + 1 (x, r) = Br\Z ε w 1 ,..., w d+1 e -ipx v(p) dp, v + 2 (x, r) = Br∩Z ε w 1 ,..., w d+1 e -ipx v(p) dp, v -(x, r) = R d \Br e -ipx v
≤ J 1 + J 2 , J 1 = 2 β c 9 ε -1 E -1 2 Br (1 + |p|) β dp, J 2 = 2c 6 ε Br∩Z w 1 ,..., w d+1 1 + 2 π s z(p) 2 -n dp, x ∈ R d , 1 ≤ r ≤ 2 √ E, E ≥ ρ 3 , (VI.93)
where ρ 3 is the same as in Proposition VI.3. In addition,

J 1 ≤ c 10 ε -1 E -1 2 r d+β , c 10 = |S d-1 | 2 d+β d+β c 9 , J 2 ≤ c 11 ε d+1 , c 11 = 1 s d |B 1 | z∈Z d 1 + 2 π s z 2 -n , (VI.94)
where

|B 1 | is the standard Euclidean volume of B 1 . Finally, if r = r 3 (E), ε = ε 3 (E)
, where r 3 (E), ε 3 (E) are defined in (VI.52), then

r d-n = (2τ ) d-n E -α 3 , ε -1 E -1 2 r d+β = (2τ ) d+β E -α 3 , ε d+1 = E -α

Proof of Lemma VI.1

Note that

w(p) = R d | q(ξ)| 2 ω ν (p -ξ) dξ, p ∈ R d , (VI.96) ω ν (x) = |x| ν 2 K ν |x| , x ∈ R d , (VI.97)
where q, ω ν are the Fourier transforms of q, ω ν . The Fourier transform ω ν can be computed explicitely :

ω ν (p) = c 12 (1 + |p| 2 ) d 2 +ν , c 12 = Γ( d 2 + ν)2 ν-1 π d 2
.

(VI.98) Indeed, formula (VI.98) follows from the Fourier inversion theorem and the following computations :

R d e -ipx dp

(1 + |p| 2 ) d 2 +ν = R R d-1
e -i|x|t dt dξ

(1 + t 2 + |ξ| 2 ) d 2 +ν = |S d-2 | R R e -i|x|t r d-2 dt dr (1 + t 2 + r 2 ) d 2 +ν r= √ 1+t 2 τ ====== |S d-2 | R e -i|x|t dt (1 + t 2 ) 1 2 +ν +∞ 0 τ d-2 dτ (1 + τ 2 ) d 2 +ν = c -1 12 |x| ν K ν (|x|), x ∈ R d .
Here, it was used that

+∞ 0 τ d-2 dτ (1 + τ 2 ) d 2 +ν = 1 2 B( d-1 2 , ν + 1 2 ) = 1 2 Γ( d-1 2 )Γ(ν + 1 2 ) Γ( d 2 + ν) , |S d-2 | = 2π d-1 2 Γ( d- 1 2 )
, where B and Γ denote the beta and gamma functions. Using (VI.96), (VI.98), we obtain the estimates Lemma VI.1 is proved.

w(p) ≥ |ξ|≤1 c 12 | q(ξ)| 2 (1 + |p -ξ|) d+2ν dξ ≥ c 1 (q, ν) (1 + |p|) d+2ν , p ∈ R d , c 1 (q, ν) = c 12 2 d+2ν |ξ|≤1 | q(ξ)| 2 dξ.
which are often called the shock-wave equations, the inviscid Burgers equations or the Riemann-Burgers equations. In this interpretation ξ 1 is the time variable and ξ 0 is the space variable.

The following Cauchy-type formula from [START_REF]Chaînes holomorphes de bord donné dans CP n[END_REF] plays the essential role in the reconstruction of X from γ :

G m (ξ) def == 1 2πi γ z m 1 d(ξ 0 + ξ 1 z 1 + z 2 ) ξ 0 + ξ 1 z 1 + z 2 = N + (ξ) j=1 h m j (ξ) + P m (ξ), m ≥ 1, (VII.2)
where N + (ξ) = |X ∩ C 1 ξ | and P m (ξ 0 , ξ 1 ) is a polynomial of degree at most m with respect to ξ 0 at fixed ξ 1 . In addition, P 0 (ξ 0 , ξ 1 ) = -N -, where

N -= |X ∩ CP 1 ∞ |, CP 1 ∞ = {w ∈ CP 2 : w 0 = 0}, and
P 1 (ξ 0 , ξ 1 ) = - N - k=1 a k ξ 0 + b k 1 -a k ξ 1 , if h k (ξ 0 , 0) ∼ a k ξ 0 + b k + O(ξ -1 0 ) at ∞. (VII.3)
In particular, it follows from (VII.2) that :

G 0 (ξ) = 1 2πi γ d(ξ 0 + ξ 1 z 1 + z 2 ) ξ 0 + ξ 1 z 1 + z 2 = N + (ξ) -N -. (VII.4)
Let π 2 : C 2 → C be the projection onto the second coordinate : π 2 (z 1 , z 2 ) = -z 2 . We have that C \ π 2 γ = ∪ L l=0 Ω l , where Ω l≥0 are connected and Ω 0 is unbounded. From the definition of N ± it follows that N + (ξ 0 , 0) = N -, ξ 0 ∈ Ω 0 .

(VII.5)

Assume that complex curve X does not contain compact components without boundary, or equivalently, satisfies the following condition of minimality :

for any complex curve X ⊂ CP 2 such that ∂ X = ∂X = γ and for almost all ξ ∈ C 2 we have | X ∩ C 1 ξ | ≥ |X ∩ C 1 ξ |. ( * )
Condition of minimality ( * ) is a condition of general position and is fulfilled for X if, for example, every irreducible component of X is a transcendental complex curve. Note that from theorems of Chow [START_REF] Chow | On compact complex analytic varieties[END_REF] and Harvey-Shiffman [START_REF] Harwey | A characterization of holomorphic chaines[END_REF] it follows that an arbitrary complex curve X ⊂ CP 2 satisfying ∂ X = ∂X admits the unique representation X = X ∪ V , where X is a curve satisfying ( * ), and V is a compact algebraic curve, possibly, with multiple components.

The main result of [START_REF] Dolbeault | Surfaces de Riemann de bord donné dans CP n[END_REF] gives a solution to the important problem of J. King [START_REF] King | Open problems in geomtric function theory[END_REF], when a real curve γ ⊂ CP 2 is the boundary of a complex curve X ⊂ CP 2 . Let γ ⊂ C 2 .

Then γ = ∂X for some open connected complex curve X in CP 2 if and only if in a neighborhood W ξ * of some point ξ * ∈ C 2 one can find mutually distinct holomorphic functions h 1 , . . ., h N (ξ * ) satisfying shock-wave equations (VII.1) and also the equation

∂ 2 ∂ξ 2 0 G 1 (ξ 0 , ξ 1 ) - p j=1
h j (ξ 0 , ξ 1 ) = 0, ξ = (ξ 0 , ξ 1 ) ∈ W ξ * .

In this work in development of [START_REF] Dolbeault | Surfaces de Riemann de bord donné dans CP n[END_REF][START_REF]Chaînes holomorphes de bord donné dans CP n[END_REF] we propose a numerically realizable algorithm for reconstruction of a complex curve X ⊂ CP 2 from the known boundary and satisfying the condition of minimality. This algorithm permits, in particular, to make applicable the result of [START_REF] Henkin | On the explicit reconstruction of a Riemann surface from its Dirichlet-to-Neumann operator[END_REF] about principal possibility to reconstruct the topology and the conformal structure of a two-dimensional bordered surface X in R 3 with constant scalar conductivity from measurements on ∂X of electric current densities, induced by three potentials in general position.

Our algorithm depends on the number of points at infinity N -of the complex curve X. It was tested on many examples and admits a simple and complete justification for N -= 0, 1, 2. Despite a cumbersome description for N -≥ 3, we show that, in principle, there are no obstacles for the justification and numerical realization for any N -≥ 0. Moreover, in Theorem VII.4 we propose a method for finding the parameter N -in terms of γ. This makes the algorithm much more applicable.

Cauchy-type formulas and Riemann-Burgers equations

We begin by giving a new proof of the Cauchy type formulas (VII.2) from [START_REF]Chaînes holomorphes de bord donné dans CP n[END_REF], which allows to obtain explicit expressions for functions P m .

Theorem VII.1. Let X ⊂ CP 2 \ (0 : 1 : 0) be a complex curve with rectifiable boundary γ ⊂ C 2 and satisfying ( * ). Suppose that for almost all ξ ∈ C 2 all the points of intersection of X with C 1 ξ have multiplicity at most one. Then the following formulas hold for almost all ξ = (ξ 0 , ξ 1 ) ∈ C 2 :

G m (ξ) = N + (ξ)
d i 1 w 1 dw i 1 0 (q s ) • • • d im w 1 dw im 0 (q s ),
where X ∩ CP 1 ∞ = {q 1 , . . . , q N -}. In particular, if N -= 0 then P m = 0, m ≥ 1.

Démonstration. Put g = ξ 0 w 0 + ξ 1 w 1 + w 2 and g = g w 0 = ξ 0 + ξ 1 z 1 + z 2 . Consider differential forms Then G m (ξ) = 1 2πi γ ω m . Let us compute this integral explicitly. Denote by p j , j = 1, . . ., N + (ξ) the points of intersection of X with CP 1 ξ , and by q s , s = 1, . . ., N -the points of intersection of X with infinity CP 1 ∞ . Denote by B ε j the intersection of X with the ball of radius ε in CP 2 centered at p j and by D ε s the intersection of X with the ball of radius ε centered at q s . The restriction of form ω m on X is meromorphic with poles at points p j and q s . Thus the following equality is valid : If N -= 0, then the second group of terms is absent. The integral bB ε j ω m can be calculated as a residue at the first order pole : dw 0 . Consider the expansion of w 1 (w 0 ) into power series in w 0 in the neighborhood of point q s : w 1 (w 0 ) = w 1 (q s ) + dw 1 dw 0 (q s )w 0 + d 2 w 1 dw 2 0 (q s )w 2 0 + • • • .

142

Note further that

w m 1 (w 0 ) = ∞ k=0 i 1 +•••+im=k d i 1 w 1 dw i 1 0 (q s ) • • • d im w 1 dw im 0 (q s )w k 0 .
The coefficient near w m 0 can be presented in the form It is a polynomial of degree at most m with respect to ξ 0 .

d i 1 w 1 dw i 1 0 (q s ) • • • d im w 1 dw im 0 (q s )w k-m 0 d g dw 0 1 g dw 0 = m-1 k=0 bD ε s i 1 +•••+im=k d i 1 w 1 dw i 1 0 (q s ) • • • d im
Remark 6. We excluded the case (0 : 1 : 0) ∈ X because the point (0 : 1 : 0) can lead to a non-polynomial contribution in ξ 0 in functions P m . Consider, for example, the projective curve X : w 2 0 = w 1 w 2 and the meromorphic 1-form Ω ξ 0 on it : Ω ξ 0 = z 1 dz 2 z 2 +ξ 0 , ξ 0 ∈ C \ 0, where z 1 = w 1 w 0 , z 2 = w 2 w 0 . The form Ω ξ 0 is holomorphic on X except the points q 1 = (0 : 0 : 1), q 2 = (0 : 1 : 0) and p ξ 0 = (-ξ 0 : 1 : ξ 2 0 ).

1. If N -= 0, then P m = 0 for all m ≥ 1. Besides, G 1 = 0 in a neighborhood of any ξ * = (ξ * 0 , 0) with ξ * 0 ∈ Ω 0 . 2. If N -= 1, then P 1 (ξ 0 , 0) = c 11 + c 12 ξ 0 , where constants c 11 and c 12 satisfy the following identity in ξ = (ξ 0 , 0) ∈ Ω 0 × 0 : We differentiate the equation (VII.6) with respect to ξ 0 and restrict this equation and its differentiated version to Ω l × 0 : h 1 (ξ 0 , 0) = G 1 (ξ 0 , 0) -c 11 -c 12 ξ 0 , ∂h 1 ∂ξ 1 (ξ 0 , 0) = ∂G 1 ∂ξ 1 (ξ 0 , 0) -Ċ11 (0) -Ċ12 (0)ξ 0 , ∂h 1 ∂ξ 0 (ξ 0 , 0) = ∂G 1 ∂ξ 0 (ξ 0 , 0) -c 12 , (VII.12)

where ξ 0 ∈ Ω 0 . By (VII.1) there is the equality ∂h 1 ∂ξ 1 (ξ 0 , 0) = h 1 (ξ 0 , 0) ∂h 1 ∂ξ 0 (ξ 0 , 0). We substitute h 1 (ξ 0 , 0), ∂h 1 ∂ξ 1 (ξ 0 , 0), ∂h 1 ∂ξ 0 (ξ 0 , 0) in this equation by their expressions (VII.12), and we obtain the equation (VII. [START_REF] Baratchart | Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation[END_REF] This equation is valid, in particular, for ξ 0 ∈ Ω 0 . We divide it by ξ 0 and tend ξ 0 → ∞.

As a result, we obtain the equality Ċ12 (0) = -c 2 12 . Taking into account this equality, we can rewrite equation (VII.13) in the form (VII.14)

Taking into account that ξ 0 ∂G 1 ∂ξ 0 (ξ 0 , 0) → 0 as ξ 0 → ∞ and passing ξ 0 → ∞ in (VII.14), we obtain the equality Ċ11 (0) = -c 11 c 12 . Substituting ths explicit expression for Ċ11 (0) into (VII.14), we get (VII.10).

3. By (VII.1) functions h 1 (ξ) and h 2 (ξ) satisfy the Riemann-Burgers equation in a neighborhood of any ξ * = (ξ * 0 , 0) ∈ Ω 0 × 0, so that the following equalities are valid : We substitute the expressions of (VII.6) for h 2 1 +h 2 2 and h 1 +h 2 into (VII.17), (VII.18), using the notations P 1 (ξ 0 , ξ 1 ) = C 11 (ξ 1 ) + C 12 (ξ 1 )ξ 0 , P 2 (ξ 0 , ξ 1 ) = C 21 (ξ 1 ) + C 22 (ξ 1 )ξ 0 + C 23 (ξ 1 )ξ 2 0 . Then, equation (VII.18) restricted to Ω 0 × 0 takes the form ∂G 2 ∂ξ 0 (ξ 0 , 0) -c 22 -2c 23 ξ 0 = 2 ∂G 1 ∂ξ 1 (ξ 0 , 0) -Ċ11 (0) -Ċ12 (0)ξ 0 .

∂(h 1 h 2 ) ∂ξ 1 =
(VII. [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF] We divide this equation by ξ 0 and tend ξ 0 → ∞. It leads to the equality Ċ12 (0) = c 23 .

Taking this equality into account and passing ξ 0 → ∞ in (VII. [START_REF] Brown | Identifiability at the boundary for first-order terms[END_REF], we obtain the equality Ċ11 (0) = 1 2 c 22 . Next, we substitute the expressions of (VII.6) for h 2 1 + h 2 2 and h 1 + h 2 into (VII.17 Taking into account that ∂G k ∂ξ 1 (ξ 0 , 0) → 0 and ∂G k+1 ∂ξ 0 (ξ 0 , 0) → 0 as ξ 0 → +∞ we obtain the equalities Ċk,m (0) = km k + 1 c k+1,m+1 , k = 1, . . . , µ 0 -1, m = 1, . . . , k + 1.

Due to the Riemann-Burgers equations (VII.1) the following identity in ξ 0 ∈ Ω 0 holds : ∂e µ 0 ∂ξ 1 (ξ 0 , 0) = e µ 0 (ξ 0 , 0) ∂p 1 ∂ξ 0 (ξ 0 , 0), (VII. [START_REF] Burov | Application of the functional-analytical Novikov algorithm for the purposes of ocean tomography[END_REF] where functions e k are given by the following formulas :

ke k (ξ 0 , ξ 1 ) = k-1 i=1
(-1) i+1 e k-i (ξ 0 , ξ 1 )p i (ξ 0 , ξ 1 ) + (-1) k+1 p k (ξ 0 , ξ 1 ), p k (ξ 0 , ξ 1 ) = G k (ξ 0 , ξ 1 ) -C k1 (ξ 1 ) -C k2 (ξ 1 )ξ 0 -• • • -C k,k+1 (ξ 1 )ξ k 0 , (VII.24)

where k = 1, . . ., µ 0 . Equality (VII.23) allows to represent constants { Ċµ 0 ,j (0)} as functions of constants {c ij }. Finally, substituting the obtained expressions for constants { Ċij (0)} via constants {c ij } into equation (VII.23) we obtain the identity in ξ 0 ∈ Ω 0 for computation of constants {c ij }.

For example, in the case N -= 3 the identity (VII.23) in ξ 0 ∈ Ω 0 for finding of constants c ij takes the form Ċ31 (0) + Ċ32 (0)ξ 0 + Ċ33 (0)ξ 2 0 + Ċ34 (0)ξ = ((G 1 -P 1 ) 2 -G 2 -P 2 ) ∂ ∂ξ 0 (G 1 -P 1 ) in W ξ * , P 1 (ξ 0 , ξ 1 ) = -a 1 ξ 0 +b 1 1-a 1 ξ 1 -a 2 ξ 0 +b 2 1-a 2 ξ 0 , P 2 (ξ 0 , ξ 1 ) = - In turn, it implies, according to [START_REF]Chaînes holomorphes de bord donné dans CP n[END_REF], the moment condition

γ z k 1 1 z k 2 2 dz 2 = 0, k 1 , k 2 ≥ 0.
Then, according to [START_REF] Wermer | The hull of a curve in C n[END_REF] and [START_REF] Harwey | Boundaries of complex analytic varieties[END_REF], for an appropriate choice of orientation, γ is the boundary of a complex curve in C 2 . 2. Set h = G 1 -P 1 in a neighborhood W ξ * of ξ * . Then h satisfies Suppose that the discriminant is non-zero at ξ * . Then, without loss of generality, it is non-zero in W ξ * (we can always choose a smaller neighborhood). We denote two different roots of this equation as h 1 = h 1 (ξ), h 2 = h 2 (ξ). Clearly, h 1 and h 2 are holomorphic in W ξ * . Furthermore, by the Viète formulas we have

h 1 + h 2 = e 1 := G 1 -P 1 in W ξ * , h 1 h 2 = e 2 := 1 2 ((G 1 -P 1 ) 2 -G 2 -P 2 ) in W ξ * .
Note that by definition It remains to consider the case when the determinant of equation (VII.28) vanishes in W ξ * . Otherwise, it vanishes on (at most) a dimension one analytic set and in any neighborhood of ξ * there are balls where it does not vanish.

The zero discriminant condition reads (G 1 -P 1 ) 2 = 2(G 2 -P 1 ) in W ξ * .

We set h = 1 2 (G 1 -P 1 ). Then by definition of P 1 and from the discriminant condition we get

∂ ∂ξ 2 0 (G 1 -2 h) = 0 in W ξ * , ∂ h ∂ξ 1 = h ∂ h ∂ξ 0 in W ξ * .
By Theorem VII.2 it implies N -≤ 2. Note also that if N -= 2, then all intersections of X with C 1 ξ , ξ ∈ W ξ * , are double. Finally, note that equation (VII.11) is the restriction of (VII.26) to the set W ξ * ∩ (Ω 0 × 0). Complement 2. The statement of Theorem VII.4 can be generalized to the case N -≥ 3 in the spirit of cases N -≤ 2. Such a generalization will be developed in a separate paper together with a statement of Theorem VII.3 for N -≥ 3, indicated in Complement 1.

We pass to the description of the algorithm of reconstruction of a complex curve X ⊂ CP 2 satisfying the minimality condition ( * ) from the known boundary γ = ∂X ⊂ C 2 .

Let {ξ k 0 } N k=1 , ξ k 0 ∈ C be an arbitrary grid in C, ξ i 0 = ξ j 0 , i = j, and ξ k 0 / ∈ π 2 γ, k = 1, . . ., N . The complex curve X intersects complex line {z 2 = -ξ k 0 } at points (h s (ξ k 0 , 0), -ξ k 0 ), 1 ≤ s ≤ N + (ξ k 0 , 0). We are going to present the formulas for finding these points.

R} in contained in Ω 0 . Without loss of generality, one can suppose that |ξ k 0 | < R for all k = 1, . . ., N .

Consider an auxiliary complex curve X R = {(z 1 , z 2 ) ∈ X | |z 2 | R}. Its boundary γ R consists of two disjoint parts (possibly, multiconnected) : the first part is γ and the second part γ R is obtained by lifting the circle S R = {z ∈ C | |z| = R} to X via the projection π 2 : X → C.

The complex curve X R does not intersect infinity. Moreover, points of the form a, -ξ k 0 , k = 1, . . ., N belong to X if and only if they belong to X R . Therefore, in order to reconstruct X it is sufficient to reconstruct γ R and then to reconstruct X R , using the algorithm for the case when N -= 0. The algorithm can be formulated as follows :

1. New boundary. Choose a sufficiently large R > 0, so that B c R ⊂ Ω 0 and all ξ k 0 belong to B R . In the case of N -= 1, by virtue of formulas (VII.6), we have that h 1 (ξ 0 , 0) = 1 2πi γ z 1 dz 2 z 2 + ξ 0 -P 1 (ξ 0 , 0), |ξ 0 | = R,

where P 1 can be found using Theorem VII.3. This formula allows to recover γ R and, as a corollary, γ R = γ γ R = ∂X R .

In the case of N -= 2 we have two equalities : where P 1 and P 2 can be found using Theorem VII.3. Applying Newton identities and Viète formulas, we find h 1 (ξ 0 , 0) and h 2 (ξ 0 , 0). Thus, we have recovered γ R and, as a corollary, γ R = γ γ R = ∂X R . 2. Reduction. In order to find the complex curve X R with boundary ∂X R = γ R we apply the algorithm of reconstruction for the case of N -= 0.

Visualization

To our knowledge, there are two known algorithms for automatic visualisation of complex curves. The first one was proposed by Trott [START_REF] Trott | Visualization of riemann surfaces of algebraic functions[END_REF] and requires the knowledge of an analytic expression for the curve. The second algorithm was proposed by Nieser-Poelke-Polthier [START_REF] Nieser | Automatic generation of riemann surface meshes[END_REF] and requires the knowledge of the branching points and their indices for some fixed projection to C. Our algorithm requires the knowledge of the unordered set of points of intersections of the curve with complex lines C 1 ξ . Using Newton identities, we determine the values of symmetric functions σ k :

σ 1 = s 1 , σ 2 = 1
2 (s 2 1 -s 2 ), σ 3 = 1 6 (s 3 1 -3s 1 s 2 + 2s 3 ), σ 4 = 1 24 (s 4 1 -6s 2 1 s 2 + 3s 2 2 + 8s 1 s 3 -6s 4 ). Finally, taking into account Viète formulas, we find h 1 (ξ 0 , 0), . . ., h 4 (ξ 0 , 0) as the roots of the following polynomial in t :

t N + (ξ 0 ) -σ 1 (ξ 0 )t N + (ξ 0 )-1 + • • • + (-1) N + (ξ 0 ) σ N + (ξ 0 ,0) (ξ 0 ) = 0.

Reconstructed surface is represented at Fig. VII.2.

2. One point at infinity. Consider the the complex curve X 2 which is the part of the curve X 2 = (w 0 : w 1 : w 2 ) ∈ CP 2 : w 2 1 = w 2 2 + w 2 0 , bounded by the real curve γ 2 = {(z 1 (t), z 2 (t)) ∈ C 2 : t ∈ [0, 2π]}, z 2 (t) = 1.9e it , z 1 (t) = z 2 2 (t) + 1, z 1 (0) > 0, intersecting infinity at (0 : 1 : 1). We suppose that the number of points at infinity is apriori unknown. We are going to reconstruct X 2,R = X 2 ∩ {z 2 < R}, where R = 3, from γ 2 . Next, taking into account the Viète formulas, we find the values h 1 (ξ 0 , 0), h 2 (ξ 0 , 0), |ξ 0 | = R, as the roots of the following polynomial in t :

t 2 -σ 1 (ξ 0 )t + σ 2 (ξ 0 ) = 0.
Then we put γ 3,R = γ 3 γ 3,R and recover X 3,R from γ 3,R using the algorithm for the case of no points at infinity. The result of reconstruction is depicted at Fig. VII.4.

I.

  Agaltsov A. D., A global uniqueness result for acoustic tomography of moving fluid, Bulletin des Sciences Mathématiques 139 (8), 2015, 937-942 II. Agaltsov A. D., Novikov R. G., Uniqueness and non-uniqueness in acoustic tomography of moving fluid, Journal of Inverse and Ill-Posed Problems 24 (3), 2016, 333-340 III. Agaltsov A. D., Finding scattering data for a time-harmonic wave equation with first order perturbation from the Dirichlet-to-Neumann map, Journal of Inverse and Ill-Posed Problems 23 (6), 2015, 627-645 IV. Agaltsov A. D., Novikov R. G., Riemann-Hilbert problem approach for twodimensional flow inverse scattering, Journal of Mathematical Physics 55 (10), 2014, id103502 V. Agaltsov A. D., On the reconstruction of parameters of a moving fluid from the Dirichlet-to-Neumann map, Eurasian Journal of Mathematical and Computer Applications 4 (1), 2016, 4-11

  Calderón) : VII. Agaltsov A. D., Henkin G. M., Explicit Reconstruction of Riemann Surface with Given Boundary in Complex Projective Space, Journal of Geometric Analysis 25 (4), 2015, 2450-2473 1 Problèmes inverses pour l'équation de Helmholtz jauge-covariante 1.1 L'opérateur de Helmholtz jauge-covariant

  Soit D ⊂ R d , d ≥ 2, un domaine borné au bord ∂D assez lisse. Nous considérons le problème de Dirichlet L A,V ψ = Eψ dans D, ψ| ∂D = f,

  ψ + sc (x, k) = c(d, |k|) e i|k||x| |x| (d-1)/2 f A,V k, |k| x |x| + o(|x| -d-1 2 ), |x| → +∞, c(d, |k|) = -πi(-2πi)

  on peut définir la fonction de Green G 0 ω et l'opérateur de Dirichlet-Neumann Λ 0 ω = Λ A,V en accord avec (20) et (21) (avec L 0 ω

Figure 1 -

 1 Figure 1 -Le schéma de résolution du problème de tomographie acoustique

Figure 2 -

 2 Figure 2 -Les champs vectorielsA, A 1 , A 2 , A = A 1 + A 2 de[START_REF] Shurup | Numerical simulation of the functional approach for recovering vector fields in acoustic tomography[END_REF]

Figure 3 -

 3 Figure 3 -Le schéma de démonstration de l'unicité

3 .

 3 Le plongement est une application injective dont la différentielle est aussi injective et qui de plus est un homéomorphisme sur son image 4. Par définition, une composante fermée est une composante connexe compacte et sans bord. 5. Plus géneralement, il existe au plus une chaine holomorphe bordée par un courant rectifiable réel donné. composantes fermées, voir [32, Proposition 1.4.1]. Localement, cela correspond à l'unicité d'une fonction analytique dans un disque avec des valeurs données sur le bord.

Figure 4 -

 4 Figure 4 -La courbe complexe reconstruit à partir de son bord (la courbe réelle rouge). La couleur désigne Im(z 1 ).

1 .

 1 Theorem II.1. Let D be an open bounded simply connected domain in R d , d ≥ 3, with path connected C 1 boundary ∂D. Let L (j) ω and Λ (j)

  ). Theorems II.1, II.2 and II.3, II.4 are proved in Sections 3 and 4, respectively. Let h be a real-valued function supported in D, h ∈ C 2 (D) and |∇h| 2 < 1 in D, (II.13) where D is an open bounded domain in R d .

2

 2

3. 1

 1 Proof of Theorem II.1

  satisfying (III.1), where ν = (ν 1 , . . . , ν d ) is the unit exterior normal to ∂D. The map Φ(E) is known as the Dirichlet-to-Neumann map for equation (III.1) in D.

  and S d-1 is the unit sphere in R d . Note that the history of functions ψ, h and ψ γ , h γ goes back to [Fa1], [Fa2]. Functions f (k, l) and h γ (k, l), where k, l ∈ R d \ 0, k 2 = l 2 = E, γ ∈ S d-1 , and h(k, l), where k, l ∈ C d \ R d , Im k = Im l, k 2 = l 2 = E, are considered as the scattering data S E for equation (III.7) at fixed E

Lemma III. 3 .

 3 Let A 1 , . . ., A d , V ∈ C 0,α c (D, M n (C)) for some 0 < α ≤ 1. Suppose that E is not a Dirichlet eigenvalue for operators L and -∆ in D. Then :1. for any ϕ ∈ C 1,β (∂D, M n (C)), 0 < β < 1, there exists the unique solutionψ ∈ C 2 (D, M n (C)) ∩ C 1 (D, M n (C)) to problem (III.79) ; 2. ψ is the unique solution of class C 1 (D, M n (C)) to equation ψ(x) = ψ 0 (x) + D Γ(x, y, k) -2i d j=1A j (y)∂ y j + V (y) ψ(y) dy, (III.80)where Γ(x, y, k) is the Green function for Dirichlet problem for operator ∆ + E in D, E = k 2 , andψ 0 ∈ C 2 (D, M n (C)) ∩ C 1 (D, M n (C)),is the unique solution to Dirichlet problem ∆ψ 0 + Eψ 0 = 0 in D, ψ 0 | ∂D = ϕ; (III.81) 3. the operator S : C 1,β (∂D, M n (C)) → C 1 (D, M n (C)), S(ϕ) = ψ, is a continuous linear operator.

  0 in D there exists the unique solution ψ ∈ C 1 (D, M n (C)) to equation (III.80) ; 2. ψ belongs to C 2 (D, M n (C)) and satisfies (III.79) with ϕ = ψ 0 | ∂D ;

Γ

  j (x, y, k)a m (y)ψ m (y) dy, j = 0, . . . , d. (III.83) System (III.83) can be considered as a Fredholm equation of second kind in space (C(D, M n (C))) d+1 .Suppose that functions ψ j ∈ C(D, M n (C)), j = 0, . . ., d, solve (III.83). Denote ψ = ψ 0 . The first equation of (III.83) implies that ψ ∈ C 1 (D, M n (C)). Differentiating the first equation with respect to x 1 , . . ., x d , we see that ∂ x j ψ = ψ j . Hence ψ satisfies (III.80).

1 .

 1 Extending a solution to equation (III.41) to a solution to equation (III.67). Let ϕ ∈ C 1,β (D, M n (C)), 0 < β < 1, be a solution to equation (III.41). We will show that ϕ gives rise to a solution ψ(x, k) = e ikx µ(x, k), µ ∈ W 1,∞ (R d , M n (C)) to equation (III.67).Letϕ + ∈ C 2 (D, M n (C)) ∩ C 1 (D, M n (C)) be the unique solution to (III.79) given by Lemma III.3. Define ϕ -by formulaϕ -(x) = ψ 0 (x, k) + ∂D A 0 (x, y, k)ϕ(y) dy, x ∈ R d \ D.(III.88)Using formulas (III.41), (III.42), (III.49), we obtain the following formula :D R 0 (x, y, k) -2i d j=1 A j (y)∂ y j + V (y) -V 0 (y) ϕ + (y) dy = ∂D A(x,y, k)ϕ(y) dy, (III.89) where x ∈ R d \ D. It follows from (III.35), (III.37) that formula (III.89) holds for x ∈ R d \ D. Using formulas (III.88), (III.89), we obtain formulaϕ -(x) = ψ 0 (x, k) + D R 0 (x, y, k)× × -2i d j=1 A j (y)∂ y j + V (y) -V 0 (y) ϕ + (y) dy, (III.90) where x ∈ R d \ D. It follows from (III.35), (III.36), (III.37), (III.90) that ϕ-∈ C 1 (R d \ D, M n (C)).Note that the following formula holds :D R 0 (x, y, k) -2i d j=1A j (y)∂ y j + V (y) -V 0 (y) ϕ + (y) dy = D δ x (y)ϕ + (y) dy + ∂D R 0 (x, y, k) ∂ϕ + (y) ∂ν y -∂R 0 ∂ν y (x, y, k)ϕ(y) dy, (III.91) where x ∈ ∂D. Formula (III.91) follows from the following chain of equalities : D R 0 (x, y, k) -2i d j=1 A j (y)∂ y j + V (y) -V 0 (y) ϕ + (y) dy (III.79) == D R 0 (x, y, k) ∆ + E -V 0 (y) ϕ + (y) dy (III.54)

(III. 96 )

 96 It follows from formulas (III.91), (III.95) together with formula (III.90) that ψ satisfies (III.67) in R d . Using (III.32), (III.35), (III.37), (III.67), we obtain that ψ ∈ C 1 (R d , M n (C)).

  , we have shown that if equation (III.41) has a solution ϕ ∈ C 1,β (∂D, M n (C)), then equation (III.67) has a solution ψ(x, k) = e ikx µ(x, k) with µ ∈ W 1,∞ (R d , M n (C)) and ψ| ∂D = ϕ. It follows from the latter property that different solutions to (III.41) give rise to different solutions to (III.67). More precisely, if ψ , ψ are two solutions to (III.67), then ψ | ∂D = ψ | ∂D = ϕ. It follows that ψ , ψ are two solutions to (III.79) and hence ψ | D = ψ | D . Finally, it follows from (III.67) that ψ = ψ . 2. Restricting a solution to equation (III.67) to a solution to equation (III.41). Let ψ

  supports in D, formula (III.67) together with formulas (III.35), (III.37), (III.36) also imply that ψ ∈ C 2 (R d \ D, M n (C)). Hence ϕ belongs to C 2 (∂D, M n (C)). Now if ψ and ψ are two solutions to (III.67) such that ψ | ∂D = ψ | ∂D , then it is clear that they give rise to different solutions to (III.41). If, otherwise, ψ | ∂D = ψ | ∂D then ψ = ψ as was shown in the end of the preceding part of this subsection.

  5d) where D = D∪∂D and C k,β (D) denotes the space of k times continuously differentiable functions in D whose k-th derivatives are β-Hölder continuous. Let c = c| ∂D , v = v| ∂D , ρ = ρ| ∂D . (V.6)

1 2

 1 is the unique solution of class C(D) to equation (V.13). For vectors a = (a 1 , . . . , a d ) and b = (b 1 , . . . , b d ) in D we put by definition

  VI.10) where j, j 1 , j 2 ∈ {1, . . . , m}. Thus, S consists of the phaseless scattering data |f | 2 , |f 1 | 2 , . . ., |f m | 2 measured sequentially, first, for the unknown scatterer v and then for v in the presence of known scatterer w j disjoint from v for j = 1, . . ., m.Actually, in the present work we continue studies of[START_REF]Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions[END_REF] on the following inverse scattering problem for equation (VI.1) :Problem VI.1. Reconstruct potential v from the phaseless scattering data S for some appropriate background scatterers w 1 , . . . , w m .

  are the Fourier transforms of v, v j , w j (defined as in (VI.6)) ; -formula (VI.11) is considered for all p ∈ R d such that the determinantζ w 1 , w 2 (p) def == Re w 1 (p) Im w 2 (p) -Im w 1 (p) Re w 2 (p) = 0.(VI.13)

4 1 /

 1 2 γ(p), |γ(p)| = 1, γ(p)p = 0, (VI.14)

(VI. 99 )

 99 Properties (VI.23) follow from (VI.20), (VI.22), (VI.96) and (VI.99).

0 s=1 i 1 +

 01 ) + P m (ξ), m ≥ 1, (VII.[START_REF] Aharonov | Significance of electromagnetic potentials in quantum theory[END_REF] where P m (ξ) is holomorphic in a neighborhood of almost all (ξ 0 , 0), ξ 0 ∈ C \ π 2 (γ), and is polynomial in ξ 0 of degree at most m for any fixed ξ 1 . Furthermore, the following explicit formulas hold :P m (ξ 0 , ξ 1 ) = 0 w 0 + ξ 1 w 1 + w 2 )| qs µ •••+im=m

  m = 0, 1, . . .

  N -> 0. Computation of integral bD ε s ω 1 will be done in two steps. Let us calculate first bD

i 1 +•••+im=m d i 1 w 1 dw i 1 0(q s ) • • • d im w 1 dw im 0 (

 110 relation d g = d g dw 0 dw 0 and expansion of w 1 (w 0 ) into power series in w 0 we obtain :

w 1 dw im 0 (From here, taking into account the relation d g dw 0 1 gd i 1 w 1 dw i 1 0

 011 = d ln g dw 0 , we obtain, finally(q s ) • • • d im w 1 dw im 0 (q s ) (m -k -1)! d m-k dw m-k 0 ln(ξ 0 w 0 + ξ 1 w 1 + w 2 )| qs .

c 11 ∂G 1 ∂ξ 0 1 ∂ξ 0 2 0+ G 2 1 -

 11101021 (ξ) + c 12 ξ 0 ∂G (ξ) + G 1 (ξ) = G 1 (ξ) ∂G 1∂ξ 0 (ξ 0 , 0) -∂G 1 ∂ξ 1 (ξ).(VII.10)3. If N -= 2, then P 1 (ξ 0 , 0) = c 11 + c 12 ξ 0 , P 2 (ξ 0 , 0) = c 21 + c 22 ξ 0 + c 23 ξ 2 0, where constants c 11 , c 12 , c 21 , c 22 , c 23 satisfy the following identity in ξ = (ξ 0 , 0) ∈ Ω 0 × 0 :ae 10 (c 2 12 + c 23 ) = ∂G 2 ∂ξ 1 -2 ∂G 1 ∂ξ 1 (G 1 -c 11 -c 12 ξ 0 ) + G 1 (c 22 + 2c 23 ξ 0 ) + ∂G 1 ∂ξ 0 • (G 1 -c 11 -c 12 ξ 0 ) 2 -G 2 + c 21 + c 22 ξ 0 + c 23 ξ 2c 11 G 1 -2c 12 G 1 ξ 0 -G 2 • (-c 12 ), (VII.11)where all the functions are evaluated at point ξ = (ξ 0 , 0). Démonstration. 1. By Theorem VII.1, if N -= 0 then P m = 0 for all m ≥ 1. It also follows from formula (VII.6) that G 1 = 0 in a neighborhood of any ξ * = (ξ * 0 , 0) with ξ * 0 ∈ Ω 0 . 2. It follows from Theorem VII.1 that P 1 (ξ 0 , ξ 1 ) = C 11 (ξ 1 ) + C 12 (ξ 1 )ξ 0 in a neighborhood of almost all ξ * = (ξ * 0 , 0) ∈ Ω l × 0, where C 11 , C 12 are holomorphic in a neighborhood of zero. We need to find constants c 11 = C 11 (0) and c 12 = C 12 (0).

∂G 1 ∂ξ 1 (

 1 ξ 0 , 0) -Ċ11 (0) -Ċ12 (0)ξ 0 = = G 1 (ξ 0 , 0) -c 11 -c 12 ξ 0 ∂G 1 ∂ξ 0 (ξ 0 , 0) -c 12 .

∂G 1 ∂ξ 1 (

 1 ξ 0 , 0) -Ċ11 (0) = = G 1 (ξ 0 , 0) -c 11 -c 12 ξ 0 ∂G 1 ∂ξ 0 (ξ 0 , 0) -G 1 (ξ 0 , 0) -c 11 c 12 .

= G 1 -c 11 -c 12 ξ 0 2 -Complement 1 .

 11121 ) and restrict the obtained formula to Ω 0 × 0. It leads to the equality2 G 1 -c 11 -c 12 ξ 0 G 2 + c 21 + c 22 ξ 0 + c 23 ξ 2 0 ∂G 1 ∂ξ 0 -c 12 .(VII.20) The statement of Theorem VII.3 admits a development for the case N -≥ 3. In this caseP k (ξ 0 , ξ 1 ) = C k1 (ξ 1 ) + C k2 (ξ 1 )ξ 0 + • • • + C k,k+1 (ξ 1 )ξ k 0 , k = 1, . . . , N -. Denote Ċij (0) = ∂C ij∂ξ 1 (0) and c ij = C ij (0) for i = 1, . . ., N -and j = 1, . . ., i + 1. Let us indicate the following general procedure for finding of constants c ij . Due to the Riemann-Burgers equations (VII.1) the following identities in ξ 0 ∈ Ω 0 hold for k = 1, . . . , N --1 :-∂G k ∂ξ 1 (ξ 0 , 0) + Ċk1 (0) + Ċk2 (0)ξ 0 + • • • + Ċk,k+1(0)ξ k , 0) + c k+1,2 + 2c k+1,3 ξ 0 + • • • + (k + 1)c k+1,k+2 ξ k 0 .

∂p 1 ∂ξ 0 ,

 0 The next theorem permits to find N -= |X ∩ CP1 ∞ | from γ = ∂X. Theorem VII.4. Let X ⊂ CP 2 \ (0 : 1 : 0) be a complex curve with rectifiable boundary γ ⊂ C 2 and satisfying ( * ). Let G m≥1 be the functions defined in (VII.2) and letN -= |X ∩ CP 1 ∞ |. Fix any ξ * 0 ∈ Ω 0 and let W ξ * be a neighborhood of ξ * = (ξ * 0 , 0) in C 2 .Then the following statements are valid :1. If G 1 = 0 in W ξ * , then either N -= 0, or γ bounds a complex curve in C 2 , where γ denotes γ with the opposite orientation.2. If there exist complex constants c 11 , c 12 such that∂ ∂ξ 1 G 1 -P 1 = (G 1 -P 1 ) ∂ ∂ξ 0 (G 1 -P 1 ) in W ξ * ,(VII.25)where P 1 (ξ 0 , ξ 1 ) = c 11 ξ 0 +c 12 1+c 11 ξ 1 , then N -≤ 1. Furthermore, c 11 , c 12 are the same constants as in Theorem VII.3.3. If there exist complex constants a1 , a 2 , b 1 , b 2 , c 1 , c 2 such that c 1 + c 2 = ae 10 , ∂ ∂ξ 1 ((G 1 -P 1 ) 2 -G 2 -P 2 )

2 j=1 a j ξ 0 +b j 1 -a j ξ 1 2 + 2a j c j 1 -a j ξ 1 ,(VII. 26 ) 2 1 + a 2 2 = 2 1 + b 2 2 + 2a 1 c 1 + 2a 2 c 2 =Démonstration. 1 .

 21211262222121 then N -≤ 2. Furthermore, these constants are related to the constants of Theorem VII.3 by the equations :a 1 + a 2 = -c 12 , b 1 + b 2 = -c 11 , a -c 23 , a 1 b 1 + a 2 b 2 = -1 2 c 22 , b -c 21 , c 1 + c 2 = ae 10 .(VII.27) The equality G 1 = 0 in W ξ * implies that G 1 = 0 for all ξ ∈ Ω 0 × C.

∂ 2 ∂ξ 2 0(G 1

 21 -h) = 0 in W ξ * , ∂h ∂ξ 1 = h ∂h ∂ξ 0 in W ξ * .It follows from Theorem VII.2 that N -≤ 1. Note also that equation (VII.10) is the restriction of (VII.25) to W ξ * ∩ (Ω 0 × 0).3. Consider the following quadratic equation in variable t :t 2 -(G 1 -P 1 )t + 1 2 ((G 1 -P 1 ) 2 -G 2 -P 2 ) = 0.(VII.28)

h 1 (2πi γ z 1 dz 2 z 2 z 2

 122 ξ 0 , 0) + h 2 (ξ 0 , 0) = 1 + ξ 0 -P 1 (ξ 0 , 0), |ξ 0 | = R, + ξ 0 -P 2 (ξ 0 , 0), |ξ 0 | = R,

Figure VII. 2 -

 2 Figure VII.2 -The given boundary γ 1 = γ r 1 γ g 1 (red and green) and the reconstructed complex curve X 1 ⊂ C 2 . The color indicates Im(z 1 ).

Figure VII. 4 -2πi γ z 1 dz 2 z 2 z 2

 422 Figure VII.4 -The real curve γ 3 (red) and the reconstructed complex curve X 3 with two points at infinity. The color indicates Im(z 1 ).

  1, II.2 of Section 2. (A2) We show that the boundary data Λ ω given for three different frequencies ω = ω 1 , ω 2 , ω 3 uniquely determine c, ρ, v, α under the assumptions that ω 1 , ω 2 , ω 3 ∈ σ(L ω ) and ζ doesn't vanish anywhere, see Theorems II.3, II.4 of Section 2. (B) We give examples of coefficients c

  2, II.3 and we have formulas (II.16), (II.45), (II.18)-(II.21) for d = 2 and formulas (II.37)-(II.43) where in (II.21), (II.40), (II.43) ω ∈ {ω 1 , ω 2 , ω 3 }. Now, separating the real and imaginary parts of (II.21) we obtain equality (II.23) and also equality (II.46) for d = 2, where ω ∈ {ω 1 , ω 2 , ω 3 }. Using equality (II.46) and the assumption that ω 1 , ω 2 , ω 3 are positive mutually different frequencies we obtain, in particular, (II.24) for d = 2. As in the proof of Theorem II.1, we use (II.20), (II.23) to obtain (II.29). Using (II.19), (II.24), (II.29) for d = 2 and (II.43) as in the proof of Theorem II.2, we obtain (II.36) for d = 2.

  1, III.2, III.1 , III.2 we reduced Problem III.1 to Problem III.2. As regards to methods of solving Problem III.2 we refer to[START_REF] Agaltsov | Riemann-Hilbert problem approach for two-dimensional flow inverse scattering[END_REF][START_REF] Arians | Geometric approach to inverse scattering for the Schrödinger equation with magnetic and electric potentials[END_REF][START_REF] Eskin | Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy[END_REF][START_REF]Inverse scattering problems for Schrödinger operators with magnetic and electric potentials[END_REF][START_REF]Inverse scattering problems for the Schrödinger operators with external Yang-Mills potentials[END_REF][START_REF] Eskin | Global uniqueness in the inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials[END_REF][START_REF] Novikov | The ∂-equation in the multidimensional inverse scattering problem[END_REF][START_REF] Henkin | A multidimensional inverse problem in quantum and acoustic scattering[END_REF][START_REF] Henkin | The Yang-Mills fields, the Radon-Penrose transform, and the Cauchy-Riemann equations[END_REF][START_REF] Nicoleau | A stationary approach to inverse scattering for Schrödinger operators with first order perturbation[END_REF][START_REF]The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF][START_REF]An iterative approach to non-overdetermined inverse scattering at fixed energy[END_REF][START_REF]Monochromatic recontruction algorithms for two-dimensional multichannel inverse problems[END_REF][START_REF] Päivärinta | Inverse scattering for the magnetic Schrödinger operator[END_REF][START_REF] Shiota | An inverse problem for the wave equation with first order perturbation[END_REF][START_REF] Xiaosheng | Inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials in two dimensions at fixed energy[END_REF] and references therein.

  , (III.23), (III.27), (III.41), (III.44), (III.47) and related proofs of Theorems III.2, III.2 . It is important to note that equation (III.11) and formula (III.13) give much more stable way to find functions ψ 0 , h 0 from A 0 1 , . . ., A 0 d , V 0 than equation (III.20) and formula (III.19) if | Im k| is sufficiently large.

  3.3 Reformulation of equations (III.8), (III.11) and formulas (III.10), (III.13) Subtracting equation (III.11) written for ψ from equation (III.11) written for ψ 0 we obtain formula

  III.1' It follows from formulas (III.53), (III.62) that we can apply identity (III.49) to (III.71) and (III.72). Applying (III.49) to (III.71) and (III.72), we obtain (III.40) and (III.46), respectively.

  Proof of Lemma III.[START_REF]A global uniqueness result for acoustic tomography of moving fluid[END_REF]. One can see that solution to (III.79) is unique since E is not a Dirichlet eigenvalue for L in D.Now let ψ 0 be the unique solution to (III.81) constructed in Lemma III.5. Then Lemma III.4 gives us the solution to (III.79) with desired properties.Property 3 is a consequence of formulas (III.82), (III.84).

	4.2 Compactness of operators in equations (III.41), (III.44),
	(III.47)

∂D) . (III.84)

Proof of Lemma III.5. The solution is unique since E is not a Dirichlet eigenvalue for -∆. To show existence, we can reduce problem (III.79) to a corresponding Fredholm integral equation of second kind as in Lemma III.

[START_REF] Agaltsov | Riemann-Hilbert problem approach for two-dimensional flow inverse scattering[END_REF]

.

Existence of such C β > 0 that (III.84) holds follows from Lemmas 2.16, 2.23 of

[START_REF] Colton | Integral equation methods in scattering theory[END_REF]

.

  and 0 < β < 1 be fixed. Suppose that assumptions of Theorem III.1 are fulfilled. Then equation (III.41) is uniquely solvable for ψ ∈ C 1,β (∂D, M n (C)) if and only if equation (III.67) is uniquely solvable for

  It follows from formulas (III.32), (III.35), (III.37) and from the corresponding properties of single and double layer potentials that the following formulas are valid :

	R 0 (x + 0ν x , y, k) where x ∈ R ∂D ∂R 0 ∂ν y (x + 0ν x , y, k)ϕ(y) dy = -ϕ(x) + ∂ϕ + ∂ν y (y) dy = R 0 (x -0ν x , y, k) ∂D ∂ν y (x -0ν x , y, k)ϕ(y) dy, ∂ϕ + ∂ν y (y) dy, ∂R 0	(III.93)
	∂D	∂D
	y, k)	∂ϕ + ∂ν y	(y) -	∂R 0 ∂ν y	(x, y, k)ϕ(y) dy,	(III.92)

d \ D.

  w d+1 , (VI.[START_REF] Colton | Integral equation methods in scattering theory[END_REF] if w 1 is defined as in (VI.24), andw 2 (x) = w 1 (x -se 1 ), . . . , w d+1 (x) = w 1 (x -se d ), (VI.29)where (e 1 , . . . , e d ) is the standard basis of R d and s > 0. Thus, in principle, for Problem VI.1 with background scatterers w 1 , . . ., w d+1 as in (VI.29), for each p ∈ R d \ π s Z d formulas (VI.11), (VI.12) can be used with appropriate w j in place of w 2 , where j = 2, . . ., d + 1.

  For Problem VI.1 for d ≥ 2, m = 2, and for the case when ζ w 1 , w 2 has no zeros (the case of (VI.27) in Section 3) we have the following result : Theorem VI.1. Let v satisfy (VI.2) and v ∈ W n,1 (R d ) for some n > d. Let w 1 , w 2 be the same as in (VI.27). Let

	u

2, (VI.38) where w 1 , w 2 , f , f 1 , f 2 are the same as in (VI.11), (VI.12), ζ w 1 , w 2 is defined by (VI.13), k E (p), l E (p) are the same as in (VI.14), (VI.32), and p ∈ B 2 √ E , d ≥ 2.

  1, (VI.41) where w 1 , w 2 and y are the same as in (VI.24)-(VI.26). One can see that Z ε w 1 , w 2 is the open ε |y| -neighborhood of Z w 1 , w 2 defined in (VI.26). Note that for any p ∈ Z ε w 1 , w 2 there exists the unique z(p) ∈ Z such that |py -πz(p)| < ε.

  is the integer number of (VI.42). For Problem VI.1 for d ≥ 2, m = 2, and for the case when ζ w 1 , w 2 has zeros on hyperplanes (the case of (VI.26) in Section 3) we have the following result :Theorem VI.2. Let v satisfy (VI.2) and v ∈ W n,1 (R d ) for some n > d. Let w 1 , w 2 be the same as in (VI.24)-(VI.26). Let

	2 ,	(VI.43)
	where z(p) u

  5.1 Proof of Theorem VI.1 Proposition VI.1. Let v satisfy (VI.2) and w 1 , w 2 be the same as in (VI.27), d ≥ 2.

  ). Proposition VI.1 follows from formulas (VI.11), (VI.14), estimates (VI.15), definitions (VI.34)-(VI.38), and the properties that

  15), c 1 , β are the same as in Lemma VI.1 ; z(p), p ⊥ are defined in (VI.42), (VI.43), x j v = x j v(x), and • n,1 is defined in (VI.31). of Proposition VI.2. It follows from formulas (VI.24), (VI.26) and (VI.35), (VI.36) that M w 1 , w 2

	Proof

  1 , p ∈ R d . (VI.72)Estimates (VI.71) follow from (VI.72), where we replace v by x j v and use that v belongs to W n,1 (R d ) and is compactly supported.

	Estimates (VI.69)-(VI.71) imply
	|ϕ ε 2

  Proposition VI.3. Let v satisfy (VI.2) and w 1 , . . ., w d+1 be the same as in (VI.24), (VI.29), d ≥ 2. Then :

	5.3 Proof of Theorem VI.3
	82)
	Formula (VI.45) follows from representation (VI.76), estimates (VI.57), (VI.77), (VI.80),
	(VI.81) and formulas (VI.82).
	Theorem VI.2 is proved.

  .., w d+1 .

		(VI.89)
	Here, the mean value theorem was used for v(ξ) on [p, ε s ϑ + π s z(p)], ϑ ∈ S d-1 . One can
	see that	estimates (VI.70) and (VI.71) hold for all ξ ∈ R d
		such that |ξ -π s z(p)| ≤ ε

s , where p ∈ Z ε w 1 ,..., w d+1 .

(VI.90) It follows from (VI.89), (VI.90) and from the upper estimate on ε of (VI.84), that

  .., w d+1 .

	(VI.91)
	Estimate (VI.84) follows from estimates (VI.88) and (VI.91).
	Proposition VI.3 is proved.

  Note that h1 h 2 = 1 2 h 1 + h 2 2 -1 2 h 2 1 + h 2 2 .Therefore the system (VII.15)-(VII.[START_REF] Berezansky | On uniqueness theorem in inverse problem of spectral analysis for Schrödinger equation[END_REF]) is equivalent to the system∂(h 1 + h 2 ) 2

					h 1	∂h 2 ∂ξ 1	+	∂h 1 ∂ξ 1	h 2 = h 1 h 2	∂(h 1 + h 2 ) ∂ξ 0	,	(VII.15)
		∂(h 2 1 + h 2 2 ) ∂ξ 0	= 2h 1	∂h 1 ∂ξ 0	+ 2h 2	∂h 2 ∂ξ 0	= 2	∂(h 1 + h 2 ) ∂ξ 1	.	(VII.16)
	∂ξ 1	-	∂(h 2 1 + h 2 2 ) ∂ξ 1	= h 1 + h 2	2 -h 2 1 + h 2 2	∂(h 1 + h 2 ) ∂ξ 0	,	(VII.17)
			∂(h 2 1 + h 2 2 ) ∂ξ 0	= 2	∂(h 1 + h 2 ) ∂ξ 1	.	(VII.18)

  ∂P 1 ∂ξ 1 = 1 2 ∂P 2∂ξ 0 . Note also that by Lemma 3.3.1 of[START_REF]Chaînes holomorphes de bord donné dans CP n[END_REF] we have Now denote by h 1 , h 2 the shock-wave extensions of h 1 (•, 0) and h 1 (•, 0) to W ξ * which exist and are unique by the Cauchy-Kowalevski theorem. Set e 1 = h 1 + h 2 , e 2 = h 1 h 2 . Due to the shock-wave equations for h 1 and h 2 , the functions e 1 and e 2Thus, e 1 , e 2 and e 1 , e 2 are holomorphic solutions to the same system with the same restrictions at ξ 1 = 0. By the Cauchy-Kowalevski theorem, e 1 = e 1 and e 2 = e 2 . It follows from the Viète formulas that h 1 , h 2 coincide with h 1 , h 2 (up to order). Hence, h 1 , h 2 satisfy the shock-wave equations.Applying Theorem VII.2, we obtain that N -≤ 2.

	satisfy
		∂ e 1 ∂ξ 1	=	1 2	∂ ∂ξ 0	( e 2 1 -2 e 2 ) in W ξ * ,
		∂ e 2 ∂ξ 1	= e 1	∂ e 2 ∂ξ 0	in W ξ
	∂G 1 ∂ξ 1 = 1 2	∂G 2 ∂ξ 0 . It leads to the equation
		∂ e 1 ∂ξ 1	=	1 2	∂ ∂ξ 0	( e 2 1 -2 e 2 ) in W ξ * .	(VII.29)
	Furthermore, equation (VII.26) can be rewritten in the form
			∂ e 2 ∂ξ 1		= e 2	∂ e 1 ∂ξ 0	in W ξ * .	(VII.30)

* .

plus précisément, la vitesse de propagation des ondes acoustiques

Si F : (Y, h) → (X, g) est une isométrie des variétés riemanniennes, on a ∆ h • F * = F * • ∆ g , où F * désigne l'image réciproque. Alors, l'opérateur Λ g est invariant par rapport aux isométries fixant le bord.

Si f est une fonction positive lisse sur X, on a ∆ f g = f -1 ∆ g (notons que cette propriété est spécifique pour le cas bidimensionnel). Cela implique que les métriques g et h = f g, où f > 0, f | ∂X = 1, ne sont pas distinguable par l'opérateur Λ g .En tenant compte de la non-unicité susmentionnée, on s'interèsse à la détermination de (X, g) à partir de Γ = ∂X et Λ g à une application conforme fixant le bord Γ près. Rappelons qu'une application conforme est une application préservant les angles entre les vecteurs tangents tandis qu'une isométrie préserve le produit scalaire. Le problème inverse à valeurs au bord pour l'opérateur Λ g est posé comme suit : Problème 7. Soit (X, g) une variété riemannienne bidimensionnelle au bord lisse Γ = ∂X et soit Λ g l'opérateur de Dirichlet-Neumann défini dans[START_REF]Transparent potentials at fixed energy in dimension two. Fixed-energy dispersion relations for the fast decaying potentials[END_REF]. Étant donnés Γ et Λ g , déterminer (X, g) à une application conforme fixant le bord Γ près.

CMAP, Ecole polytechnique, CNRS, Université Paris-Saclay, 91128, Palaiseau, France ; IEPT RAS, 117997 Moscow, Russia ; and Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia ; email : novikov@cmap.polytechnique.fr

Université Pierre et Marie Curie, case 247, 4 place du Jussieu, 75252, Paris, France, email : henkin@math.jussieu.fr

Remerciements

Article IV

Riemann-Hilbert problem approach for two-dimensional flow inverse scattering A. D. Agaltsov, R. G. Novikov 1 We consider inverse scattering for the time-harmonic wave equation with first-order perturbation in two dimensions. This problem arises in particular in the acoustic tomography of moving fluid. We consider linearized and nonlinearized reconstruction algorithms for this problem of inverse scattering. Our nonlinearized reconstruction algorithm is based on the non-local Riemann-Hilbert problem approach. Comparisons with preceding results are given.

Article V

On the reconstruction of parameters of a moving fluid from the Dirichlet-to-Neumann map

A. D. Agaltsov

We consider an inverse boundary value problem for a model time-harmonic equation of acoustic tomography of moving fluid with variable current velocity, sound speed, density and absorption. In the present article it is assumed that at fixed frequency the coefficients of this equation are already recovered modulo an appropriate gauge transformation using some reconstruction method from boundary measurements presented in the literature. Our main result consists in formulas and equations that allow to get rid of this gauge non-uniqueness and recover the fluid parameters using boundary measurements at several frequencies.

Introduction

We consider a model equation for a time-harmonic acoustic pressure ψ = ψ(x) (time dependence e -iωt ) in a moving fluid with sound speed c = c(x), current velocity v = v(x), density ρ = ρ(x) and absorption α = α(x, ω) = ω ζ(x) α 0 (x) at fixed frequency ω :

Article VI

Error estimates for phaseless inverse scattering in the Born approximation at high energies

We study explicit formulas for phaseless inverse scattering in the Born approximation at high energies for the Schrödinger equation with compactly supported potential in dimension d ≥ 2. We obtain error estimates for these formulas in the configuration space.

Article VII Explicit reconstruction of Riemann surface with given boundary in complex projective space

In this paper we propose a numerically realizable method for reconstruction of a complex curve with known boundary and without compact components in complex projective space.

Introduction

Let CP 2 be the complex projective plane with homogeneous coordinates (w 0 : w 1 : w 2 ). Let X ⊂ CP 2 be a complex curve with rectifiable boundary γ = ∂X. Without loss of generality, we suppose that the following conditions of general position hold :

By Darboux lemma [START_REF] Darboux | Théorie des surfaces[END_REF][START_REF]Chaînes holomorphes de bord donné dans CP n[END_REF] functions {h j } satisfy the following equations

One can see that the residues of Ω ξ 0 at q 1 , q 2 are equal to 0 and 1/ξ 0 , respectively. It follows that the residue at p ξ 0 is equal to -1/ξ 0 . Now, consider the part X of X obtained by cutting off some small neighborhood of the point q 1 . Let γ denote the boundary of X. Choose any ξ = (ξ 0 , 0) such that

} and, as far as the residue of Ω ξ 0 at q 1 is zero, we have that

We will also use the following result of [START_REF] Henkin | On the explicit reconstruction of a Riemann surface from its Dirichlet-to-Neumann operator[END_REF], which gives an effective characterization of functions h j≥1 of Theorem VII.1.

Theorem VII.2 (Remark 4 to Theorem 3a of [START_REF] Henkin | On the explicit reconstruction of a Riemann surface from its Dirichlet-to-Neumann operator[END_REF]). Let X ⊂ CP 2 \ (0 : 1 : 0) be a complex curve with rectifiable boundary γ ⊂ C 2 and satisfying ( * ). Fix any ξ * 0 ∈ Ω 0 and let W ξ * be a neighborhood of ξ * = (ξ * 0 , 0) in C 2 . Let G k≥1 , h j≥1 be defined as in (VII.2) and (VII.6). Suppose that there exist functions h 1 , . . ., h p , holomorphic in W ξ * and satisfying

Then p ≥ N -. Furthermore, if p = N -:= |X ∩ CP 1 ∞ |, then h 1 , . . ., h N -coincide with h 1 , . . ., h N -in W ξ * (up to order).

Reconstruction algorithm

We now pass to the reconstruction algorithm for a complex curve X ⊆ CP 2 with given boundary ∂X and satisfying the condition of minimality ( * ). Let us consider the cases N -= 0, 1, 2.

The reconstruction algorithm is based on formulas (VII.6). The next theorem permits to find the functions P m of (VII.6). We will use the notation

Theorem VII.3. Let X ⊂ CP 2 , (0 : 1 : 0) ∈ X, be a complex curve with rectifiable boundary γ ⊂ C 2 and satisfying ( * ). Let h j≥1 and P m≥1 be the functions defined in Theorem VII.1. Then the following statements are valid :

We divide this equation by ξ 2 0 and tend ξ 0 → ∞. This leads to the equality

Taking this equality into account, dividing (VII.20) by ξ 0 and passing ξ 0 → ∞, we obtain the equality 2c 11 Ċ12 (0) + 2c 12 Ċ11 (0) + Ċ22 (0) = -2c 11 c 12 + c 22 c 12 .

Using the obtained equalities, one can rewrite (VII.20) in the form

We pass ξ 0 → ∞ and note that the following relations are valid lim

As a result, we obtain Due to the obtained relations, we can express constants Ċij (0) as functions of c ij : The algorithm takes as input the points {ξ k 0 } N k=1 and the curve γ (for example, represented as a finite number of points belonging to γ). On the output of the algorithm we obtain the set of points (h

No points at infinity

1. Computation of N + . According to formula (VII.4), for every domain Ω l≥0 , the number µ l = N + (ξ 0 , 0), ξ 0 ∈ Ω l , is equal to the winding number of the curve π 2 γ with respect to a point ξ 0 ∈ Ω l :

2. Computation of power sums. If N -= 0 then, according to Theorem VII.3, for every point ξ k 0 ∈ Ω l≥0 we have that P m (ξ k 0 , 0) = 0. Using formula (VII.6) we obtain the following formulas for the power sums of the functions to be determined :

3. Computation of symmetric functions. For every point ξ k 0 ∈ Ω ≥1 , the Newton identities

allow to reconstruct the elementary symmetric functions :

Desymmetrisation.

For every point ξ k 0 ∈ Ω l , using Viète formulas, one can find the complex numbers h 1 (ξ k 0 , 0), . . ., h µ l (ξ k 0 , 0) (up to order). The points

are the required points of the complex curve X.

One or two points at infinity

These cases can be reduced to the case N -= 0 in the following way. Since π 2 γ ⊂ C is a compact real curve, there exists such R > 0, that the set

Let us describe in few words the algorithm of visualisation of complex curves that we use in our examples. Denote by π 1 : C 2 → C the projection into the first factor : π 1 (z 1 , z 2 ) = z 1 . Suppose that X is a complex curve in C 2 such that the covering π 1 : X \ {ramification points} → C has multiplicity L. Consider, for simplicity, a rectangular grid Λ in C :

where N is a natural number. Suppose now that we are given the set X Λ = π -1 1 (Λ)∩X and we need to visualize the part of X lying above the rectangle 0

Let us introduce some terminology. We define a path in Λ as a map γ :

N for all admissable k, where M is some natural number.

Let γ : {1, . . . , M } → Λ be a path in Λ and let i :

] is linear for all admissable k. It is clear that i * γ is a continuous function and hence it can be lifted to X by the map π 1 .

We define a path in X Λ as a map Γ :

, where i * is the pullback map with respect to i and

We also say that Γ is obtained by lifting of γ.

We will call subsets of Λ and X Λ path-connected if every two points of these sets can be connected by a path in Λ and X Λ , respectively.

Let us describe the practical way to lift paths in Λ to paths in X Λ . Suppose that N is sufficiently large. Let γ : {1, . . . , M } → Λ be a path in Λ and let Γ(1) ∈ π -1 (γ(1))∩X be an arbitrary point. We select Γ(k) ∈ π -1 (γ(k)) ∩ X in such a way that

Then Γ is a path in X Λ obtained by lifting of γ. All possible lifts of γ may be obtained by varying Γ(1). Note that if γ is closed, i. e. γ(1) = γ(M ), Γ need not to be closed.

Finding of ramification points and making branch cuts. The first step in visualization procedure consists in finding of ramification points of X with respect to projection π 1 . Since we have only a finite number of points on X we can find ramification points only approximately. More precisely, we will localize them in small circles.

Without restriction of generality we suppose that all ramification points are projected by π 1 into interior points of Λ. Take any interior point z 1 ∈ Λ and select a small closed path γ : {1, . . . , M } → Λ around z 1 so that there is at most one ramification point inside the polygon γ(1) . . . γ(M ). For example, one can take as γ the following path :

where i is the imaginary unit. Now consider different lifts of γ to X Λ . If at least one lift is not closed, mark z 1 as a possible ramification point (meaning that it is situated near the projection of some ramification point of X). Now vary z 1 and mark all possible ramification points. The resulting set will consist of several path-connected components each of which localizes the position of one ramification point of X with respect to π 1 .

Next, we need to cut the grid Λ with branching points removed into simply connected domains. The natural choice is the shortest cut graph, which can be computed by the algorithm proposed in [START_REF] Kälberer | Stripe parametrization of tubular surfaces[END_REF], which, in turn, generalizes the algorithm for the closed surfaces of [START_REF] Erickson | Greedy optiman homotopy and homology generators[END_REF].

Denote the cut graph by Λ c . An important observation is that every closed path in Λ \ Λ c always lifts to a closed path in X Λ since it doesn't contain π 1 -projections of ramification points inside.

Visualization. Now denote Λ \ Λ c = ∪ S s=1 Λ s , where Λ s are different path-connected components. Take any z s 1 ∈ Λ s and z s 2 ∈ π -1 1 (z s 1 ) ∩ X. Now take other z 1 ∈ Λ s and connect z s 1 with z 1 by some path γ. Then γ lifts to a path Γ with Γ(1) = (z s 1 , z s 2 ) and Γ(2) = (z 1 , z 2 ) for some z 2 ∈ π -1 1 (z 1 ) ∩ X and z 2 doesn't depend on γ. Varying z 1 we thus obtain the map Σ(z s 1 , z s 2 ) : Λ s → X Λ which allows us to visualize the part of X. Varying z s 2 ∈ π -1 1 (z s 1 ) ∩ X (the latter is the finite set, namely, it consists of L elements) we obtain the other maps Σ(z s 1 , z s 2 ) which allow us to visualize other parts of X. Clearly, the set of obtained maps doesn't depend on the choice of z s 1 ∈ Λ s . Hence we can denote the obtained maps by Σ l s , l = 1, . . ., L. It is clear that

The part π -1 (Λ c ) ∩ X Λ consists of cuts and preimages of possible ramification points. The cuts can be visualized as the already visualized part of the surface. The only problem is the visualization of π 1 -preimages of possible ramification points. But the latter take a little part of the surface when N is large and one can just forget about their visualization. On the other hand, in our examples they were visualized using low-level graphics approach.

Examples of application of this algorithm are given at

Examples

1. No points at infinity. We consider the complex curve given by the equation

The boundary γ 1 = ∂X 1 of this curve is the disjoint union of two real curves γ r 1 and γ g 1 , see Fig. VII.2. We are going to recover X 1 from γ 1 . First, note that the real curve |z 2 | = 1.9 in C divides C into two parts Ω 0 = {|z 2 | > 1.9} and Ω 1 = {|z 2 | < 1.9}. We check numerically that G 1 (ξ) is zero in a neighborhood W ξ * of some ξ * = (ξ * 0 , 0), ξ 0 ∈ Ω 0 , and thus N -= 0 by Theorem VII.4. Next, we compute the value

for some ξ 0 ∈ Ω 1 (it will be the same for any ξ 0 ∈ Ω 1 ). It turns out that µ 1 = 4 and thus π 2 :

j=1 , ξ = (ξ 0 , 0), |ξ 0 | < 1.9. Next, we compute the values of the power sum functions s k :

Note that the curve π 2 γ 2 = {|z 2 | = 1.9} divides C into two connected components Ω 0 = {z 2 > 1.9} and Ω 1 = {z 2 < 1.9}.

According to the algorithm, the first step is to determine the number of points at infinity. We choose some ξ * = (ξ * 0 , 0) with ξ * 0 ∈ Ω 0 and check that in some small neighborhood W ξ * of ξ * we have G 1 ≡ 0 and thus N -≥ 1 by Theorem VII. [START_REF] Agaltsov | Riemann-Hilbert problem approach for two-dimensional flow inverse scattering[END_REF].

Next, we fix two generic ξ 1 0 , ξ 2 0 ∈ Ω 0 and determine c 11 , c 12 from the linear system

(VII.32) This system is uniquely solvable and gives c 11 ≈ 1, c 12 ≈ 0.

Next, we check numerically that in the neighborhood W ξ * of ξ * the following identity holds :

, where P 1 (ξ 0 , ξ 1 ) = c 11 ξ 0 +c 12 1+c 11 ξ 1 .

(VII.33)

Theorem VII.4 together with the estimate N -≥ 1 imply that N -= 1.

Next, according to the algorithm, in order to reduce the problem to the case of no points at infinity, we need to recover the auxilary curve γ

∞ . Thus, it remains to recover X 2,R from the known boundary γ 2,R using the algorithm for the case of no points at infinity, see the previous example. The reconstructed curve is depicted at Fig. VII.3.

3. Two points at infinity. Consider the complex curve X 3 , which is the part of the algebraic curve

bounded by the real curve

The curve X 3 has two points at infinity, namely, q 1 = (0 : 1 : e

3 ), q 2 = (0 : 1 : e -2πi 3 ). We suppose that the number of points at infinity is a priori unknown. We are going to reconstruct X 3 from γ 3 . As in the previous examples, the real curve

The first step is to determine the number of points at infinity. Choose a generic ξ * = (ξ * 0 , 0), ξ * 0 ∈ Ω 0 , and some neighborhood W ξ * of ξ * . 1. We check that G 1 ≡ 0 in W ξ * so that N -≥ 1 by Theorem VII.4.

2. We choose two generic ξ 1 0 , ξ 2 0 ∈ Ω 0 and solve (VII.32) for c 11 , c 12 . It turns out that system (VII.32) is uniquely solvable. However, identity (VII.33) does not hold in W ξ * . Using Theorem VII.4, we obtain that N -≥ 2.

3. We consider (VII.11) as an identity in a neighborhood of ξ * 0 . We find numerically that c 11 ≈ 0.9974, c 12 ≈ -0.3335, c 21 ≈ 1.0133, c 22 ≈ -0.6699, c 23 ≈ 0.5567.

We check that (VII.26) holds in a neighborhood of

given by equations (VII.27). If follows from Theorem VII.4 and from the inequality

Suppose that we want to recover X 3,R = X 3 ∩ {|z 2 | < R}, R = 3. According to the algorithm, we need to find the auxilary boundary γ

Conclusion

The main results of the present work can be summarized as follows :

1. We obtain global uniqueness theorems for acoustic tomography of moving fluid at one, two or three fixed frequences depending on a choosen model.

2.

We present examples of moving fluids which are not distinguishable in the setting of acoustic tomography even if the data is measured at all frequences.

3. We give formulas and equations which reduce the inverse Dirichlet-to-Neumann problem for a gauge-covariant Helmholtz equation to an appropriate inverse scattering problem.

4. We propose a functional-analytic reconstruction algorithm for the above-mentioned inverse scattering problem, which allows to recover the gauge class of the operator from its scattering amplitude at fixed frequency.

5.

We present equations which allow to get rid of the above gauge non-uniqueness and uniquely recover the operator in the setting of acoustic tomography of moving fluid, using measurements at several frequencies.

6. We obtain error estimates in the coordinate space for phaseless reconstructions in the Bord approximation at high energies for the Schrödinger equation.

7.

We give an explicit version of the Harwey-Shiffman result on the unique determination of a complex curve by its boundary, developing the algorithm of Henkin-Michel for the inverse boundary value problem for two-dimensional Riemannian manifolds.

One can underline the following principal directions of possible development of the subjects studied in the thesis :

1. Adapt the algorithm of Articles III-V for acoustic tomography of moving fluid to the case of partial data. In particular, the case where only the imaginary part of the Green function is known is of great importance for applications with cross-correlation-type measurements, e.g., in passive ocean tomography [START_REF] Burov | The use of low-frequency noise in passive tomography of the ocean[END_REF], in helioseismology [START_REF] Gizon | Solving the forward problem of helioseismology in the frequency domain[END_REF] and in other settings, see, e.g., [START_REF] Snieder | Unified Green's function retrieval by cross-correlation ; connection with energy principles[END_REF].

2. Study the stability and obtain the error estimates for the functional-analytic reconstruction algorithm of Article IV. Note that the stability was studied numerically in many particular cases by O. D. Rumyantseva, A. S. Shurup and D. I. Zotov (acoustic department of Moscow State University), see, e.g., [START_REF] Shurup | Numerical simulation of the functional approach for recovering vector fields in acoustic tomography[END_REF].

3. Generalize the functional-analytic reconstruction algorithm of Article IV to the case of the two-dimensional gauge-covariant Helmholtz equation with matrix coefficients. This generalization is motivated by applications in mode tomography of moving ocean, see, e.g., [START_REF] Baykov | Mode tomograpgy of moving ocean[END_REF].

4. Find an efficient and noise stable algorithm for the phaseless inverse scattering problem for the Schrödinger equation at fixed frequency using the error estimates of Article VI.

5. Find compact formulas and equations for recovering the complex curve in the complex projective plane from its boundary for the case of arbitrary number of points at infinity. Specify easy verifiable conditions for the unique solvability of these equations. Abstract : This work is devoted to study of some inverse problems for the gauge-covariant Helmholtz equation, whose particular cases include the Schrödinger equation for a charged elementary particle in a magnetic field and the time-harmonic wave equation describing sound waves in a moving fluid. These problems are mainly motivated by applications in different tomographies, including acoustic tomography, tomography using elementary particles and electrical impedance tomography. In particular, we study inverse problems motivated by applications in acoustic tomography of moving fluid. We present formulas and equations which allow to reduce the acoustic tomography problem to an appropriate inverse scattering problem. Next, we develop a functional-analytic algorithm for solving this inverse scattering problem. However, in general, the solution to the latter problem is unique only up to an appropriate gauge transformation. In this connection, we give formulas and equations which allow to get rid of this gauge non-uniqueness and recover the fluid parameters, by measuring acoustic fields at several frequencies. We also present examples of fluids which are not distinguishable in this acoustic tomography setting. Next, we consider the inverse scattering problem without phase information. This problem is motivated by applications in tomography using elementary particles, where only the absolute value of the scattering amplitude can be measured relatively easily. We give estimates in the configuration space for the phaseless Born-type reconstructions, which are needed for the further development of precise inverse scattering algorithms. Finally, we consider the problem of determination of a Riemann surface in the complex projective plane from its boundary. This problem arises as a part of the inverse Dirichlet-to-Neumann problem for the Laplace equation on an unknown 2-dimensional surface, and is motivated by applications in electrical impedance tomography.