Mourad Bellassoued

Mourad Choulli
email: mourad.choulli@univ-lorraine.fr

David Dos

Santos Ferreira

AND PLAMEN STEFANOV Yavar Kian
email: yavar.kian@univ-amu.fr

A BORG-LEVINSON THEOREM FOR MAGNETIC SCHRÖDINGER OPERATORS ON A RIEMANNIAN MANIFOLD

Keywords: 2010 Mathematics Subject Classification. Primary 35R30, 35J10, Secondary: 35P99 Borg-Levinson type theorem, magnetic Schrödinger operator, simple Riemannian manifold, uniqueness, stability estimate

This article is concerned with uniqueness and stability issues for the inverse spectral problem of recovering the magnetic field and the electric potential in a Riemannian manifold from some asymptotic knowledge of the boundary spectral data of the corresponding Schrödinger operator under Dirichlet boundary conditions. The spectral data consist of some asymptotic knowledge of a subset of eigenvalues and Neumann traces of the associated eigenfunctions of the magnetic Laplacian. We also address the same question for Schrödinger operators under Neumann boundary conditions, in which case we measure the Dirichlet traces of eigenfunctions. In our results we characterize the uniqueness of the magnetic field from a rate of growth of the eigenvalues, combined with suitable asymptotic properties of boundary observation of eigenfunctions, of the associated magnetic Schrödinger operator. To our best knowledge this is the first result proving uniqueness from such general asymptotic behavior of boundary spectral data.

Introduction and main results

1.1. Statement of the problem. Let M " pM, gq be a smooth and compact Riemannian manifold with boundary BM. We denote the Laplace-Beltrami operator associated to the Riemannian metric g by ∆. In local coordinates, the metric reads g " pg jk q, and the Laplace-Beltrami operator ∆ is given by

∆ " 1 a |g| n ÿ j,k"1 B Bx j ˆa|g| g jk B Bx k ˙.
Here pg jk q is the inverse of the metric g and |g| " detpg jk q. Given a couple of magnetic and electric potentials B " pA, qq, where q P L 8 pMq is real-valued, and A " a j dx j is a covector field (1-form) with real-valued coefficients, a j P W 1,8 pMq, we consider the magnetic Laplacian

H B " 1 a |g| n ÿ j,k"1 ˆ1 i B Bx j `aj ˙a|g| g jk ˆ1 i B Bx k `ak ˙`q
" ´∆ ´2i A ¨∇ ´i δA `|A| 2 `q :" ´∆A `q. (1.1)

Here, the dot product is in the metric with A and ∇ considered as covectors, δ is the coderivative (codifferential) operator, corresponding to the divergence with identifying vectors and covectors, which sends 1-forms to a functions by the formula

δA " 1 a |g| n ÿ j,k"1 B Bx j ´gjk a |g|a k ¯,
and we recall that, for A " a j dx j , we have |A| 2 " g jk a j a k . For B " pA, qq with q P L 8 pMq and A " a j dx j , a j P W 1,8 pMq, define on L 2 pMq the unbounded self-adjoint operator H B as follows

(1.2) H B u " H B u
and

(1.3) DpH B q " u P H 1 0 pMq, ´∆A u `qu P L 2 pMq ( .

Here H k pMq, denotes the standard definition of the Sobolev spaces.

The operator H B is self adjoint and has compact resolvent, therefore its spectrum σpH B q consists in a sequence λ B " pλ B,k q of real eigenvalues, counted according to their multiplicities, so that (1.4) ´8 ă λ B,1 ď λ B,2 ď . . . ď λ B,k Ñ `8 as k Ñ 8.

In the sequel φ B " pφ B,k q denotes an orthonormal basis of L 2 pMq consisting in eigenfunctions with φ B,k associated to λ B,k , for each k.

In the rest of this text, we often use the following notation, where k ě 1, ψ B,k " pB ν `iApνqq φ B,k , on BM and ψ B " pψ B,k q, where ν the outward unit normal vector field on BM with respect to the metric g. We address the question of whether one can recover, in some suitable sense, the magnetic field A and the potential q from some asymptotic knowledge of the boundary spectral data pλ B , ψ B q with B " pA, qq. As for most inverse problems, the main issues are uniqueness and stability.

1.2. Obstruction to uniqueness. We recall that there is an obstruction to the recovery of the electromagnetic potential B from the boundary spectral data pλ B , ψ B q. Indeed, let B " pA, qq, and let V P C 1 pMq be such that V |BM " 0 and set q B " pA `dV, qq. Then it is straightforward to check that (1.5) e ´iV H B e iV " H q B , pλ B , ψ B q " pλ q B , ψ q B q. Therefore, the magnetic potential A cannot be uniquely determined by the boundary spectral data pλ B , ψ B q and our inverse problem needs to be stated differently.

According to [START_REF] Sharafutdinov | Integral Geometry of Tensor Fields[END_REF], for every covector A P H k pM, T ˚Mq, there exist uniquely determined A s P H k pM, T ˚Mq and V P H k`1 pMq such that (1.6)

A " A s `dV, δA s " 0, V | BM " 0.

Following the well established terminology, A s and dV are called respectively the solenoidal and potential parts of the covector A. In view of the obstruction described above, the best one can expect is the simultaneous recovery of A s and q from some knowledge of the boundary spectral data pλ B , ψ B q. From now on, we focus our attention on this problem.

Known results.

There is a vast literature devoted to inverse spectral problems in one dimension. We refer for instance to the pioneer works by Ambartsumian [START_REF] Ambartsumian | Über eine Frage der Eigenwerttheorie[END_REF], Borg [START_REF] Borg | Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe[END_REF], Levinson [31], Gel'fand and Levitan [START_REF] Gel | On the determination of a differential equation from its spectral function[END_REF]. In the flat case, the first uniqueness results for inverse spectral problems in dimensions greater or equal to two is due to Nachman, Sylvester and Uhlmann [START_REF] Nachman | An n-dimensional Borg-Levinson theorem[END_REF]. At almost the same time, Isozaki [START_REF] Isozaki | Some remarks on the multi-dimensional Borg-Levinson theorem[END_REF] improved the result by Nachman, Sylvester and Uhlamann by using the Born approximation. He proved that the potential in a Schrödinger equation is uniquely determined by the large eigenvalues and the trace of the normal derivatives of the corresponding eigenfunctions. Developing further Isozaki's approach, Choulli and Stefanov [START_REF] Choulli | Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data[END_REF] gave a generalization of Isozaki's uniqueness result together with a Hölder stability estimate with respect to appropriate metrics for the spectral data. We mention that, following a remark of Isozaki which goes back to [START_REF] Isozaki | Some remarks on the multi-dimensional Borg-Levinson theorem[END_REF], the uniqueness and stability results of [START_REF] Choulli | Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data[END_REF] were stated with only some asymptotic closeness of the boundary spectral data. We mention also the work of [START_REF] Canuto | Determining Coefficients in a Class of Heat Equations via Boundary Measurements[END_REF][START_REF] Canuto | Determining Two Coefficients in Elliptic Operators via Boundary Spectral Data: a Uniqueness Result, Bolletino Unione Mat[END_REF], dealing with recovery of general non-smooth coefficients from the full boundary spectral data, and the work [START_REF] Kavian | Uniqueness and stability results for an inverse spectral problem in a periodic waveguide[END_REF] who have considered a similar inverse spectral problem for Schrödinger operators in an infinite cylindrical waveguide.

Another approach for getting uniqueness in the spectral inverse problem for the Laplace-Beltrami operator was introduced by Belishev [START_REF] Belishev | An approach to multidimensional inverse problems for the wave equation[END_REF] and Belishev and Kurylev [6]. This approach consists in reducing the inverse spectral problem under consideration into an inverse hyperbolic problem for which one can apply the so called boundary control method. This method allows to consider the trace of the normal derivative of eigenfunctions only in a part of the boundary. We refer to [START_REF] Belishev | To the reconstruction of a Riemannian manifold via its spectral data (BC-method)[END_REF][START_REF] Katchalov | Inverse boundary spectral problems[END_REF][START_REF] Katchalov | Multidimensional inverse problem with incomplete boundary spectral data[END_REF][START_REF] Lassas | An inverse problem for a wave equation with sources and observations on disjoint sets[END_REF][START_REF] Lassas | Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets[END_REF] and [START_REF] Kian | Application of the boundary control method to partial data Borg-Levinson inverse spectral problem[END_REF] in the case of non-smooth coefficients. We mention that none of these papers considered this problem with observations corresponding to some asymptotic knowledge of the boundary spectral data. Actually, to our best knowledge, beside the present paper, there is no other results dealing with inverse spectral problem on non flat manifolds with data similar to the one considered by [START_REF] Choulli | Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data[END_REF][START_REF] Kavian | Uniqueness and stability results for an inverse spectral problem in a periodic waveguide[END_REF].

One of the first stability estimate for inverse spectral problems was established by Alessandrini and Sylvester [START_REF] Alessandrini | Stability for multidimensional inverse spectral problem[END_REF]. This result was reformulated by the second author in a more precise way in [START_REF] Choulli | Une introduction aux problèmes inverses elliptiques et paraboliques[END_REF]. A similar result in the case of the Laplace-Beltrami operator was proved by the first and the third authors in [START_REF] Bellassoued | Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map[END_REF] using the idea introduced in [START_REF] Alessandrini | Stability for multidimensional inverse spectral problem[END_REF]. With the help of a result quantifying the uniqueness of continuation for a Cauchy problem with data on a part of the boundary for a wave equation, the first two authors and Yamamoto [START_REF] Bellassoued | Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem[END_REF] proved a double logarithmic stability estimate under the assumption that the potential is known near the boundary. In [START_REF] Choulli | Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data[END_REF], the second and the last authors provided one of the first Hölder type stability estimate for the multi-dimensional Borg-Levinson theorem of determining the potential from some asymptotic knowledge of the boundary spectral data of the associated Schrödinger operator. In [START_REF] Kavian | Uniqueness and stability results for an inverse spectral problem in a periodic waveguide[END_REF], the fourth author, Kavian and Soccorsi proved a similar result for an inverse spectral problem in an infinite cylindrical waveguide. 1.4. Preliminaries. We briefly recall some notations and known results in Riemannian geometry. We refer for instance to [START_REF] Jost | Riemannian Geometry and Geometric Analysis[END_REF] for more details. By Riemannian manifold with boundary, we mean a C 8 -smooth manifold with boundary in the usual sense, endowed with a metric g.

As before M denotes a compact Riemannian manifold of dimension n ě 2. Fix a local coordinate system x " `x1 , . . . , x n ˘and let pB 1 , . . . , B n q be the corresponding tangent vector fields. For x P M, the inner product and the norm on the tangent space T x M are given by gpX, Y q " xX, Y y "

n ÿ j,k"1 g jk X j Y k , |X| " xX, Xy 1{2 , X " n ÿ i"1 X i B i , Y " n ÿ i"1 Y i B i .
The cotangent space T x M is the dual of T x M. Its elements are called covectors or one-forms. The disjoint union of the tangent spaces

T M " ď xPM T x M
is called the tangent bundle of M. Similarly, the cotangent bundle T ˚M is the disjoint union of the spaces T x M, x P M. A 1-form A on the manifold M is a function that assigns to each point x P M a covector Apxq P T x M. An example of a 1-form is the differential of a function f P C 8 pMq, which is defined by

df x pXq " n ÿ j"1 X j Bf Bx j , X " n ÿ j"1 X j B j .
Hence f defines the mapping df : T M Ñ R, which is called the differential of f given by df px, Xq " df x pXq. In local coordinates,

df " n ÿ j"1 B j f dx j .
where pdx 1 , . . . , dx n q is the basis in the space T x M, dual to the basis pB 1 , . . . , B n q.

The Riemannian metric g induces a natural isomorphism ı : T x M Ñ T x M given by ιpXq " xX, ¨y. For X P T x M denote X 5 " ıpXq, and similarly for A P T x M we denote A 7 " ı ´1pAq, ı and ı ´1 are called musical isomorphisms. The sharp operator is given by

(1.7) T x M ÝÑ T x M, A Þ ÝÑ A 7 ,
given in local coordinates by (1.8) pa j dx j q 7 " a j B j , a j "

n ÿ k"1 g jk a k .
Define the inner product of 1-forms in T x M by (1.9) xA, By " xA 7 , B 7 y "

n ÿ j,k"1 g jk a j b k " n ÿ j,k"1 g jk a j b k .
The metric tensor g induces the Riemannian volume

dv n " |g| 1{2 dx 1 ^¨¨¨^dx n .
We denote by L 2 pMq the completion of C 8 pMq endowed with the usual inner product

pf 1 , f 2 q " ż M f 1 pxqf 2 pxq dv n , f 1 , f 2 P C 8 pMq.
A section of a vector bundle E over the Riemannian manifold M is a C 8 map s : M Ñ E such that for each x P M, spxq belongs to the fiber over x. We denote by C 8 pM, Eq the space of smooth sections of the vector bundle E. According to this definition, C 8 pM, T Mq denotes the space of vector fields on M and C 8 pM, T ˚Mq denotes the space of 1-forms on M. Similarly, we may define the spaces L 2 pM, T ˚Mq (resp. L 2 pM, T Mq) of square integrable 1-forms (resp. vectors) by using the inner product

(1.10) pA, Bq " ż M xA, By dv n , A, B P T ˚M.
Define the Sobolev space H k pMq as the completion of C 8 pMq with respect to the norm

}f } 2 H k pMq " }f } 2 L 2 pMq `n ÿ k"1 }∇ k f } 2 L 2 pM,T k Mq ,
where ∇ k is the covariant differential of f in the metric g. If f is a C 8 function on M, then ∇f is the vector field defined by Xpf q " x∇f, Xy, for every vector field X on M. In the local coordinates system, the last identity can be rewritten in the form

(1.11) ∇f " n ÿ i,j"1 g ij Bf Bx i B j " pdf q 7 .
The normal derivative of a function u is given by the formula (1.12)

B ν u :" x∇u, νy " n ÿ j,k"1 g jk ν j Bu Bx k ,
where ν is the unit outward vector field to BM. Likewise, we say that a 1-form A " a j dx j belongs to H k pM, T ˚Mq if each a j P H k pMq. The space H k pM, T ˚Mq is a Hilbert space when it is endowed with the norm

}A} H k pM,T ˚Mq " ˜n ÿ j"1 }a j } 2 H k pMq ¸1 2 .
As usual, the vector space of smooth 2-forms on M is denoted by Ω 2 pMq. In local coordinates, a 2-form ω is represented as

ω " n ÿ j,k"1 ω jk dx j ^dx k ,
where ω jk are real-valued functions on M. Similarly as before, ω is in H s pM, Ω 2 pMqq, s P R, if ω jk P H s pMq for each j, k. Additionally, H s pM, Ω 2 pMqq is a Hilbert space for the norm

}ω} H s pM,Ω 2 pMqq " ˜ÿ j,k }ω jk } 2 H s pMq ¸1 2 .
In the rest of this text, the scalar product of L 2 pBMq is also denoted by x¨, ¨y:

(1.13) xf 1 , f 2 y " ż BM f 1 pxq f 2 pxq dσ n´1
where dσ n´1 is the volume form of BM.

1.5. Main results. Prior to the statement of our main results, we introduce the notion of simple manifolds [START_REF] Stefanov | Stability estimates for the X-ray transform of tensor fields and boundary rigidity[END_REF]. We say that the boundary BM is strictly convex if the second fundamental form is positive-definite for any x P BM.

Definition 1.1. A manifold M is simple if BM is strictly convex and, for any x P M, the exponential map exp x : exp ´1 x pMq ÝÑ M is a diffeomorphism.

Note that if M is simple, then it is diffeomorphic to a ball, and every two points can be connected by a unique minimizing geodesic depending smoothly on its endpoints. Also, one can extend it to a simple manifold M 1 such that M int 1 Ą M. We now introduce the admissible sets of magnetic potentials A and electric potentials q. Set B " W 2,8 pM, T ˚Mq ' L 8 pMq. We endow B with its natural norm }B} B " }A} W 2,8 pM,T ˚Mq `}q} L 8 pMq .

For r ą 0, set (1.14) B r " tB " pA, qq P B, }B} B ď ru .

Let B P B r , " 1, 2, we denote by pλ ,k , φ ,k q, k ě 1, the eigenvalues and normalized eigenfunctions of the operator H B .

For " 1 or " 2, let

(1.15) ψ ,k " pB ν `iA pνqq φ ,k , k ě 1.
At this point we remark that when A 1 " A 2 it is clear that H B1 ´HB2 " q 1 ´q2 whence by the min-max principle, sup kě1 |λ 1,k ´λ2,k | ď }q 1 ´q2 } L 8 pM q ă 8.

Assume now that A 1 ‰ A 2 and δA 1 " δA 2 . Then we have

H B1 ´HB2 " ´2ipA 1 ´A2 q∇ `|A 1 | 2 ´|A 2 | 2 `q1 ´q2 .
Thus, H B1 ´HB2 R BpL 2 pM qq. Therefore, we can reasonably expect that

sup kě1 |λ 1,k ´λ2,k | " `8.
Keeping in mind this property and the obstruction described in Section 1.2, it seems natural to expect the recovery of the solenoidal part of the magnetic potential from a rate of growth of the eigenvalues. Our first result give a positive answer to this issue together with the recovery of the electric potential. Theorem 1.2. Assume that M is simple. Let B " pA , q q P B r , " 1, 2, such that

(1.16) B α x A 1 pxq " B α
x A 2 pxq, x P BM, |α| ď 1. Furthermore, assume that there exists t P r0, 1{2q so that (1.17) sup

kě1 k ´t{n |λ 1,k ´λ2,k | `ÿ kě1 k ´2t{n }ψ 1,k ´ψ2,k } 2 L 2 pBMq ă 8.
Then A s 1 " A s 2 . Moreover, under the additional conditions (1.18) lim kÑ`8

|λ 1,k ´λ2,k | " 0, and

ÿ kě1 }ψ 1,k ´ψ2,k } 2 L 2 pBMq ă 8,
we have q 1 " q 2 .

In the spirit of [START_REF] Choulli | Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data[END_REF][START_REF] Kavian | Uniqueness and stability results for an inverse spectral problem in a periodic waveguide[END_REF], we consider also the stability issue for this problem stated as follows.

Theorem 1.3. Assume that M is simple. Let B " pA , q q P B r , " 1, 2, such that A 1 and A 2 satisfies (1.16) and q 1 ´q2 P H 1 0 pMq satisfies }q 1 ´q2 } H 1 0 pMq ď r. Furthermore, assume that there exists t P p0, 1{2q so that (1.19) sup

kě1 k ´t{n |λ 1,k ´λ2,k | `ÿ kě1 }ψ 1,k ´ψ2,k } 2 L 2 pBMq ă 8. Then A s 1 " A s 2 and
(1.20)

}q 1 ´q2 } L 2 pMq ď C ˆlim sup kÑ8 |λ 1,k ´λ2,k | ˙1 2 ă 8,
the constant C only depends on r and M.

To our best knowledge Theorems 1.2 and 1.3 are the first results dealing with inverse spectral problems for Schrödinger operators, with non-constant leading coefficients, from asymptotic knowledge boundary spectral data similar to the one considered by [START_REF] Choulli | Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data[END_REF][START_REF] Kavian | Uniqueness and stability results for an inverse spectral problem in a periodic waveguide[END_REF]. Note also that Theorem 1.3 seems to be the first stability result of recovering the electric potential from partial boundary spectral data in such general context (the only other similar results can be found in [START_REF] Bellassoued | Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem[END_REF][START_REF] Choulli | Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data[END_REF][START_REF] Kavian | Uniqueness and stability results for an inverse spectral problem in a periodic waveguide[END_REF] where stable recovery of Schrödinger operators on a bounded domain, with an Euclidean metric and without magnetic potential, have been considered).

We recall that multi-dimensional Borg-Levinson type theorems for magnetic Schrödinger operators have been already considered in [START_REF] Katchalov | Multidimensional inverse problem with incomplete boundary spectral data[END_REF][START_REF] Kian | A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data[END_REF][START_REF] Serov | Borg-Levinson theorem for magnetic Schrödinger operator[END_REF]. Among them, only [START_REF] Kian | A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data[END_REF] considered the uniqueness issue from boundary spectral data similar to (1.18). The results in the present work can be seen as an improvement of that in [START_REF] Kian | A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data[END_REF] in four directions. First of all, we prove for the first time the extension of such results to a general simple Riemanian manifold by proving the connection between our problem and the injectivity of the so called geodesic ray transform borrowed from [START_REF] Yu | On uniqueness of determination of a form of first degree by its integrals along geodesics[END_REF][START_REF] Frigyik | The X-ray transform for a generic family of curves and weights[END_REF][START_REF] Sharafutdinov | Integral Geometry of Tensor Fields[END_REF][START_REF] Stefanov | Stability estimates for the X-ray transform of tensor fields and boundary rigidity[END_REF]. In addition, by using some results of [START_REF] Stefanov | Stability estimates for the X-ray transform of tensor fields and boundary rigidity[END_REF], we establish stability estimates for this problem where [START_REF] Kian | A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data[END_REF] treated only the uniqueness. In contrast to [START_REF] Kian | A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data[END_REF], we do not require the knowledge of the magnetic potentials on the neighborhood of the boundary. This condition is relaxed, by considering only some knowledge of the magnetic potentials at the boundary given by (1.16). Finally, we show, for what seems to be the first time, that even a rate of growth of the difference of eigenvalues like (1.17), (1.19) can determine the magnetic potential appearing in a magnetic Schrödinger operator.

The main ingredient in our analysis is a suitable representation formula that involves the magnetic potential A and the electric potential q in terms of the Dirichletto-Neumann map associated to the equations H B u´λu " 0, for a well chosen set of complex λ's. In [START_REF] Isozaki | Some remarks on the multi-dimensional Borg-Levinson theorem[END_REF][START_REF] Kian | A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data[END_REF], the authors considered such a representation for a bounded domain with flat metric. Using a construction inspired by [START_REF] Bellassoued | Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field[END_REF][START_REF] Bellassoued | Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map[END_REF][START_REF] Santos Ferreira | Determining a magnetic Schrödinger operator from partial Cauchy data[END_REF][START_REF] Santos Ferreira | Limiting Carleman weights and anisotropic inverse problems[END_REF][START_REF] Santos Ferreira | The Calderón problem in transversally anisotropic geometries[END_REF][START_REF] Stefanov | Stable determination of the hyperbolic Dirichlet-to-Neumann map for generic simple metrics[END_REF] we show how one can extend such an approach to more general manifolds. Note that this construction differs from the one considered by [START_REF] Santos Ferreira | Determining a magnetic Schrödinger operator from partial Cauchy data[END_REF][START_REF] Santos Ferreira | Limiting Carleman weights and anisotropic inverse problems[END_REF][START_REF] Santos Ferreira | The Calderón problem in transversally anisotropic geometries[END_REF] for recovering the magnetic Schrödinger operators from boundary measurements. Actually, our results hold for a general simple manifold even in the case n ě 3, whereas the determination of Schrödinger operators from boundary measurements in the same context is still an open problem (see [START_REF] Santos Ferreira | Limiting Carleman weights and anisotropic inverse problems[END_REF][START_REF] Santos Ferreira | The Calderón problem in transversally anisotropic geometries[END_REF]).

In this paper we treat also the problem of determining the Neumann realization of magnetic Schrödinger operator. For simplicity and in order to avoid any confusion between the results for the different operators, we give the statement of the result for the Neumann realization of magnetic Schrödinger operator in Theorem 6.1 of Section 6. the result of Theorem 6.1 is stated with an optimal growth of the difference of eigenvalues (see the discussion just after Theorem 6.1).

We believe that following the idea of [START_REF] Bellassoued | An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains[END_REF][START_REF] Kian | A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data[END_REF][START_REF] Krupchyk | Uniqueness in an inverse boundary problem for a magnetic Schrodinger operator with a bounded magnetic potential[END_REF][START_REF] Salo | Inverse problems for nonsmooth first order perturbations of the Laplacian[END_REF], one can relax the regularity condition imposed to the magnetic potentials as well as condition (1.16). This approach requires the construction of anzats depending on an approximation of the magnetic potential instead of the magnetic potential itself. In order to avoid the inadequate expense of the size of the paper, we do not consider this issue.

1.6. Outline. The outline of the paper is as follows. We review in Section 2 the geodesic ray transform for 1-one forms and functions on a manifold. Section 3 is devoted to an asymptotic spectral analysis. We construct in Section 4 geometrical optics solutions for magnetic Schrödinger equations. We particularly focus our attention on the solvability of the eikonal and the transport equations which are essential in the construction of geometric optic solutions. Additionally, we provide a representation formula. The proof of Theorems 1.2 and 1.3 are given in Section 5. The Neumann case is briefly discussed in Section 6. Finally, we prove some uniform estimates related to the Weyl's formula for the magnetic Schrödinger operator in appendix A.

A short review on the geodesic ray transform on a simple manifold

We collect in this section some known results on the geodesic ray transform for functions and 1-forms on a smooth simple Riemannian manifold pM, gq. These results will be used later in this text.

Denote by divX the divergence of a vector field X P H 1 pM, T Mq on M, i.e. in local coordinates ([24, page 42]), (2.1) divX "

1 a |g| n ÿ i"1 B i ´a|g| X i ¯, X " n ÿ i"1 X i B i .
Using the inner product of a 1-form, we can define the coderivative operator δ as the adjoint of the exterior derivative via the relation

(2.2) pδA, vq " pA, dvq , A P C 8 pM, T ˚M q, v P C 8 pMq.
Then δA is related to the divergence of vector fields by δA " divpA 7 q, where the divergence is given by (2.1). If X P H 1 pM, T Mq the divergence formula reads

(2.3) ż M divX dv n " ż BM xX, νy dσ n´1 .
For f P H 1 pMq, we have the following Green formula

(2.4) ż M divX f dv n " ´żM xX, ∇f y dv n `żBM xX, νyf dσ n´1 .
Therefore, for u, w P H 2 pMq, the following identity holds ż

M ∆ A uw dv n " ´żM x∇ A u, ∇ A wy dv n `żBM pB ν u `iApνquqw dσ n´1 (2.5) " ż M u∆ A w dv n `żBM ´pB ν u `iApνquqw ´upB ν w `iApνqwq ¯dσ n´1 ,
where ∇ A u " ∇u `iuA 7 . For x P M and θ P T x M, denote by γ x,θ the unique geodesic starting from x and directed by θ.

Recall that the sphere bundle and co-sphere bundle of M are respectively given by SM " tpx, θq P T M; |θ| " 1u , S ˚M " tpx, pq P T ˚M; |p| " 1u , The exponential map exp x : T x M ÝÑ M is defined as follows

(2.6) exp x pvq " γ x,θ p|v|q, θ " v |v| .
We assume in the rest of this section that M is simple and we point out that any arbitrary pair of points in M can be joined by an unique geodesic of finite length.

Given px, θq P SM and denote by γ x,θ the unique geodesic γ x,θ satisfying the initial conditions γ x,θ p0q " x and 9 γ x,θ p0q " θ, which is defined on the maximal interval r ´px, θq, `px, θqs, with γ x,θ p ˘px, θqq P BM.

" B `SM Y SBM Y B ´SM. It is straightforward to check that ˘: SM Ñ R satisfy ´px, θq ď 0, `px, θq ě 0,
`px, θq " ´ ´px, ´θq, ´px, θq " 0, px, θq P B `SM, ´pϕ t px, θqq " ´px, θq ´t, `pϕ t px, θqq " `px, θq `t. To each 1-form A P C 8 pM, T ˚Mq, with A " a j dx j , associate the smooth symbol σ A P C 8 pSMq given by (2.9)

σ A px, θq " n ÿ j"1
a j pxqθ j " xA 7 pxq, θy, px, θq P SM.

Recall that the Riemannian scalar product on T x M induces the volume form on S x M given by

dω x pθq " a |g| n ÿ k"1 p´1q k θ k dθ 1 ^¨¨¨^y dθ k ^¨¨¨^dθ n .
As usual, the notation p ¨means that the corresponding factor has been dropped. We also consider the volume form dv 2n´1 on the manifold SM defined as follows dv 2n´1 px, θq " dω x pθq ^dv n , where dv n is the Riemannnian volume form on M. By Liouville's theorem, the form dv 2n´1 is preserved by the geodesic flow. The corresponding volume form on the boundary BSM " tpx, θq P SM, x P BMu is given by dσ 2n´2 " dω x pθq ^dσ n´1 , where dσ n´1 is the volume form of BM.

Santaló's formula will be useful in the sequel: (2. Until the end of this section, we assume that M is simple.

2.1. Geodesic ray transform of 1-forms. The ray transform of 1-forms on M is defined as the linear operator

I 1 : C 8 pM, T ˚Mq ÝÑ C 8 pB `SMq
acting as follows

I 1 pAqpx, θq " ż γ x,θ A " n ÿ j"1 ż `px,θq 0 a j pγ x,θ ptqq 9 γ j x,θ ptqdt " ż `px,θq 0 σ A pϕ t px, θqqdt.
It is easy to check that I 1 pdϕq " 0 for any ϕ P C 8 pMq satisfying ϕ |BM " 0. On the other hand, it is known that I 1 is injective on the space of solenoidal 1-forms satisfying δA " 0. Therefore, if A P H 1 pM, T ˚Mq is so that I 1 pAq " 0, then A s " 0. Whence, there exists ϕ P H 1 0 pM q X H 2 pM q such that A " dϕ. As a consequence of this observation, we have (2.12)

|I 1 pAqpx, θq| " |I 1 pA s qpx, θq| ď C}A s } C 0 , A P C 0 pM, T ˚Mq.
With reference to [START_REF] Sharafutdinov | Integral Geometry of Tensor Fields[END_REF], we recall that

I 1 : L 2 µ pB `SMq ÝÑ L 2 pM, T ˚Mq is given by (2.13) pI 1 Ψpxqq j " ż SxM θ j q Ψpx, θq dω x pθq.
Here q Ψ is the extension of Ψ from B `SM to SM, which is constant on every orbit of the geodesic flow. That is q Ψpx, θq " Ψ `γx,θ p ´px, θqq, 9 γ x,θ p ´px, θqq ˘" ΨpΦ ´px,θq px, θqq, px, θq P SM.

One can check [START_REF] Sharafutdinov | Integral Geometry of Tensor Fields[END_REF] that I 1 has a bounded extension, still denoted by I 1 ,

I 1 : H k pM, T ˚Mq ÝÑ H k pB `SMq.
We complete this subsection by results borrowed from [START_REF] Stefanov | Stability estimates for the X-ray transform of tensor fields and boundary rigidity[END_REF]. We extend pM, gq to a smooth Riemannian manifold pM 1 , gq such that M int 1 Ą M and we consider the normal operator N 1 " I 1 I 1 . Then there exist

C 1 ą 0, C 2 ą 0 such that (2.14) C 1 }A s } L 2 pMq ď }N 1 pAq} H 1 pM1q ď C 2 }A s } L 2 pMq , for any A P L 2 pM, T ˚Mq. If O is an open set of M 1 , N 1 is
I 0 f px, θq " ż `px,θq 0 f pγ x,θ ptqq dt.
Similarly to I 1 , I 0 has an extension, still denoted by I 0 :

(2.18)

I 0 : H k pMq ÝÑ H k pB `SMq
for every integer k ě 0. We refer to [36, Theorem 4.2.1] for details.

Considering I 0 as a bounded operator from L 2 pMq into L 2 µ pB `SMq, we can compute its adjoint

I 0 : L 2 µ pB `SMq Ñ L 2 pMq (2.19) I 0 Ψpxq " ż SxM q Ψpx, θq dω x pθq,
where q Ψ is the extension of Ψ from B `SM to SM which is constant on every orbit of the geodesic flow: q Ψpx, θq " Ψpγ x,θ p `px, θqqq.

Let M 1 be a simple manifold so that M int 1 Ą M and consider the normal operator N 0 " I 0 I 0 . Then there exist two constants

C 1 ą 0, C 2 ą 0 such that (2.20) C 1 }f } L 2 pMq ď }N 0 pf q} H 1 pM1q ď C 2 }f } L 2 pMq
for any f P L 2 pMq, see [START_REF] Stefanov | Stability estimates for the X-ray transform of tensor fields and boundary rigidity[END_REF].

If O is an open set of M 1 , N 0 is an elliptic pseudo-differential operator of order ´1 on Ω, whose principal symbol is a multiple of |ξ| ´1, see [START_REF] Stefanov | Stability estimates for the X-ray transform of tensor fields and boundary rigidity[END_REF]. Therefore there exists a constant C k ą 0 such that, for all f P H k pOq compactly supported in O,

(2.21) }N 0 pf q} H k`1 pM1q ď C k }f } H k pOq .

Asymptotic spectral analysis

We fix in all of this section B " pA , q q P B r , " 1, 2, satisfying the assumptions of Theorem 1.2. As in Section 1, H B , " 1, 2, is the operator defined by (1.2) and (1.3) when B " B . Furthermore, for λ P ρpH B q, denote by R B pλq the resolvent of H B and, for s P r0, 1{2q, recall the following classical resolvent estimate

(3.1) }R B pλq} L pL 2 pMq;H 2s pMqq ď C s | λ| 1´s , " 1, 2.
For f P H 3{2 pBMq and λ P ρpH B q, " 1, 2, consider the Dirichlet problem

(3.2) $ & % pH B ´λqu " 0 in M, u " f on BM.
Let κ be a boundary defining function, that is a smooth function κ : M Ñ R such that ' κpxq ą 0 for all x P M int , ' κ| BM " 0 and dκ| BM ‰ 0.

We recall that one can construct such a function by combining local coordinates with boundary distance functions or by considering the first eigenvalue of the Dirichlet Laplacian. We can now state the following result.

Lemma 3.1. If f P H 3{2 pBMq and λ P ρpH B q, then the BVP (3.2) has a unique solution u pλq " u f pλq P H 2 pMq given by the series

(3.3) u pλq " ÿ kě1 xf, ψ ,k y λ ´λ ,k φ ,k ,
the convergence takes place in H 1 pMq. Moreover, for any neighborhood V of BM in M, we have

(3.4) lim λÑ´8 `}u pλq} L 2 pMq `}κdu pλq} L 2 pMq ˘" 0.
Proof. The proof of (3. xκupλqA, κdupλqy dv n `żM `2ixA, κdκy `p|A| 2 `q ´λqκ 2 ˘|upλq| 2 dv n .

An application of Cauchy-Schwarz's inequality yields

}κdupλq} 2 L 2 pMq ´λ}κupλq} 2 L 2 pMq ď C}upλq} L 2 pMq }κdupλq} L 2 pMq `C}upλq} 2 L 2 pMq ď C 1 }upλq} 2 L 2 pMq `1 2 }κdupλq} 2 L 2 pMq . Then, it follows (3.7) 1 2 }κdupλq} 2 L 2 pMq ´λ}κupλq} 2 L 2 pMq ď C}upλq} 2 L 2 pMq
and since ´λ ą 0

(3.8) }κdupλq} 2 L 2 pMq ď C}upλq} 2 L 2 pMq , implying Caccioppoli's inequality (3.5).
Lemma 3.2. Let f P H 3{2 pBMq and, for µ P ρpH B1 q X ρpH B2 q, set w 1,2 pµq " u 1 pµq ´u2 pµq P H 2 pMq, where u pµq is the corresponding solution to (3.2) with B and λ are substituted by B and µ. Then we have that w 1,2 pµq converges to 0 in H 2 pMq as µ Ñ ´8. In particular, B ν w 1,2 pµq Ñ 0 in L 2 pBMq as µ Ñ ´8.

Proof. For the sake of simplicity, we use in this proof wpµq instead of w 1,2 pµq. Since the trace map v Þ Ñ B ν v is continuous from H 2 pMq into L 2 pBMq, it is enough to show that }wpµq} H 2 pMq Ñ 0 when µ Ñ ´8. Let µ ă µ ˚ă ´2}q} 8 , for some fixed µ ˚ă 0. It is straightforward to check that wpµq is the solution of the boundary value problem (3.9)

$ & % pH B1 ´µq wpµq " hpµq in M, wpµq " 0 on BM.
Here hpµq is given by (3.10) hpµq " ´2ixA 2 ´A1 , du 2 pµqy `pV 2 ´V1 q u 2 pµq with V j " ´iδA j `|A j | 2 `qj , j " 1, 2. Multiplying the first equation of (3.9) by wpµq, we apply Green's formula (2.5) This completes the proof of the lemma.

The following lemma will be useful in the sequel. We omit its proof since it is quite similar to that in [START_REF] Kavian | Uniqueness and stability results for an inverse spectral problem in a periodic waveguide[END_REF][START_REF] Kian | A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data[END_REF]. Lemma 3.3. Let f P H 3{2 pBMq and, for µ, λ P ρpH B q, set w pλ, µq " u pλq ú pµq, where u pµq is the solution of (3.2) when λ is substituted by µ. Then we have (3.19) pB ν `iA pνqq w pλ, µq "

ÿ kě1 pµ ´λqxf, ψ ,k y pλ ´λ ,k qpµ ´λ ,k q ψ ,k ,
the convergence takes place in H 1{2 pBMq.

Isozaki's representation formula

In the present section we provide a version of Isozaki's approach [START_REF] Isozaki | Some remarks on the multi-dimensional Borg-Levinson theorem[END_REF], based on the so-called Born approximation method. The usual anzats used to solve the problem of determining the coefficients of a magnetic Laplace-Beltrami operator, from the corresponding Dirichlet-to-Neumann map will be useful in our analysis. Let us describe briefly this method.

In all of this section B " pA , q q P B r , " 1, 2, with A satisfying (1.16). We extend the covector A 1 to a W 2,8 covector on M 1 supported in the interior of M 1 and still denoted by A 1 . Then, we consider the extension of A 2 to M 1 , still denoted by A 2 , defined by (4.1)

A 1 pxq " A 2 pxq, x P M 1 zM.

Then, (1.16) implies that A 2 P W 2,8 pM 1 ; T ˚M1 q. We fix also A " A 1 ´A2 .

4.1. Representation formula. If u pλq, λ P ρpH B1 q X ρpH B2 q, is the solution of (3.2) when B " B , define the Dirichlet-to-Neumann map by

(4.2) Λ B pλq : f P H 3{2 pBMq Þ Ñ pB ν `iA pνqq u pλq |BM , " 1, 2.
We fix ψ P C 2 pMq a function satisfying the eikonal equation

(4.3) |dψ| 2 " n ÿ i,j"1 g ij Bψ Bx i Bψ Bx j " 1.
We set also two functions α P H 2 pMq solving the transport equations

(4.4) xdψ, dα y `1 2 p∆ψqα " 0, " 1, 2.
This function will be given in Section 4.2. Consider also two functions β A P H 2 pMq, " 1, 2, solutions of the transport equations

(4.5) xdψ, dβ A y `ixA , dψyβ A " 0, @ x P M, " 1, 2.
Henceforth τ ą 1 and λ τ " τ `i. Let (4.6) ϕ 1,τ pxq " e iλτ ψpxq α 1 β A1 pxq :" e iλτ ψpxq β 1 pxq, ϕ 2,τ pxq " e iλτ ψpxq α 2 β A2 pxq :" e iλτ ψpxq β 2 pxq, where, for " 1, 2, α is a solution of (4.4) and β A is a solution of (4.5). Define (4.7)

S B pτ q " xΛ B pλ 2 τ qϕ 1,τ , ϕ 2,τ y " ż BM Λ B pλ 2 τ qϕ 1,τ ϕ 2,τ dσ n´1 , " 1, 2.
Lemma 4.1. We have

(4.8) S B1 pτ q " ż BM β 1 ´Bν β 2 ´iA 1 pνqβ 2 ´iλ τ β 2 B ν ψ ¯dσ n´1 `żM β 1 H B1 pβ 2 q dv n ´2λ τ ż M β 1 β 2 xA, dψy dv n ´żM R B1 pλ 2 τ q `eiλτ ψ H B1 pβ 1 q ˘´e ´iλτ ψ H B1 pβ 2 q
´2λ τ e ´iλτ ψ β 2 xA, dψy ¯dv n . and (4.9)

S B2 pτ q " ż BM β 1 ´Bν β 2 ´iA 2 pνqβ 2 ´iλ τ β 2 B ν ψ ¯dσ n´1 `żM β 1 H B2 pβ 2 q dv n ´żM R B2 pλ 2 τ q `eiλτ ψ pH B2 pβ 1 q ´2λ τ xA, dψyβ 1 q ˘´e ´iλτ ψ H B2 pβ 2 q ¯dv n .
Here R B pλ 2 τ q is the resolvent of H B .

Proof. Direct computations yield

`HB1 ´λ2 τ ˘ϕ1 ,τ " e iλτ ψ H B1 pβ 1 q (4.10) `eiλτ ψ ˆλ2 τ β 1 `|dψ| 2 ´1˘´2 iλ τ β A1 ´xdψ, dα 1 y `α1 2 ∆ψ 2iλ τ α 1 pxdψ, dβ A1 y `ixA 1 , dψyβ A1 q
Ṫaking into account (4.3) and (4.4)-(4.5), with " 1, the right-hand side of (4.10) becomes (4.11) `HB1 ´λ2 τ ˘ϕ1 ,τ " e iλτ ψ H B1 pβ 1 q " e iλτ ψ k 1 .

Denote by u 1 the solution of the BVP

$ & % `HB1 ´λ2 τ ˘u1 " 0 in M, u 1 " ϕ 1,τ on BM.
We split u 1 into two terms, u 1 " ϕ 1,τ `v1 , where v 1 is the solution of the boundary value problem

$ & % `HB1 ´λ2 τ ˘v1 " ´eiλτ ψ k 1 in M, v 1 " 0 on BM. Therefore (4.12) u 1 " ϕ 1,τ ´`H B1 ´λ2 τ ˘´1 pe iλτ ψ k 1 q " ϕ 1,τ ´RB1 pλ 2 τ q `eiλτ ψ k 1 ˘. As (4.13) S B1 pτ q " ż BM pB ν u 1 `iA 1 pνqu 1 q ϕ 2,τ dσ n´1 ,
we get by applying formula (2.5)

S B1 pτ q " ż M ∆ A1 u 1 ϕ 2,τ dv n ´żM u 1 ∆ A1 ϕ 2,τ dv n (4.14) `żBM ϕ 1,τ ´Bν ϕ 2,τ `iA 1 pνqϕ 2,τ ¯dσ n´1 .
On the other hand, by a simple computation and using (4.3), (4.4) and (4.5), we get

∆ A1 ϕ 2,τ " ∆ A1 pe iλτ ψ β 2 q " ´λ2 τ ϕ 2,τ `eiλτ ψ ∆ A1 pβ 2 q ´2iλ τ e iλτ ψ α 2 pxdψ, dβ 2 y `ixA 1 , dψyβ 2 q `2iλ τ β 2 e iλτ ψ ´xdψ, dα 2 y `α2 2 ∆ψ " ´λ2 τ ϕ 2,τ `eiλτ ψ ∆ A1 pβ 2 q ´2iλ τ e iλτ ψ α 2 p´ixA 2 , dψyβ 2 `ixA 1 , dψyβ 2 q " ´λ2 τ ϕ 2,τ `eiλτ ψ ∆ A1 pβ 2 q `2λ τ e iλτ ψ β 2 xA, dψy.
Whence, in light of (4.12), we find

ż M u 1 ∆ A1 ϕ 2,τ dv n " ż M `ϕ1 ,τ ´RB1 pλ 2 τ q `eiλτ ψ k 1 ˘λ 2 τ ϕ 2,τ `e´iλτ ψ ∆ A1 pβ 2 q `2λ τ e ´iλτ ψ β 2 xA, dψy ¯dv n ,
and, using again (4.12), we get

ż M ∆ A1 u 1 ϕ 2,τ dv n " ´żM H B1 u 1 ϕ 2,τ dv n `żM q 1 u 1 ϕ 2,τ dv n " ż M `ϕ1 ,τ ´RB1 pλ 2 τ qpe iλτ ψ k 1 q ˘´´λ 2 τ ϕ 2,τ `q1 ϕ 2,τ ¯dv n .
We deduce that

ż M ∆ A1 u 1 ϕ 2,τ dv n ´żM u 1 ∆ A1 ϕ 2,τ dv n (4.15) " ż M `ϕ1 ,τ ´RB1 pλ 2 τ qpe iλτ ψ k 1 q ȇ´iλτ ψ H B1 pβ 2 q ´2λ τ e ´iλτ ψ β 2 xA, dψy ¯dv n " ż M β 1 H B1 pβ 2 q dv n ´2λ τ ż M β 1 β 2 xA, dψy dv n ´żM R B1 pλ 2 τ qpe iλτ ψ k 1 q ´e´iλτ ψ H B1 pβ 2 q ´2λ τ e ´iλτ ψ β 2 xA, dψy ¯dv n . Moreover ż BM ϕ 1,τ pB ν ϕ 2,τ `iA 1 pνqϕ 2,τ q dσ n´1 (4.16) " ż BM β 1 ´Bν β 2 ´iA 1 pνqβ 2 ´iλ τ β 2 B ν ψ ¯dσ n´1 .
Finally, we get (4.8) by combining (4.14), (4.15) and (4.16).

The proof of (4.9) is quite similar to that of (4.8). But, for the reader's convenience, we detail the proof of (4.9). By a simple computation we find `HB2 ´λ2 τ ˘ϕ1 ,τ " e iλτ ψ H B2 pβ 1 q (4.17)

`eiλτ ψ ˆλ2 τ β 1 `|dψ| 2 ´1˘´2 iλ τ β A1 ´xdψ,
u 2 " ϕ 1,τ ´`H B2 ´λ2 τ ˘´1 pe iλτ ψ k 2 q " ϕ 1,τ ´RB2 pλ 2 τ qpe iλτ ψ k 2 q. Since (4.20) S B2 pτ q " ż BM pB ν u 2 `iA 2 pνqu 2 q ϕ 2,τ dσ n´1 ,
we obtain, by applying formula (2.5),

S B2 pτ q " ż M ∆ A2 u 2 ϕ 2,τ dv n ´żM u 2 ∆ A2 ϕ 2,τ dv n (4.21) `żBM ϕ 1,τ pB ν ϕ 2,τ `iA 2 pνqϕ 2,τ q dσ n´1 .
On the other hand, by using (4.3), (4.4) and (4.5), we find

(4.22) ∆ A2 ϕ 2,τ " ∆ A2 pe iλτ ψ β 2 q " ´λ2 τ ϕ 2,τ `eiλτ ψ ∆ A2 pβ 2 q. Whence ż M u 2 ∆ A2 ϕ 2,τ dv n " ż M pϕ 1,τ ´RB2 pλ 2 τ qpe iλτ ψ k 2 qq (4.23) ˆ´´λ 2 τ ϕ 2,τ `e´iλτ ψ ∆ A2 pβ 2 q ¯dv n and ż M ∆ A2 u 2 ϕ 2,τ dv n " ´żM H B2 u 2 ϕ 2,τ dv n `żM q 2 u 2 ϕ 2,τ dv n " ż M `ϕ1 ,τ ´RB2 pλ 2 τ qpe iλτ ψ k 2 q ˘´´λ 2 τ ϕ 2,τ `q2 ϕ 2,τ ¯dv n .
Thus,

ż M ∆ A2 u 2 ϕ 2,τ dv n ´żM u 2 ∆ A2 ϕ 2,τ dv n (4.24) " ż M `ϕ1 ,τ ´RB2 pλ 2 τ qpe iλτ ψ k 2 q ˘´e ´iλτ ψ H B2 pβ 2 q ¯dv n " ż M β 1 H B2 pβ 2 q dv n ´żM R B2 pλ 2 τ q `eiλτ ψ k 2 ˘´e ´iλτ ψ H B1 pβ 2 q ¯dv n " ż M β 1 H B2 pβ 2 q dv n ´żM R B2 pλ 2 τ q `eiλτ ψ pH B2 pβ 1 q ´2λ τ xA, dψyβ 1 q ˘´e ´iλτ ψ H B1 pβ 2 q ¯dv n .
Moreover, we have

ż BM ϕ 1,τ pB ν ϕ 2,τ `iA 2 pνqϕ 2,τ q dσ n´1 (4.25) " ż BM β 1 ´Bν β 2 ´iA 2 pνqβ 2 ´iλ τ β 2 B ν ψ ¯dσ n´1 .
Inserting (4.25) and (4.24) in (4.21), we obtain

S B2 pτ q " ż BM β 1 ´Bν β 2 ´iA 2 pνqβ 2 ´iλ τ β 2 B ν ψ ¯dσ n´1 (4.26) `żM β 1 H B2 pβ 2 q dv n ´żM R B2 pλ 2 τ q `eiλτ ψ pH B2 pβ 1 q ´2λ τ xA, dψyβ 1 q ˘´e ´iλτ ψ H B2 pβ 2 q ¯dv n .
This completes the proof of the Lemma.

Subtracting side by side (4.8) and (4.9), and using the fact that A 1 " A 2 on BM, we obtain the following identity, that we will use later in the text. (4.27)

S B1 pτ q ´SB2 pτ q " ´2λ τ ż M β 1 β 2 xA, dψy dv n `żM β 1 pH B1 ´HB2 qpβ 2 q dv n ´żM R B1 pλ 2 τ q `eiλτ ψ H B1 pβ 1 q ˘´e ´iλτ ψ ´HB1 pβ 2 q ´2λ τ β 2 xA, dψy ¯¯dv n `żM R B2 pλ 2 τ q `eiλτ ψ pH B2 pβ 1 q ´2λ τ xA, dψyβ 1 q ˘´e ´iλτ ψ H B2 pβ 2 q ¯dv n .

Solving the eikonal and tranpsort equations.

We construct the phase function ψ solution to the eikonal equation (4.3) and the amplitudes α and β , " 1, 2, solutions to the transport equations (4.4)-(4.5).

Let y P BM 1 . Denote points in M 1 by pr, θq where pr, θq are polar normal coordinates in M 1 with center y. That is, x " exp y prθq, where r ą 0 and

θ P S ỳ M 1 " tθ P T y M 1 , |θ| " 1, xθ, νy ă 0u .
In these coordinates (depending on the choice of y) the metric has the form r gpr, θq " dr 2 `g0 pr, θq.

If u is a function in M, set, for r ą 0 and θ P S y M 1 , r upr, θq " upexp y prθqq, If u is compactly supported, r u is naturally extended by 0 outside M. The geodesic distance to y provide an explicit solution of the eikonal equation (4.3): (4. [START_REF] Krupchyk | Uniqueness in an inverse boundary problem for a magnetic Schrodinger operator with a bounded magnetic potential[END_REF] ψpxq " d g px, yq.

Since y P M 1 zM, we have ψ P C 8 pMq and (4.29) r ψpr, θq " r " d g px, yq.

We now solve the transport equation (4.4). To this and, recall that if f prq is any function of the geodesic distance r, then

(4.30) ∆ r g f prq " f 2 prq ` ´1 2 B Br f 1 prq.
Here " pr, θq denotes the square of the volume element in geodesic polar coordinates. In the new coordinates system, equation ( 4

Asymptotic behavior of the boundary representation formula.

We discuss in this subsection the asymptotic behavior of S B1 pτ q ´SB2 pτ q, as well as the asymptotic behavior of rS B1 pτ q ´SB2 pτ qs{τ , as τ Ñ 8.

As before, B " pA , q q P B r , " 1, 2 are so that A satisfy (1.16). Set

Apxq " pA 1 ´A2 qpxq, qpxq " pq 1 ´q2 qpxq.

Note that A, extended by 0 outside M, belongs to C 0 pM 1 , T ˚M1 q. We also extend q by 0 outside M. This extension, still denoted by q, is an element of L 8 pM 1 q.

Lemma 4.2. For any η P H 2 pS ỳ M 1 q, we have

(4.40) lim τ Ñ`8 S B1 pτ q ´SB2 pτ q τ " 2i ż S ỳ M1
´eiI1Apy,θq ´1¯η py, θq dω y pθq.

Proof. By the resolvent estimate, we have

(4.41) }R B pλ 2 τ q} L pL 2 pMqq ď 1 | pλ 2 τ q| " 1 2τ , " 1
S B1 pτ q ´SB2 pτ q τ " 2 ż M β 1 β 2 xA, dψy dv n (4.42) " 2 ż M α 1 α 2 β A1 β A2 xA, dψy dv n .
Applying (4.1) and making the change variable x " exp y prθq, with r ą 0 and θ P S y M 1 , we get This in (4.42) gives the expected inequality.

Lemma 4.3. Assume that A 1 " A 2 and q 1 ´q2 P H 1 0 pM q. Then, for any η P H 2 pS ỳ M 1 q, we have

(4.45) lim τ Ñ`8 pS B1 pτ q ´SB2 pτ qq " ż S ỳ M1
I 0 pqqpy, θqηpy, θqdω y pθq.

Proof. Since A 1 " A 2 , (4.27) is reduced to the following formula

S B1 pτ q ´SB2 pτ q " ż M qpxqβ 1 pxqβ 2 pxq dv n (4.46) ´żM R B1 pλ 2 τ q `eiλτ ψ H B1 pβ 1 q ˘´e ´iλτ ψ H B1 pβ 2 q ¯dv n `żM R B2 pλ 2 τ q `eiλτ ψ H B2 pβ 1 q ˘´e ´iλτ ψ H B2 pβ 2 q ¯dv n .
Once again the resolvent estimate enables us to get

(4.47) lim τ Ñ`8 pS B1 pτ q ´SB2 pτ qq " ż M qpxqpα 1 α 2 qpxq dv n .
We complete the proof by mimicking the end of the previous proof in order to obtain

(4.48) ż M qpxqpα 1 α 2 qpxq dv n " ż S ỳ M1
I 0 pqqpy, θqηpy, θqdω y pθq.

This completes the proof.

Proof of the main results

Asymptotic behavior of the spectral data.

Prior to the completion of the proof of Theorems 1.2 and 1.3, we establish some technicals lemmas. Assumptions and notations are the same as in the preceding one.

Lemma 5.1. For t P r0, 1{2q and " 1, 2, we have

(5.1) ÿ kě1 k 2t{n ˇˇˇx ϕ 1,τ , ψ ,k y λ ,k ´λ2 τ ˇˇˇ2 ď C τ 2t }η} 2 H 2 pS ỳ M1q

and

(5.2)

ÿ kě1 k 2t{n ˇˇˇx ϕ 2,τ , ψ 2,k y λ ,k ´λ2 τ ˇˇˇ2 ď C τ 2t ,
the constant C depends on t, M, r and B if t ą 0, and it is independent on B when t " 0.

Proof. By Lemma 3.1 the solution of the boundary value problem (3.2), with f " ϕ 1,τ , λ " λ τ and B " B 1 , is given by the series

(5.3) u 1 pλ τ q " ÿ kě1 xϕ 1,τ , ψ 1,k y λ 2 τ ´λ1,k φ 1,k .
If µ " 2r `1, then the operator H B `µ is positive. Indeed, for u P H 1 0 pMq, we have ż

M pH B `µq uu dv n " ż M |∇ A u| 2 dv n `żM pq `µq|u| 2 dv n ě ż M |du| 2 dv n `pµ ´}q } 8 ´2}A } 8 q ż M |u| 2 dv n .
Since DppH B `µq

1 2
q " H 1 0 pMq we have, by interpolation, DppH B `µq t 2 q " H t 0 pMq " H t pMq (e.g. [33, Chapter 1, Theorems 11.1 and 11.6]). Whence, for w P H t pMq, we have (5.4)

ÿ kě1 p1 `|λ ,k |q t |pw, φ ,k q| 2 ď C }w} 2 H t pMq , " 1, 2,
the constant C only depends on t, r and M and B . On the other hand, we get from (4.12)

}u 1 pλ τ q} H t pMq ď }ϕ 1,τ } H t pMq `}R B1 pλ 2 τ qpe iλτ ψ H B1 β 1 q} H t pMq (5.5) ď Cτ t }η} H 2 pS ỳ M1q .
Here again the constant C only depends on t, r, M and B 1 , where we used that exp ´1 y pM q Ă trθ : r ą 0, θ P S ỳ pM 1 qu in order to restrict the norm of η to S ỳ M 1 . This estimate and (5.4) with w " u 1 pλ τ q and " 1 entail (5.6)

ÿ kě1 p1 `|λ 1,k |q t |pu 1 pλ τ q, φ 1,k q| 2 ď C 1 τ 2t }η} 2 H 2 pS ỳ M1q .
We get the first estimate (5.1) for " 1, by using (A.1) in Appendix A and the identity (5.7)

pu 1 pλ τ q, φ 1,k q " xϕ 1,τ , ψ 1,k y λ 2 τ ´λ1,k .
To prove the first inequality (5.1) for " 2, we consider u 2 pλ τ q, the solution of the BVP (3.2) when λ " λ τ , f " ϕ 1,τ and B " B 2 . By Lemma 3.1, this solution is given by the series (5.8)

u 2 pλ τ q " ÿ kě1 xϕ 1,τ , ψ 2,k y λ 2 τ ´λ2,k φ 2,k .
On the other hand, we get from (4.19) and (3.1)

}u 2 pλ τ q} H t pMq ď }ϕ 1,τ } H t pMq (5.9) `}R B2 pλ 2 τ qpe iλτ ψ pH B2 pβ 1 q ´2λ τ xA, dψyβ 2 q } H t pMq ď C ˆτ t `|λ τ | τ 1´t ˙}η} H 2 pS ỳ M1q ď Cτ t }η} H 2 pS ỳ M1q .
Applying again (5.4) with w " u 2 pλ τ q and " 2 entail (5.10)

ÿ kě1 p1 `|λ 2,k |q t |pu 2 pλ τ q, φ 2,k q| 2 ď C 2 τ 2t }η} 2 H 2 pS ỳ M1q . Since (5.11) pu 2 pλ τ q, φ 2,k q " xϕ 1,τ , ψ 2,k y λ 2 τ ´λ2,k
.

we obtain (5.1) with " 2.

The second inequality of (5.2) is proved similarly.

Let us recall some notations that we introduced in Section 3. For f P H 3{2 pBMq fixed and λ, µ P ρpH B1 q X ρpH B2 q, if u pλq (resp. u pµq) is the solution of the boundary value problem (3.2) for B " B (resp. B " B and λ " µ), " 1, 2, we have posed w pλ, µq " u pλq ´u pµq, w 1,2 pµq " u 1 pµq ´u2 pµq.

(5.12) Let (5.13) Kpτ, µ, f q " pB ν `iA 1 pνqq w 1 pλ τ , µq ´pB ν `iA 2 pνqq w 2 pλ τ , µq on BM.

Then, by (3.19), we obtain

(5.14) Kpτ, µ, f q " ÿ kě1 " pµ ´λ2 τ qxf, ψ 1,k y pλ 2 τ ´λ1,k qpµ ´λ1,k q ψ 1,k ´pτ ´λ2 τ qxf, ψ 2,k y pλ 2 τ ´λ2,k qpµ ´λ2,k q ψ 2,k  .
We define (5.15) Lpτ, µq " xKpτ, µ, ϕ 1,τ q, ϕ 2,τ y.

From (5.14), we get (5.16) Lpτ, µq "

ÿ kě1 pµ ´λ2 τ q " xϕ 1,τ , ψ 1,k yxψ 1,k , ϕ 2,τ y pλ 2 τ ´λ1,k qpµ ´λ1,k q ´xϕ 1,τ , ψ 2,k yxψ 2,k , ϕ 2,τ y pλ 2 τ ´λ2,k qpµ ´λ2,k q  .

Define

(5.17)

L ˚pτ q " ÿ kě1 L 1,k pτ q `ÿ kě1 L 2,k pτ q `ÿ kě1 L 3,k pτ q, with L 1,k pτ q " @ ϕ 1,τ , ψ 1,k ´ψ2,k D xψ 1,k , ϕ 2,τ y λ 2 τ ´λ1,k L 2,k pτ q " @ ϕ 1,τ , ψ 2,k D xψ 1,k ´ψ2,k , ϕ 2,τ y λ 2 τ ´λ1,k , L 3,k pτ q " @ ϕ 1,τ , ψ 2,k D xψ 2,k , ϕ 2,τ y ˆ1 pλ 2 τ ´λ1,k q ´1 pλ 2 τ ´λ2,k q ˙.
Lemma 5.2. Under assumption (1.17), Lpτ, µq converge to L ˚pτ q as µ Ñ ´8 and, for t P r0, 1{2q, we have (5.18) lim sup

τ Ñ8 τ ´t|L ˚pτ q| ď C}η} H 2 pS ỳ M1q lim sup kÑ8 k ´t{n |λ 1,k ´λ2,k |.
Proof. We split Lpτ, µq into three series Lpτ, µq "

ÿ kě1 L 1,k pµ, τ q `ÿ kě1 L 2,k pµ, τ q `ÿ kě1 L 3,k pµ, τ q, with L 1,k pτ, µq " pµ ´λ2 τ q xϕ 1,τ , ψ 1,k ´ψ2,k yxψ 1,k , ϕ 2,τ y pλ 2 τ ´λ1,k qpµ ´λ1,k q , L 2,k pτ, µq " pµ ´λ2 τ q xϕ 1,τ , ψ 2,k yxψ 1,k ´ψ2,k , ϕ 2,τ y pλ 2 τ ´λ1,k qpµ ´λ1,k q , L 3,k pτ, µq " pµ ´λ2 τ qxϕ 1,τ , ψ 2,k yxψ 2,k , ϕ 2,τ y ˆˆ1 pλ 2 τ ´λ1,k qpµ ´λ1,k q ´1 pλ 2 τ ´λ2,k qpµ ´λ2,k q ˙.
Under assumption (1.17) and in light of (5.1), we can see that the series in L 1,k pτ, µq, L 2,k pτ, µq and L 3,k pτ, µq converge uniformly with respect to µ ! ´1. Therefore, Lpτ, µq converge to L ˚pτ q as µ Ñ ´8.

We have

(5.19) |L 1,k pτ q| ď }ϕ 1,τ } L 2 pBMq }ψ 1,k ´ψ2,k } L 2 pBMq ˇˇˇx ψ 1,k , ϕ 2,τ y λ 2 τ ´λ1,k ˇˇˇ, (5.20) |L 2,k pτ q| ď }ϕ 1,τ } L 2 pBMq }ϕ 2,τ } L 2 pBMq }ψ 1,k ´ψ2,k } 2 L 2 pBMq |λ 2 τ ´λ1,k | `}ϕ 2,τ } L 2 pBMq }ψ 1,k ´ψ2,k } L 2 pBMq ˇˇˇx ϕ 1,τ , ψ 1,k y λ 2 τ ´λ1,k ˇˇˇ, (5.21) |L 3,k pτ q| ď }ϕ 1,τ } L 2 pBMq }ψ 1,k ´ψ2,k } L 2 pBMq |λ 2,k ´λ1,k | |λ 2 τ ´λ2,k | ˇˇˇx ψ 2,k , ϕ 2,τ y λ 2 τ ´λ2,k ˇˇ| λ 2,k ´λ1,k | ˇˇˇx ϕ 1,τ , ψ 1,k y λ 2 τ ´λ1,k ˇˇˇˇˇˇx ψ 2,k , ϕ
k 2t{n ˇˇˇx ψ 1,k , ϕ 2,τ y λ 2 τ ´λ1,k ˇˇˇ2 ¸1{2 ˆ˜8 ÿ k"n1 k ´2t{n }ψ 1,k ´ψ2,k } 2 L 2 pBMq ¸1{2 ď C ˜8 ÿ k"n1 k ´2t{n }ψ 1,k ´ψ2,k } 2 L 2 pBMq ¸1{2 ,
the constant C is independent on τ . Since the last term goes to zero as n 1 tends to 8 by (1.18), we easily get

k ´1{n }ψ 1,k ´ψ2,k } 2 L 2 pBMq `C ˜sup τ ą1 τ ´2t 8 ÿ k"1 k 2t{n ˇˇˇx ψ 1,k , ϕ 1,τ y λ 2 τ ´λ1,k ˇˇˇ2 ¸1{2 ˜8 ÿ k"n1 k ´2{n }ψ 1,k ´ψ2,k } 2 L 2 pBMq ¸1{2 ď C 8 ÿ k"n1 k ´2t{n }ψ 1,k ´ψ2,k } 2 L 2 pBMq `C ˜8 ÿ k"n1 k ´2t{n }ψ 1,k ´ψ2,k } 2 L 2 pBMq ¸1{2 .
Then, using again the fact that n 1 is arbitrary and (1.17 " S B1 pτ q ´SB2 pτ q ´żBM B ν w 1,2 pµqϕ 2,τ dσ n´1 .

According to Lemmas 3.2 and 5.2, formula (5.28) and passing to the limit as µ goes to ´8, we get (5.29) S B1 pτ q ´SB2 pτ q " L ˚pτ q.

Furthermore, from (5.18) we have τ ´t pS B1 pτ q ´SB2 pτ qq is bounded for τ ą 1 and t P r0, 1{2q. Then τ ´1 pS B1 pτ q ´SB2 pτ qq goes to zero as τ tends to 8. This in (4.40) yields,

(5.30)

ż S ỳ M1
´eiI1Apy,θq ´1¯η py, θq dω y pθq " 0.

Since η is arbitrary in H 2 pS y Mq, we obtain that I 1 Apy, θq P 2πZ for any θ P S ỳ M 1 .

On the other hand, since BM 1 is strictly convex, which implies that I 1 A " 0, because y P BM 1 is arbitrary. From (2.14), we deduce that the solenoidal part A s in the Hodge decomposition of the 1-form A is equal to zero. This completes the proof of the first part of Theorem 1.2. Now let us consider the second part of the theorem. For this purpose, we assume that condition (1.18) is fulfilled and we would like to show that q 1 " q 2 . Note first that the condition A s " 0 implies dA " 0 and, since M 1 is simply connected, there exists ϕ P W 3,8 pM 1 q such that dϕ " A. Since A " 0 on M 1 zM by eventually extracting a constant to ϕ we may assume that ϕ " 0 on M 1 zM . In particular we have ϕ |BM " B ν ϕ |BM " 0. Let B 3 " pA 1 , q 2 q. Applying (1.5), we deduce that e ´iϕ H B2 e iϕ " H B3 .

S ỳ M 1 Q θ Þ Ñ `
In particular, for λ 3,k , k ě 1, the non-decreasing sequence of eigenvalues of H B3 we have λ 3,k " λ 2,k and φ 3,k " e ´iϕ φ 2,k corresponds to an orthonormal basis of eigenfunctions of H B3 . Moreover, fixing ψ 3,k " pB ν `iA 2 pνqq φ 3,k , we deduce that ψ 3,k pxq " pB ν `iA 1 pνqq e ´iϕ φ 2,k pxq " e ´iϕ pB ν `iA 1 pνq ´iB ν ϕq φ 2,k pxq " pB ν `iA 2 pνqq φ 2,k pxq " ψ 2,k pxq, x P BM.

Combining this with (1.18), we deduce that lim kÑ`8 |λ 1,k ´λ3,k | " 0, and

ÿ kě1 }ψ 1,k ´ψ3,k } 2 L 2 pBMq ă 8.
In view of this gauge invariance property, from now on, without lost of generality, we may assume that A 1 " A 2 . According to (1.18), with t " 0, the right hand side of (5.18) is equal to zero. Since q P H 1 pM 1 q, by the smoothing effect of the normal operator N 0 " I 0 I 0 (see (2.21)), N 0 q P H 2 pM q and

(5.36) }N 0 pqq} H 2 pM1q ď C}q} H 1 pMq ď Cr 1 .

Since I 0 : H 2 pM 1 q Ñ H 2 pB `SM 1 q is bounded, we can take η " I 0 N 0 pqq. We integrate with respect to y P BM 1 the left hand side (5.35) the constant C only depends on M and r 1 . We complete the proof by using the interpolation inequality

}N 0 pqq} H 1 pM1q ď C}N 0 pqq} 1 2 L 2 pM1q }N 0 pqq} 1 2 H 2 pM1q ď C 1 }N 0 pqq} 1 2 L 2 pM1q ,
the constants C and C 1 only depend on M, r. We then apply (2.20) to get (1.20).

Extension to the Neumann case

We explain in this section how to adapt the preceding analysis to obtain an uniqueness result for an inverse spectral problem fo the Schrödinger operator under Neumann boundary condition.

For B " pA, qq P B, define the unbounded self-adjoint operator H B , acting in L 2 pMq as follows (6.1) H B u " H B u " ´∆A u `qu, u P DpH B q, with domain (6.2) DpH B q " u P H 1 pMq, ´∆A u `qu P L 2 pMq, pB ν `iApνqqu |BM " 0 ( .

Fix B P B r , " 1, 2 and denote by pµ ,k , χ ,k q, k ě 1, the eigenvalues and normalized eigenfunctions of H B .

We aim in this section to prove the following uniqueness result. Theorem 6.1. Assume that (1.16) and the conditions

(6.3) `8 ÿ k"1 }χ 1,k ´χ2,k } 2 L 2 pBM q ă 8, (6.4) lim kÑ`8 k ´1 n |µ 1,k ´µ2,k | " 0, are fulfilled. Then A s 1 " A s 2 .
Note that, according to Weyl's formula in [11, page 114], we have that lim kÑ`8

k ´1 n |µ 1,k ´µ2,k | ă 8.
Therefore, condition (6.4) seems to be the optimal rate of growth of the difference of eigenvalues that guaranty the uniqueness of the magnetic potential. Similarly to the Dirichlet case, for " 1, 2, define the N-to-D map

N ,λ : g P H 1 2 pBMq Þ Ñ v j pλq |BM ,
where v j pλq P H 2 pMq is the solution of the BVP (6.5). Define, For " 1, 2,

Q j pτ q " @ N j,λ 2 τ pB ν `iA j νqϕ 1,τ , pB ν `iA j νqϕ 2,τ D (6.6) " ż BM pB ν ´iA j νqϕ 2,τ N j,λ 2 τ pB ν `iA j νqϕ 1,τ dσ n´1 ,
with ϕ j,τ , j " 1, 2, given in (4.6). Proposition 6.2. We have

Q 1 pτ q " ż BM piλ τ qB ν ψβ 1 `ipA 1 νqβ 1 `Bν β 1 qβ 2 pxq dσ n´1 pxq (6.7) ´2λ τ ż SypM1q ż `py,θq 0 r σ A pr, y, θq r β 1 r β 2 1{2 dr dω y pθq ´żM β 1 H B1 pβ 2 q dv n `żM " pH B1 ´λ2 τ q ´1pe iλτ ψ H A1,q1 β 1 q ‰ e ´iλτ ψ " 2λ τ pA∇ψqβ 2 `HB1 β 2 ı dv n and Q 2 pτ q " ż BM piλ τ qB ν ψβ 1 `ipA 1 νqβ 1 `Bν β 1 qβ 2 pxq dσ n´1 pxq (6.8) ´żM β 1 H B2 pβ 2 q dv n `żM " pH B2 ´λ2 τ q ´1e iλτ ψ p2λ τ p´A∇ψqβ 1 `HB2 β 1 q ‰ ´e´iλτ ψ H B2 β 2 ¯dv n .
Proof. Applying Green's formula, we get

Q 1 pτ q " ż M divpv 1 pλ 2 τ q∇ A1 ϕ 2,τ q dv n " ż M A ∇ A1 v 1 pλ 2 τ q, ∇ A1 ϕ 2,τ E g dv n `żM v 1 pλ 2 τ q∆ A1 ϕ 2,τ dv n " ´żM ∆ A1 v 1 pλ 2 τ qϕ 2,τ dv n `żBM pB ν `iA 1 νqv 1 pλ 2 τ qϕ 2,τ dσ g `żM v 1 pλ 2 τ q∆ A1 ϕ 2,τ dv n
where v 1 pλ 2 τ q the solution of the BVP (6.5), with g " pB ν `iA 1 νqϕ 1,τ , λ " λ 2 τ , A " A 1 , q " q 1 . Using the fact that pB ν `iA 1 νqv 1 pλ 2 τ qpxq " gpxq " pB ν `iA 1 νqϕ 1,τ pxq, x P BM, we deduce that

Q 1 pτ q " ż BM piλ τ qB ν ψβ 1 `ipA 1 νqβ 1 `Bν β 1 qβ 2 pxq dσ n´1 pxq ´żM ∆ A1 v 1 pλ 2 τ qϕ 2,τ dv n `żM v 1 pλ 2 τ q∆ A1 ϕ 2,τ dv n .
This identity at hand, we proceed as in Lemma 4.1 to get (6.7). Similar arguments allows us to derive (6.8).

As for the derivation of (4.40), we obtain from (6.7) and (6.8) the following identity Q 2 pτ q ´Q1 pτ q (6.9) `żM " pH B2 ´λ2 τ q ´1e iλτ ψ p2λ τ p´A∇ψqβ 1 `HB2 β 1 q ‰ ´e´iλτ ψ H B2 β 2 ¯dv n , from which we deduce that, for all y P BM 1 and all η P H 2 pS ỳ M 1 q, (6.10) 2i ż S ỳ pM1q

"
´eiI1Apy,θq ´1¯η py, θqdω y pθq " lim τ Ñ`8

Q 2 pτ q ´Q1 pτ q τ .

The following lemma is needed in the proof of Theorem 6.1.

Lemma 6.3. For " 1, 2, consider ϕ j,τ , j " 1, 2, given by (4.6). Then, we have with C ą 0 independent of τ .

Proof. Let τ " }q 1 } L 8 pM q `}q 2 } L 8 pM q `1 and note that DppH B `τ q 1{2 q " H 1 pM q since it coincides with the domain of the form associated to the operator H B `τ . Whence, for any w P H 1 pM q, we have

8 ÿ k"1 p1 `|µ ,k |q|pw, χ ,k q L 2 pM q | 2 ď C}w} 2 H 1 pM q ,
the constant C only depends on τ , A , q and M. Combining this estimate with a Weyl's formula for Neumann magnetic operators, similar to that in Lemma A.1, we get (6.11).

6.2. End of the proof of Theorem 6.1. The following lemma is useful in the sequel Lemma 6.4. Let g P H 1{2 pBMq, B P B, λ P ρ pH B q and denote by vpλq the solution of the BVP (6.5). Then In light of this lemma, we have Q 2 pτ q ´Q1 pτ q (6.13)

" 8 ÿ k"1 @ pB ν `iA 1 νqϕ 1,τ , χ 2,k D @ χ 2,k , pB ν `iA 1 νqϕ 2,τ D λ 2 τ ´µ2,k
´@pB ν `iA 1 νqϕ 1,τ , χ G k pτ q.

We combine these inequalities, estimates (6.11) and Weyl's formula in order to get, by repeating the arguments used to prove Lemma 5.2, that lim sup τ Ñ`8

ˇˇˇQ 2 pτ q ´Q1 pτ q τ ˇˇˇď Cp1 `}η} H 2 pS ỳ pM1qq q 2 plim sup kÑ`8 k ´1 n |µ 2,k ´µ1,k |q.

Then, from (6.4) and (6.10) we deduce that I 1 A P 2πZ. We proceed similarly to the proof of Theorem 1.2 to get that A s 1 " A s 2 . This completes the proof of Theorem 6.1.

Appendix A. Weyl's formula

We establish some uniform estimates related to the Weyl's formula of magnetic Schrödinger operators. Our estimates, which are also valid for the Neuman realization of magnetic Schrödinger operators, can be stated as follows.

Lemma A.1. Let B " pA, qq P B. Then there exists a constant C ą 1, only depending on M and r ě }A} 2 L 8 pM,T ˚Mq `}q} L 8 pMq so that (A.1) C ´1k 2{n ď 1 `|λ k B | ď Ck 2{n , k ě 1 Proof. Let pλ k q be the sequence of eigenvalues, counted according to their multiplicities, of the Laplace-Belrami operator under Dirichlet boundary condition. By Weyl's asymptotic formula [11, We get the expected two-sided inequalities (A.1) by using (A.2) and the minmax principle.

3 ) and lim λÑ´8 }u pλq} 2 L 2 ( 3

 3223 pMq " 0 is quite similar to that of[START_REF] Kian | A multidimensional Borg-Levinson theorem for magnetic Schrödinger operators with partial spectral data[END_REF] Lemma 2.1]. The proof of (3.4) is then completed by establishing the following Caccioppoli's type inequality, where λ ă 0:(3.5) }κdu pλq} L 2 pMq ď C}u pλq} L 2 pMq ,the constant C only depends on r and M.For the sake of simplicity, we omit the subscript in u pλq and B . Multiplying the first equation of (3.2) by κ 2 upλq, using the fact that κ |BM " 0 and applying Green's formula, we obtain 0 " ´żM ∆ A upλqκ 2 upλq dv n `żM pq ´λqκ 2 |upλq| 2 dv n

( 4 . 2 ż 2 ż

 422 43) 2 ż M xA, dψypα 1 α 2 qpxqpβ A1 β A2 qpxq dv n " pr, y, θqpr α 1 r α 2 qpr, θqp r β A1 r β A2 qpr, θq 1{2 dr dω y pθq " pr, y, θq r β A1 pr, θq r β A2 pr, θqηpy, θqdr dω y pθq " ż S ỳ M1 ż `py,θq 0 r σ A pr, y, θq exp ˜i ż `py,θq 0 r σ A pr `s, y, θqds ¸ηpy, θqdr dω y pθq. dψypα 1 α 2 qpxqpβ A1 β A2 qpxq dv n " 2i ż S ỳ M1 pexp piI 1 Apy, θqq ´1q ηpy, θqdω y pθq.

1 |L 1 , 2 τ ´r| ě 2 ?

 1122 k pτ q| " 0.In the sequel, we use the following useful observation: for r ą 1 the map τ Þ Ñ |λ 2 τ ´r| reach its minimum at τ " ? r ´1. Hence |λ r ´1, τ ą 0.This observation together with (5.1), (5.20) and (A.1) in Appendix A

6. 1 .

 1 Boundary representation formulae for the Neumann problem. For g P H 1{2 pBMq and ρpH B q, pB ν `iAνqv " g on BM.

2 nˇˇˇˇ2 ă C}η} 2 H 2 2 n

 2222 ˇˇˇˇ@ ϕ 1,τ , χ ,k D µ ,k ´λ2 τ pS ỳ pM1qq τ 2 , ˇˇˇˇˇA ϕ 2,τ , χ ,k E µ ,k ´λ2 τ ˇˇˇˇˇ2 ď Cτ 2 , " 1, 2

(6. 12 )

 12 vpλq |BM " ÿ kě1 xg, χ k y λ ´µk χ k ,the convergence takes place in H 1{2 pBMq.

  an elliptic pseudodifferential operator of order ´1 on O having as principal symbol px, ξq " p jk px, ξqq 1ďj,kďn , where Therefore, for each integer k ě 0, there exists a constant C k ą 0 such that, for any A P H k pM, T ˚Mq compactly supported in O, we have(2.15) }N 1 pAq} H k`1 pM1q ď C k }A s } H k pOq .

	j,k px, ξq "	c n |ξ|	ˆgjk	|ξ| 2 ´ξj ξ k	˙.

2.2. Geodesic ray transform of functions. Following [36, Lemma 4.1.1], the ray transform of functions is the linear operator (2.16) I 0 : C 8 pMq ÝÑ C 8 pB `SMq acting as follows (2.17)

  Using that pH B1´µ˚q´1 is an isomorphism from L 2 pMq onto H 2 pMq, there exists a constant C, depending on M and B 1 , so that}wpµq} H 2 pMq ď C}hpµq `pµ ´µ˚q wpµqq} L 2 pMq ď C `}hpµq} L 2 pMq `|µ ´µ˚| }wpµq} L 2 pMq ď C `}hpµq} L 2 pMq `2|µ|}wpµq} L 2 pMq ˘, (3.13)where the positive constant C is not dependent on µ. Using now the estimate (3.11), we obtain(3.14) }wpµq} H 2 pMq ď 4C}hpµq} L 2 pMq .

	We deduce that, for ´µ sufficiently large,
	p´}q} 8	´µ 2	q}wpµq} 2 L 2 pMq	`|µ| 4	}wpµq} 2 L 2 pMq ď C}hpµq} 2 L 2 pMq ,
	for some positive constant C, not dependent on µ, and then we conclude that
	(3.11)				|µ|}wpµq} 2 L 2 pMq ď C}hpµq} 2 L 2 pMq .
	Moreover we have
		$
	(3.12)	&	pH B1 ´µ˚q wpµq " hpµq `pµ ´µ˚q wpµq in M,
		%	wpµq " 0	on BM.
						1, we
	get
	(3.17)				lim sup µÑ´8	}hpµq} L 2 pMq " 0,
	entailing by (3.14)
	(3.18)				lim sup
						in
	order to obtain
	ż

M hpµqwpµq dv n " ż M H B1 wpµqwpµq dv n ´żM µ|wpµq| 2 dv n " ż M |∇ A1 w| 2 dv n `żM pq ´µq|w| 2 dv n . On the other hand, in view of (1.16) there exists C ą 0 such that (3.15) |A 1 pxq ´A2 pxq| ď Cκpxq, x P M. Applying (3.15), we obtain (3.16) }hpµq} L 2 pMq ď C 2 `}κdu 2 pµq} L 2 pMq `}u 2 pµq} L 2 pMq for some constant C 2 independent of µ. Then, according to (3.4) in Lemma 3.µÑ´8 }wpµq} H 2 pMq " 0.

  " e iλτ ψpxq pH B2 pβ 1 q ´2λ τ xA, dψyβ 1 q " e iλτ ψpxq k 2 .

					2iλ
					dα 1 y	`α1 2	∆ψ
					Ṫaking
			τ α 1 pxdψ, dβ A1 y `ixA 2 , dψyβ A1 q
	into account (4.3)-(4.4) and (4.5), the right-hand side of (4.17) takes the
	form
	(4.18) ,τ Let u 2 be the solution of the BVP `HB2 ˘ϕ1 ´λ2 τ
			$
			&	`HB2	´λ2 τ ˘u2 " 0	in M,
			%	u 2 " ϕ 1,τ	on BM.
					solution of
	the BVP	$
		&	`HB2	´λ2 τ ˘v2 " ´eiλτ ψ k 2	in M
		%	v 2 " 0	on BM.
	Therefore
	(4.19)

As for u 1 , we split u 2 into two terms, u 2 " ϕ 1,τ `v2 , where v 2 is the

  } L 2 pBMq ď }β 1 } L 2 pBMq ď C}η} H 2 pSyM1q } L 2 pBMq ď }β 1 } L 2 pBMq ď C,the constant C only depends on M. This estimate entails in particular that

	But (5.22) }ϕ 1,τ and sup τ ą1 (5.23) sup τ Ñ`8 Thus, for an arbitrary positive integer n 1 , we get τ ´t|L 1,k pτ q| " 0, k ě 1. lim sup τ Ñ`8 τ ´t 8 ÿ k"1 |L 1,k pτ q| " lim sup τ Ñ`8 τ ´t 8 ÿ k"n1 This estimate together with (5.1), (5.19), (5.22) and (5.23) imply |L 1,k pτ q|. τ ´t 8 ÿ |L 1,k pτ q| ď C ˜sup τ ´2t 8 ÿ }ϕ 2,τ lim sup k"n1 τ ą1 k"1	2,τ y τ ´λ2,k λ 2	ˇˇˇ.
	τ ą1

  |L 3,k pτ q| ď C}η} H 2 pS ỳ M1q lim sup Kpτ, µ, ϕ 1,τ q " pB ν `iA 1 pνqqu 1 pλq´pB ν `iA 2 pνqqu 2 pλq´B ν w 1,2 pµq on BM. `iA 1 pνqq u 1 pλqϕ 2,τ dσ n´1 ´żBM pB ν `iA 2 pνqq u 2 pλqϕ 2,τ dσ n´1 ´żBM B ν w 1,2 pµqϕ 2,τ dσ n´1 B1 pλ 2 τ qϕ 1,τ ϕ 2,τ dσ n´1 ´żBM Λ B2 pλ 2 τ qϕ 1,τ ϕ 2,τ dσ n´1 ´żBM B ν w 1,2 pµqϕ 2,τ dσ n´1

	The expected result follows from (5.24), (5.25) and (5.26).
	5.2. End of the proof of the main results. We are now ready to complete the
	proof of Theorems 1.2 and 1.3.
	Proof of Theorem 1.2. Since A , " 1, 2, satisfy (1.16) and w 1,2 pµq " 0 on BM, we
	easily obtain the following identity, useful in the sequel,
	(5.27) By formula (5.15) we get
			ż
	(5.28) Lpτ, µq "		Kpτ, µ, ϕ 1,τ qϕ 2,τ dσ n´1
			BM
	ż
	"	pB ν
	BM
						), we find
	(5.25)			lim sup	τ	´t 8 ÿ	|L 2,k pτ q| " 0.
				τ Ñ`8		k"1
	The same argument as before enables us to obtain
	(5.26)	lim sup	τ	´t 8 ÿ		k ´t{n |λ 1,k ´λ2,k |.
		τ Ñ`8		k"1		kÑ`8

" ż BM Λ

  py, θq is continuous, and letting θ tend to a tangent direction θ 0 P S y BM 1 we get

		lim θÑθ0	`py, θq " 0
	hence
		2πm " lim θÑθ0	I 1 Apy, θq " 0
	and therefore
	(5.31)	I 1 Apy, θq " 0, θ P S ỳ M 1

  Proof of Theorem 1.3. We already proved that dA 1 " dA 2 in Theorem 1.2 and according to the gauge invariance property of the boundary spectral data, without lost of generality, we may assume that A 1 " A 2 . Then a straightforward application of the min-max principle yields(5.32) |λ 1,k ´λ2,k | ď }q 1 ´q2 } L 8 pMq . 0 pqqpy, θqηpy, θq dσ 2n´2 ˇˇˇď C}η} H 2 pS ỳ M1q lim sup

	In that case (1.19) is reduced to
	(5.33)		ÿ	}ψ 1,k ´ψ2,k } 2 L 2 pBMq ă 8.
			kě1
	Combining this with (4.45), (5.18) for t " 0 (which is valid in the present case) and
	taking into account that
	(5.34)	lim sup	|S B1 pτ q ´SB2 pτ q| " lim sup	|L ˚pτ q|,
		τ Ñ`8		τ Ñ`8
	we obtain, for any η P H 2 pS ỳ M 1 q real valued, that
	(5.35) ˇˇˇż	S ỳ M1			kÑ`8	|λ 1,k ´λB k 2 |.

I

  }I 0 N 0 pqq} H 2 pB`SM1q ď C}N 0 pqq} H 2 pM1q ď C 1 ,

	in order to get |N 0 pqq| 2 dv n " }N 0 pqq} 2 L 2 pM1q . the constants C and C 1 only depend on M and r. This (5.37) and (5.38), give ż B`SM1 I 0 pqqpy, θqηpy, θq dσ 2n´2 " ż M1 Combined with (5.35), this inequality entails (5.37) }N 0 pqq} 2 (5.38) L On the other hand, it follows from (5.36) (5.39) }N 0 pqq} 2 kÑ`8 L 2 pM1q ď C lim sup |λ 1,k ´λ2,k |,

2 pM1q ď C}I 0 N 0 pqq} H 2 pB`SM1q lim sup kÑ`8 |λ 1,k ´λ2,k |.

  2λ τ 1{2 dr dω y pθq `żM pq 1 ´q2 qβ 1 β 2 dv n pxq ´żM β 1 p∆ A1 β 2 ´∆A2 β 2 q dv n

	ż	ż	`py,θq
	SypM1q β 2 ´żM 0 r σ A pr, y, θq r β 1 r " pH B1 ´λ2

τ q ´1pe iλτ ψ H B1 β 1 q ‰ e ´iλτ ψ " 2λ τ pA∇ψqβ 2 `HB1 β 2 ı dv n .

  1,k D @ χ 1,k , pB ν `iA 1 νqϕ 2,τ DObserve that, according to (1.16), A 1 can be substituted by A 2 in the identity above.On the other hand, we have from (4.6)}pB ν `iA 1 νqϕ j,τ } L 2 pBMq ď |λ τ |}B ν ψβ j } L 2 pBMq `}pB ν `iA 1 νqβ j } L 2 pBMq (6.14)ď Cτ p1 `}η} H 2 pSyM1q q, the constant C being independent of τ . Thus,|Q 2 pτ q ´Q1 pτ q| ď pτ q " }pB ν `iA 1 νqϕ 1,τ } L 2 pBMq }χ 2,k ´χ1,k } L 2 pBMq | @ χ 2,k , pB ν `iA 1 νqϕ 2,τ C}χ 2,k ´χ1,k } L 2 pBMq τ | @ χ 2,k , pB ν `iA 1 νqϕ 2,τ F k pτ q " }pB ν `iA 1 νqϕ 2,τ } L 2 pBMq }χ 2,k ´χ1,k } L 2 pBMq | @ pB ν `iA 1 νqϕ 1,τ , χ 1,k C}χ 2,k ´χ1,k } L 2 pBMq τ | @ pB ν `iA 1 νqϕ 1,τ , χ 2,k Ck ´1 n |µ 2,k ´µ1,k |}χ 1,k ´χ2,k } L 2 pBMq k , pB ν `iA 1 νqϕ 2,τ

	the constant C ą 0 being independent on τ and k.
	Noting that
				sup τ ą1	τ 2 τ ´µ ,k | |λ 2	ă 8,	" 1, 2, k ě 1,
	we deduce that we have, for all k ě 1,
			lim sup	τ ´1E k pτ q " lim sup	τ ´1F k pτ q " lim sup	τ ´1G k pτ q " 0.
			τ Ñ`8							τ Ñ`8	τ Ñ`8
	Then, for any arbitrary integer n 1 ě 1, we get
				lim sup	τ	´1 8 ÿ	E k pτ q ď lim sup	τ	ÿ ´1 8	E k pτ q,
				τ Ñ`8					k"1	τ Ñ`8	k"n1
				lim sup	τ	´1 8 ÿ	F k pτ q ď lim sup	τ	ÿ ´1 8	F k pτ q,
				τ Ñ`8					k"1	τ Ñ`8	k"n1
				lim sup τ Ñ`8	τ	´1 8 ÿ k"1	λ 2 τ ´µ1,k G k pτ q ď lim sup τ ÿ ´1 8 τ Ñ`8 k"n1	.
											8
											ÿ	E k pτ q	`8 ÿ	F k pτ q	`8 ÿ
											k"1	k"1	k"1
											D	|
											|λ 2 τ ´µ2,k |
	ď D |λ 2 τ ´µ2,k |	|	,
											D	|
											|λ 2 τ ´µ2,k |
	ď D |λ 2 τ ´µ2,k |	|	`Cτ 2 }χ 1,k ´χ2,k } 2 L 2 pBMq
	and
	G k pτ q "	|	@ pB ν `iA 1 νqϕ 1,τ , χ 1,k |λ 2 D τ ´µ2,k ||λ 2 || @ χ 1,k , pB ν `iA 1 νqϕ 2,τ τ ´µ1,k |	D	||µ 2,k ´µ1,k |
	ď 1 n	|	@ pB ν `iA 1 νqϕ 1,τ , χ 1,k |λ 2 τ ´µ1,k |	D	|
	`k´1 n |µ 2,k ´µ1,k |k	1 n	|	@ pB ν `iA 1 νqϕ 1,τ , χ 1,k |λ 2 τ ´µ1,k |	D	|	|	A χ 2,k E |λ 2 τ ´µ2,k |	|	,

G k pτ q, with E k

  A vy dv n `żM quv dv n , u, v P H 1 0 pMq. Then it is not hard to check that apu, uq ď }du} 2 L 2 pMq `2? r}u} L 2 pMq }du} L 2 pMq `r}u} 2 ´2? r}u} L 2 pMq }du} L 2 pMq ´r}u} 2

	and
	apu, uq ě }du} 2 L 2 pMq	L 2 pMq
	ě	1 2	}du} 2 L 2 pM q ´3r}u} 2 L 2 pMq .
			page 114]
	(A.2)		λ k " O	´k 2 n ¯, k ě 1.
	The sesquilinear form associated to H B is given by
	ż
	apu, vq "
				L 2 pMq
	ď	3 2	}du} 2 L 2 pMq `r}u} 2 L 2 pMq

M

x∇ A u, ∇

MC and YK are supported by the grant ANR-17-CE40-0029 of the French National Research Agency ANR (project MultiOnde).

DDSF is partly supported by ANR grant iproblems. PS partly supported by NSF Grant DMS-1600327.