Damien Gobin
email: damien.gobin@mcgill.ca.

Inverse scattering at fixed energy for radial magnetic Schrödinger operators with obstacle in dimension two

Keywords: Inverse Scattering, radial magnetic Schrödinger operators, phase shifts. 2010 Mathematics Subject Classification. Primaries 81U40, 35P25; Secondary 58J50

We study an inverse scattering problem at fixed energy for radial magnetic Schrödinger operators on R 2 \ B(0, r0), where r0 is a positive and arbitrarily small radius. We assume that the magnetic potential A satisfies a gauge condition and we consider the class C of smooth, radial and compactly supported electric potentials and magnetic fields denoted by V and B respectively. If (V, B) and ( Ṽ , B) are two couples belonging to C, we then show that if the corresponding phase shifts δ l and δl (i.e. the scattering data at fixed energy) coincide for all l ∈ L, where L ⊂ N ⋆ satisfies the Müntz condition l∈L 1 l = +∞, then V (x) = Ṽ (x) and B(x) = B(x) outside the obstacle B(0, r0). The proof use the Complex Angular Momentum method and is close in spirit to the celebrated Börg-Marchenko uniqueness Theorem.

Introduction and statement of the main result

In this work we are interested in an inverse scattering problem at fixed energy for magnetic Schrödinger operators on R 2 \ B(0, r 0 ) where B(0, r 0 ) is an obstacle whose radius r 0 is positive and arbitrarily small. In quantum scattering theory, we study a pair of Hamiltonians (-∆, H A,V ) on L 2 (R 2 ), where H A,V denotes the quantum magnetic Schrödinger Hamiltonian which describes the interaction of a charged particle with an electric field ∇V and a magnetic field B = dA. The object of main interest is then the scattering operator S. Since the scattering operator S commutes with the free operator -∆ it can be reduced to a multiplication by an operator-function S(λ), called the scattering matrix at energy λ, in the spectral representation of the Hamiltonian -∆. The question we usually adress in an inverse scattering problem at fixed energy is the following:

Does the scattering matrix S(λ) at a fixed energy λ > 0 uniquely determine the electric potential V and the magnetic field B outside the obstacle?

The aim of this paper is then to prove that, up to a gauge choice, we can answer positively to this question for radial, smooth and compactly supported electric potentials and magnetic fields. Roughly speaking, we will thus show that, if the magnetic potential A satisfies a particular gauge condition, for such electric potentials and magnetic fields, if λ > 0 is a fixed energy: S(λ) = S(λ) ⇒ V (x) = Ṽ (x) and B(x) = B(x), ∀x ∈ R 2 \ B(0, r 0 ).

Inverse scattering at fixed energy for (magnetic) Schrödinger operators has been largely studied since the end of the twentieth century and is a tricky question. For instance, concerning Schrödinger operators (with no magnetic field), even if for exponentially decreasing potentials (see Novikov's papers [START_REF] Novikov | The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator[END_REF][START_REF] Novikov | The inverse scattering problem on a fixed energy level for the three-dimensional Schrödinger equation with an exponentially decreasing potential[END_REF]) or for particular classes of radial potentials (see Daudé and Nicoleau's paper [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF]), we can answer positively to this question, we emphasize that, in general, the answer is negative. Indeed, in dimension two, Grinevich and Novikov construct in [START_REF] Grinevich | Transparent potentials at fixed energy in dimension two. Fixed energy dispersion relations for the fast decaying potentials[END_REF] a family of real spherically symmetric potentials in the Schwartz space such that the associated scattering matrices are equal to the identity (transparent potentials). We also mention the work [START_REF] Sabatier | Asymptotic properties of the potentials in the inverse scattering problem at fixed energy[END_REF] of Sabatier where a class of radial transparent potentials is obtained in the three-dimensional case.

The main tool of this paper consists in complexifying the angular momentum that appears in the reduction of the Schrödinger operator into a countable family of one-dimensional radial equations, i.e. in the separation of variables procedure. Indeed, thanks to variables separation, the scattering matrix at fixed energy can be decomposed onto a family of scattering coefficients δ l , called the phase shifts, indexed by a discrete set of angular momentum l ∈ Z. The approach consisting in complexifying the angular momentum is called the Complex Angular Momentum (CAM) method. The idea of this method is the following. We first allow the angular momentum l ∈ Z to be a complex number ν ∈ C. In some cases it is then possible to extend the equality δ l = δl for all l ∈ Z into the equality δ(ν) = δ(ν) for all ν ∈ C \ {poles}. Indeed, functions in some particular classes of holomorphic functions, are completely determined by its values on a sufficiently large subset of the integers (Nevanlinna's class). We then use this new amount of informations to get the equality between the electric potentials and the magnetic fields. The general idea of considering complex angular momentum originates from a paper of Regge (see [START_REF] Regge | Introduction to complex orbita momenta[END_REF]) as a tool in the analysis of the scattering matrix of Schrödinger operators in R 3 with spherically symmetric potentials. We also refer to [START_REF] De Alfaro | Potential scattering[END_REF][START_REF] Newton | Scattering theory of waves and particles[END_REF] for books dealing with this method. We mention that this tool was already used in the field of inverse problems for one angular momentum in [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF][START_REF] Loeffel | On an inverse problem in potential scattering theory[END_REF][START_REF] Ramm | An inverse scattering problem with part of the fixed-energy phase shifts[END_REF] for Schrödinger operators, in [START_REF] Daudé | Direct and inverse scattering at fixed energy for massless charged Dirac fields by Kerr-Newman-De Sitter black holes[END_REF][START_REF] Daudé | Inverse scattering at fixed energy in de Sitter-Reissner-Nordström black holes[END_REF] in the context of general relativity, in [START_REF] Daudé | Local inverse scattering results at fixed energy in spherically symmetric asymptotically hyperbolic manifolds[END_REF][START_REF] Daudé | Inverse scattering at fixed energy on asymptotically hyperbolic Liouville surfaces[END_REF] on asymptotically hyperbolic manifolds and in [START_REF] Daudé | Non-uniqueness results for the anisotropic Calderon problem with data measured on disjoint sets[END_REF] to study counterexamples for the Calderón problem which is closely related to inverse scattering problems at fixed energy on asymptotically hyperbolic manifolds. Moreover, this method was also used for two angular momenta in [START_REF] Gobin | Inverse scattering at fixed energy on three-dimensional asymptotically hyperbolic Stäckel manifolds[END_REF] and we note that it is also a useful tool in high energy physics (see [START_REF] Collins | An introduction to Regge theory and high energy physics[END_REF]). This work is a continuation and is really close to the spirit of the paper [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF] of Daudé and Nicoleau in which the authors treat the same question for the Schrödinger operators with no magnetic fields in all dimensions and for particular classes of radial potentials.

Description of our framework

In this work, we study an inverse scattering problem for magnetic Schrödinger operators on the region Ω = R 2 \ B(0, r 0 ), where r 0 > 0 is fixed. We thus study the Hamiltonian,

H = (D -A) 2 + V = -∆ + 2iA.∇ + A 2 + idiv(A) + V.
where D = 1 i ∇, V is the electric potential and A = A 1 dx 1 + A 2 dx 2 is the 1-form corresponding to the magnetic potential. The magnetic field B is the 2-form defined as B = dA which can be identified with the antisymmetric 2 × 2 matrix

B = dA = 0 b -b 0 , b j,k = ∂ xj A k -∂ x k A j .
We then need to add a transversal gauge condition on the magnetic potential A given by the following Definition.

Definition 1.1 (Gauge choice). We say that a magnetic potential A belongs to the class A if it is smooth on R 2 and satisfies the gauge condition

A(x) = - 1 0 sB(sx).x ds.
From now on, we assume that A ∈ A. Thanks to this gauge condition we obtain the following Lemma.

Lemma 1.1. Assume that A ∈ A, then

(Transversality condition)

A(x).x = 0.

A satisfies

A(x) = γ(r, θ) r (-sin(θ), cos(θ)), (1.1)
where

γ(r, θ) = r 0 b(τ cos(θ), τ sin(θ))τ dτ. (1.2)
We now summarize the assumptions we need on the electric potential V and the magnetic field B in the following Definition.

Definition 1.2. We say that a couple (V, B) where V is an electric potential and B is a magnetic field associated with a magnetic potential A ∈ A belongs to the class C if:

1. V and B are both radial.

2. V and B are both compactly supported.

3.

V is piecewise continuous and B is smooth.

From now on, we will always assume that (V, B) is a couple of C. For such a couple we can prove the following Lemma.

Lemma 1.2. Assume that (V, B) ∈ C, then:

1. γ(r, θ) = γ(r).

2. div(A) = 0.

3. A 2 = γ(r) 2 r 2
is a radial function.

A(x)

.∇ = γ(r) r 2 D θ , where D θ = -i∂ θ . Thanks to these properties we obtain that the Hamiltonian H can be rewritten as

H = -∆ - 2γ(r) r 2 D θ + γ(r) 2 r 2 + V (r). (1.3)
We then know (see [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF][START_REF] Reed | Methods of modern mathematical physics[END_REF]) that the Hamiltonian H can be reduced into a countable family of radial Hamiltonians. Indeed, let us introduce the decomposition

L 2 (R 2 ) = L 2 (R + , rdr) ⊗ L 2 (S 1 , dσ).
For functions of the form u(x) = f (r)g(θ), where r = |x| > 0 and θ = x r ∈ S 1 , we then obtain

Hf (r)g(θ) = - d 2 dr 2 - 1 r d dr + γ(r) 2 r 2 + V (r) + ∆ S 1 - 2γ(r) r 2 D θ f (r)g(θ), (1.4)
where the operator ∆ S 1 is the Laplace-Beltrami operator on the circle S 1 . Its eigenvalues are given by -l 2 , for l ∈ Z. It then follows that

L 2 (R + , rdr) ⊗ L 2 (S 1 , dσ) = ⊕ l∈Z L 2 (R + , rdr) ⊗ K l ,
where K l is the invariant subspace of the eigenspace of ∆ S 1 , associated with the eigenvalue -l 2 for l ∈ Z, generated by e ilθ . Therefore, by introducing

L l = L 2 (R + , rdr) ⊗ e ilθ , l ∈ Z,
we thus obtain that the restriction of the Hamiltonian H on each subspace L l is given by

H |L l = - d 2 dr 2 - 1 r d dr + l 2 r 2 - 2lγ(r) r 2 + γ(r) 2 r 2 + V (r).
Finally, we can get rid of the first order term by defining the unitary operator

U : L 2 (R + , rdr) → L 2 (R + , dr) f → r 1 2 f (r) .
Indeed, conjugating H |L l by U , we then obtain a new family of radial Hamiltonians U H |L l U -1 , denoted by H l , given by

H l := - d 2 dr 2 + l 2 -1 4 r 2 - 2lγ(r) r 2 + γ(r) 2 r 2 + V (r). (1.5) Remark 1.3.
In comparison with the non-magnetic case studied in [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF] we note that the new term in the potential is -2lγ(r)

r 2 + γ(r) 2 r 2 .
The main fact to note is the presence of l in the potential that makes it necessary to obtain better estimates in terms of l on the Green kernels than the ones obtained in [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF].

Since we assumed that the magnetic field B and the electric potential V have compact supports included in a ball B(0, R), R > 0, we note (see (1.2)) that

γ(r) = γ(R) = Flux(B) 2π , ∀r ≥ R, (1.6)
where Flux(B) denotes the magnetic flux of B. Therefore,

H l = - d 2 dr 2 + l 2 -1 4 r 2 - 2lγ(R) r 2 + γ(R) 2 r 2 = - d 2 dr 2 + (l -γ(R)) 2 -1 4 r 2 , ∀r ≥ R.
Remark 1.4. We note that even if R ≤ r 0 , i.e. in particular that the support of the magnetic field B is included in the obstacle B(0, r 0 ), it still has an influence outside the obstacle since the magnetic flux appears in the corresponding Hamiltonian. This is a consequence of the Aharonov-Bohm effect. In this case we can only hope to reconstruct the flux of the magnetic field.

Since we work on Ω = R 2 \ B(0, r 0 ), the previous equality tells us that the reference operator we should use to study the dynamic of H l is given by

H 0 l = - d 2 dr 2 + (l -γ(R)) 2 -1 4 r 2 , ∀r ≥ r 0 .
For the sake of clarity we introduce two arguments

ν := l and ν R := ν -γ(R) = l -γ(R).
We can then rewrite H ν in terms of H 0 ν by the following way

H ν = H 0 ν + q ν (r) (1.7)
where the potential

q ν (r) := - 2ν(γ(r) -γ(R)) r 2 + γ(r) 2 -γ(R) 2 r 2 + V (r),
vanishes for all r ≥ R.

Remark 1.5. Here is a reason why it is easier to work on Ω instead of R 2 . Indeed, the potential q ν has a quadratic singularity as r tends to 0 and thus doesnot satisfy the hypothesis (H 1 ) introduced in [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF].

It is this singularity that prevent us from obtain the corresponding inverse scattering result on the whole plane R 2 .

Statement of the main result

The aim of this section is to introduce the scattering data we will study in the following in order to state our main result. We refer the reader to [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF][START_REF] Loeffel | On an inverse problem in potential scattering theory[END_REF] for more details. First of all we recall that we study the family of Hamiltonians H ν given by (1.5) on Ω = R 2 \ B(0, r 0 ) where r 0 > 0 is fixed. Following Regge's idea, we consider the radial Schrödinger equation on (r 0 , +∞) at the fixed energy λ = 1, where the angular momentum ν = l is now assumed to be a complex number,

-u ′′ + (ν R ) 2 -1 4 r 2 + q ν (r) u = u. (1.8)
We note that when ν R = νγ(R) = lγ(R) we recover the family of Hamiltonians (1.5) obtained previously in the separation of variables procedure. First, since the potential q ν is compactly supported, we can define the Jost solutions F ± (r, ν) as the unique solutions of (1.8) satisfying the boundary conditions at r = +∞ F ± (r, ν) ∼ e ±ir , r → +∞.

(1.9)

For every fixed r ≥ r 0 , the maps ν → F ± (r, ν) and ν → F ± ′ (r, ν) are holomorphic on the whole complex plane C. Moreover,

F ± (r, ν) = F ∓ (r, ν), ∀ν ∈ C,
and we note that the pair of Jost solutions is a fondamental system of solutions of (1.8). Secondly, we define the regular solution, denoted by Φ(r, ν), as the solution of (1.8) satisfying the Dirichlet condition at r = r 0 : Φ(r, ν) |r=r0 = 0.

Since the pair of Jost solutions is a fondamental system of solutions of (1.8) there exists two functions of ν, α and β, such that Φ(r, ν) = α(ν)F + (r, ν) + β(ν)F -(r, ν).

(1.10) Moreover, by uniqueness, Φ(r, ν) is in fact given, up to a multiplicative constant, by

Φ(r, ν) = i F -(r 0 , ν)F + (r, ν) -F + (r 0 , ν)F -(r, ν) .
We thus note that for every fixed r ≥ r 0 the maps ν → Φ(r, ν) and ν → Φ ′ (r, ν) are holomorphic on the complex plane C and moreover Φ(r, ν) = Φ(r, ν), ∀ν ∈ C.

The functions

α(ν) = iF -(r 0 , ν) and β(ν) = -iF + (r 0 , ν)
are then called the Jost functions. It follows from (1.9) that

W (F + (r, ν), F -(r, ν)) = -2i,
where the Wronskian of two functions u and v is given by W

(u, v) = uv ′ -u ′ v. Hence, α(ν) = i 2 W (Φ(r, ν), F -(r, ν)), and
β(ν) = - i 2 W (Φ(r, ν), F + (r, ν)).
From the definition of the Jost functions we immediately deduce that these functions are holomorphic on the complex plane C and satisfy α(ν) = β(ν).

(1.11)

We can now introduce the so-called Regge interpolation function

σ(ν) = e iπ(ν+ 1 2 ) α(ν) β(ν) .
We note that, when ν ∈ R, it follows from (1.11) that |σ(ν)| = 1. We can thus define the generalized phase shifts δ(ν) as a continuous function on R through the relation ν) .

σ(ν) = e 2iδ(
(1.12)

We emphasize that the quantities δ(ν) for ν = l, where l ∈ Z, are related to the physical phase shifts and are quantities that we can measure by a scattering experiment. The question we adress is then the following:

Does the phase shifts δ l uniquely determine the electric potential V and the magnetic field B?

The aim of this paper is thus to prove the following Theorem.

Theorem 1.6. Let (V, B) and ( Ṽ , B) be two couples belonging to the class C (see Definition 1.2). Let L ⊂ N ⋆ be a subset satisfying the Müntz condition

l∈L 1 l = +∞.
Assume that δ l = δl , ∀l ∈ L.

Then V (x) = Ṽ (x) and B(x) = B(x), ∀|x| ≥ r 0 .
Remark 1.7.

1. We emphasize that in the proof of Theorem 1.6 we first reconstruct the function γ(r) for all r ≥ r 0 and we can thus reconstruct the magnetic potential A. It is not surprising that we can actually reconstruct the magnetic potential A since we assumed that it satisfies a gauge condition given by Definition 1.1.

2. If we assume that the support of B is included in the ball B(0, r 0 ) we reconstruct the electric potential and the magnetic flux and we cannot hope to obtain more informations. We emphasize that this is a consequence of the Aharonov-Bohm effect which roughly speaking tells us that the magnetic field has an influence outside the ball B(0, r 0 ) even if its support is included in B(0, r 0 ).

3.

As mentionned previously we really need to introduce a parameter r 0 to avoid the neighborhood of r = 0. Indeed, our potential

q ν (r) = - 2ν(γ(r) -γ(R)) r 2 + γ(r) 2 -γ(R) 2 r 2 + V (r),
has a quadratic singularity as r tends to 0 and thus doesnot satisfy the hypothesis (H 1 ) introduced in [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF]. It is this singularity that prevent us from obtain the corresponding inverse scattering result on R 2 . Actually, we can note that the free dynamics on the whole half line (0, +∞) have to be

H 0,+ = - d 2 dr 2 + (l -γ(R)) 2 -1 4 r 2 , as r → +∞, and
H 0,-= - d 2 dr 2 + l 2 -1 4
r 2 , as r → 0, and we have to collapse these two dynamics to be able to compare our objects with the "free" ones. We still work on this question.

4. We really need to work with an electric potential V and a magnetic field B compactly supported.

Indeed, it allows us to use uniform asymptotics of Bessel functions and then to obtain estimates on the Green kernels sufficiently good in terms of ν to conclude despite the presence of ν in the potential q ν (see Appendix A).

5. In [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF] the authors obtain a local result in nature since they prove (for particular classes of electric potentials) that if the phase shifts are super-exponentially close, that is to say δ l -δl = O(e -Al ) for all A > 0, then the corresponding electric potentials coincide. To do that the authors use a uniqueness result for functions in the Hardy class. We expect that the same result is also true in our framework and we emphasize that we just need to adapt the proof of Section 3.2 to obtain it.

As mentionned previously direct and inverse scattering for magnetic Schrödinger operators are subjects of main interest. Let us give some important references in these fields.

• Direct scattering for magnetic Schrödinger operators. First, let us recall that the Aharonov-Bohm effect was introduced in [START_REF] De Aharonov | Significance of electromagnetic potentials in the quantum theory[END_REF] and was for instance also studied in [START_REF] Ruisjenaars | The Aharonov-Bohm effect and scattering theory[END_REF]. The direct scattering for magnetic Schrödinger operators was for instance the object of the papers [START_REF] Nicoleau | Matrices de diffusion pour l'opérateur de Schrödinger avec champ magnétique. Phénomène de Aharonov-Bohm[END_REF][START_REF] De Oliveira | Scattering and self-adjoint extensions of the Aharonov-Bohm Hamiltonian[END_REF][START_REF] Pankrashkin | Spectral and scattering theory for the Aharonov-Bohm operators[END_REF][START_REF] Ph | On the mathematical theory of the Aharonov-Bohm effect[END_REF][START_REF] Ruisjenaars | The Aharonov-Bohm effect and scattering theory[END_REF][START_REF] Tamura | Shadow scattering by magnetic fields in two dimensions[END_REF][START_REF] Yafaev | Scattering by magnetic fields[END_REF].

• Inverse scattering for magnetic Schrödinger operators in dimension n ≥ 3. Let us first recall some inverse scattering results from the knowledge of the scattering matrix for more than one fixed energy.

In [START_REF] Eskin | Inverse scattering problems for Schrödinger operators with magnetic and electric potentials[END_REF] the authors reconstruct the electric potential and the magnetic field from the knowledge of the scattering matrix at high energies if the magnetic and electric potentials are exponentially decreasing. In [START_REF] Nicoleau | An inverse scattering problem with the Aharonov-Bohm effect[END_REF] the authors solve the inverse scattering problem from the knowledge of the scattering matrix at all energies for electric potentials with short-range and compactly supported magnetic fields. There also exist some inverse scattering results at fixed energy. In [START_REF] Eskin | Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy[END_REF] the authors solve the inverse scattering problem at fixed energy if the electric and magnetic potentials and their derivatives of all order are exponentially decreasing. In [START_REF] Päivärinta | Inverse scattering for the magnetic Schrödinger operator[END_REF] the authors prove that the electric and the magnetic potentials are uniquely determined by the scattering matrix at one fixed energy if these potentials and their first derivatives are exponentially decreasing. In [START_REF] Weder | On inverse scattering at a fixed energy for potentials with a regular behaviour at infinity[END_REF] the authors reconstruct the asymptotics of the electric potential and the magnetic field from the knowledge of the scattering matrix at one fixed energy in the regular case (i.e. if the potentials are sums of homogeneous terms at infinity). Finally, in the very recent paper [START_REF] Krupchyk | Inverse problems for magnetic Schrödinger operators in transversally anisotropic geometries[END_REF], the authors study an inverse boundary problem for magnetic Schödinger operators on compact Riemannian manifolds with boundary for bounded magnetic and electric potentials. In this work they study the case of admissible geometries (i.e. compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold) where they reconstruct the magnetic and electric potentials from the knowledge of the Cauchy data on the boundary of the manifolds.

• Inverse scattering for magnetic Schrödinger operators in dimension two. This is the framework we study in this paper. In [START_REF] Eskin | Inverse scattering problems for Schrödinger operators with magnetic and electric potentials[END_REF] the authors solve the inverse scattering problem at high energies, in [START_REF] Weder | The Aharonov-Bohm effect and time-dependent inverse scattering theory[END_REF] the author reconstructs the magnetic field and the magnetic flux modulo 2 from the knowledge of the scattering matrix at all energies with or without obstacle. In [START_REF] Nicoleau | A stationary approach to inverse scattering for Schrödinger operators with first order perturbation[END_REF] Nicoleau uses a stationary approach to determine the asymptotic of the scattering operator and he reconstructs the electric potential and the magnetic field from the first two terms of this asymptotic expansion. In [START_REF] Nicoleau | An inverse scattering problem with the Aharonov-Bohm effect[END_REF] he solves the inverse scattering problem at all energies with Aharonov-Bohm effect for short-range potentials and compactly supported magnetic fields.

Overview of the proof

The proof of Theorem 1.6 is divided into four steps that we describe here.

Step 1: The first step of the proof consists in solving the direct problem. This will be done in Section 2. In this Section we first study the free case, i.e. when the potential q ν (r) is assumed to be zero for all r ≥ r 0 . In this case we obtain explicit expressions of the scattering objects. Indeed, we can write the free Jost solutions in terms of Bessel functions for all r ≥ r 0 :

F + 0 (r, ν) = e i(νR+ 1 2 ) π 2 πr 2 H (1) νR (r), ∀r ≥ r 0 and F - 0 (r, ν) = e -i(νR+ 1 2 ) π 2 πr 2 H (2) νR (r), ∀r ≥ r 0 .
By definition, we can then also rewrite the free regular solution and the free Jost functions in terms of Bessel functions and we also obtain an explicit formula for the free Regge interpolation function given by σ 0 (ν) = -e iπ(ν-νR) H

(2)

νR (r 0 ) H (1) νR (r 0 ) .
The aim in the following of Section 2 is then to compare the general scattering objects with the free ones. First, in Section 2.2 we prove using good estimates on the Green kernel obtained in Section A.3 that the Jost solutions are holomorphic functions with respect to ν on the whole complex plane C satisfying the following properties:

1. They are of order 1 with infinite type.

2. There exists a positive constant C such that

F ± (r, ν) ∼ CF ± 0 (r, ν), as ν → +∞.
3. They are bounded on iR + γ(R).

Secondly, in Section 2.3 we study the regular solution and we show that it is a holomorphic function with respect to ν ∈ C such that there exists a positive constant C such that for all r 0 ≤ r ≤ R and for all Re

(ν R ) = Re(ν) -γ(R) ≥ 0 |Φ(r, ν)| ≤ C 1 + |ν R | r r 0 Re(νR)
.

(1.13) Finally, in Section 2.4 we study the Jost functions α(ν) and β(ν) and we show they are holomorphic functions of order 1 with infinite type on the whole complex plane C such that if (α, β) and (α, β) are two couples of Jost functions associated with two couples (V, B) and ( Ṽ , B) belonging to the class C then, for all Re(ν R ) ≥ 0,

α(ν) β(ν) -α(ν)β(ν) = 1 2i +∞ r0 (q ν (r) -qν (r))Φ(r, ν) Φ(r, ν)dr. (1.14)
We finally show that

σ(ν) = e iπ(ν+ 1 2 ) α(ν) β(ν) → e -iπγ(R) , ν → +∞, (1.15)
and this result is interesting because it was predicted by the work [START_REF] Ph | On the mathematical theory of the Aharonov-Bohm effect[END_REF] of Roux and Yafaev.

Step 2: From now on, we solve the inverse problem and we thus introduce two couples (V, B) and ( Ṽ , B) belonging to the class C. The first step of the proof consists in reconstructing the magnetic flux. Precisely, we show, using (1.15) and (1.12), that if

δ l = δl , ∀l ∈ L,
where L ⊂ N satisfies the Müntz condition, then

γ(R) = γ(R), mod 2.
We then use the invariance of the Regge interpolation function with respect to translations of the magnetic flux by 2k, k ∈ Z, to conclude that γ(R) = γ(R).

Step 3: The third step consists in proving the uniqueness of the Regge interpolation function on almost the whole complex plane C. To do this we use an idea dued to Ramm [START_REF] Ramm | An inverse scattering problem with part of the fixed-energy phase shifts[END_REF] and we thus introduce the function

F (ν) = 2i(α(ν) β(ν) -α(ν)β(ν)).
Thanks to (1.14) we know that

F (ν) = +∞ r0 p ν (r)Φ(r, ν) Φ(r, ν)dr,
where p ν (r) = q ν (r) -qν (r) = 0, ∀r ≥ R.

We then prove, using the estimate (1.13) we obtained before on the regular solution, that the function F belongs to a particular class of functions, called the Nevanlinna class N (Π + ), when restricted to the half plane

Π + = {ν R ∈ C, Re(ν R ) > 0}.
The result we are interested in on the Nevanlinna class is the following:

Theorem 1.8 ([32], Thm. 1.3). Let h ∈ N (Π + ) satisfying h(n) = 0 for all n ∈ L where L ⊂ N ⋆ with n∈L 1 n = ∞. Then h ≡ 0 in Π + .
We recall that our main assumption is

δ l = δl , ∀l ∈ L, (1.16)
where

L ⊂ N ⋆ satisfies l∈L 1 l = +∞.
From (1.16) we easily deduce that σ(l) = e 2iδ l = e 2i δl = σ(l), ∀l ∈ L.

Therefore, since

σ(l) = e iπ(l+γ(R)+ 1 2 ) α(l) β(l)
and σ(l) = e iπ(l+γ(R)+ 1 2 ) α(l) β(l) , using that we previously showed that γ(R) = γ(R) and the definition of the function F , we obtain that

F (l) = 0, ∀l ∈ L.
So, thanks to the previous Theorem 1. [START_REF] Daudé | Non-uniqueness results for the anisotropic Calderon problem with data measured on disjoint sets[END_REF], we obtain that the map ν → F (ν) is identically zero on Π + . Finally F is identically zero on C since it is a holomorphic function. Therefore, for all ν ∈ C

F (ν) = 2i(α(ν) β(ν) -α(ν)β(ν)) = 0
and we can then conclude that

σ(ν) = σ(ν), ∀ν ∈ C \ {β(ν) = 0}.
Step 4: In the last step of the proof we use an argument close in spirit to the Börg -Marchenko Theorem to conclude the proof of Theorem 1.6. We fix r ≥ r 0 and we define F (r, ν) as an application of the complex variable ν by

F (r, ν) = F + (r, ν) F -(r, ν) -F -(r, ν) F + (r, ν),
where F ± (r, ν) and F ± (r, ν) are the Jost solutions associated with the potentials q ν and qν respectively. We are able to prove thanks to our previous study of the Jost solutions that this application is holomorphic on the whole complex plane C and of order 1 with infinite type. Moreover, F (r, ν) is bounded on the imaginary axis iR + γ(R) and using that

F (r, ν) = Ψ(r, ν)F + (r, ν) -Ψ(r, ν) F + (r, ν) + e -iπ(ν+ 1 2 ) (σ(ν) -σ(ν)) F + (r, ν) F + (r, ν),
where

Ψ(r, ν) = Φ(r, ν) β(ν) ,
we can show, using the equality of the Regge interpolation functions on the real line and the previous estimates obtained in the direct scattering problem, that F (r, ν) → 0 when ν → +∞. Finally, using a symmetry property we also obtain that F (r, ν) → 0 as ν → -∞. So, using the Phragmén-Lindelöf Theorem on each quadrant of the complex plane, we deduce that F (r, ν) is bounded on the whole complex plane C. Using the Louville's Theorem and the limit F (r, ν) → 0 as ν → +∞ we conclude that F (r, ν) is identically zero on the whole complex plane. In other words, we know that

F + (r, ν) F -(r, ν) = F -(r, ν) F + (r, ν), ∀r ≥ r 0 , ∀ν ∈ C.
From this equality we thus obtain that the Jost solutions are then so closed that q ν (r) = qν (r), ∀r ≥ r 0 , ∀ν ∈ C.

Since this equality is satisfied for all ν we can thus decouple the potentials and we can then conclude that V (x) = Ṽ (x) and B(x) = B(x), ∀|x| ≥ r 0 .

The same approach has been used recently to study inverse scattering problems on asymptotically hyperbolic manifolds (see [START_REF] Daudé | Local inverse scattering results at fixed energy in spherically symmetric asymptotically hyperbolic manifolds[END_REF][START_REF] Daudé | Inverse scattering at fixed energy on asymptotically hyperbolic Liouville surfaces[END_REF][START_REF] Daudé | Inverse scattering at fixed energy in de Sitter-Reissner-Nordström black holes[END_REF]). In the hyperbolic setting, we can prove that the Jost solutions are perturbations of the modified Bessel functions I ν (z). Moreover, in the hyperbolic context, the variable ν is fixed and depends only on the geometry of the manifolds whereas the variable z ranges over C. However, in the Euclidean setting of the paper, the situation is really different. Indeed, as in the hyperbolic case, the Jost solutions are close to the Hankel functions H (j) ν (r), but the complex angular momentum ν can be as large as possible and the radial variable r ranges over the compact set (r 0 , R).

The paper is organized as follows. In Section 2 we will study the direct scattering problem. In Section 3 we solve the inverse scattering problem at fixed energy. Finally, in Appendix A we give some useful estimates on the Bessel functions and the corresponding Green kernels.

Direct scattering

In this Section we study the direct scattering problem. It means that we will recall the definition and the basic properties of the scattering objects we are interested in. Moreover, we will compare them with the ones we obtain in the free case.

The free case

The aim of the Section is to study the free case and to give the explicit expressions of the scattering objects in this case. We first recall that the free Hamiltonian is given by

H 0 ν = - d 2 dr 2 + (ν R ) 2 -1 4 r 2 , ∀r ≥ r 0 .
Therefore, the free version of Equation (1.8) is

-u ′′ + (ν R ) 2 -1 4 r 2 u = u, ∀r ≥ r 0 , (2.17)
which is a modified Bessel equation (see (5.4.11) in [START_REF] Lebedev | Special functions and their applications[END_REF])

u ′′ + 1 -2α X u ′ + (βγX γ-1 ) 2 + α 2 -µ 2 γ 2 X 2 u = 0, is we choose α = 1 2 , γ = 1, β = 1 and µ = ν R .
Thus the free Jost solutions, which are the solutions of (2.17) satisfying F ± 0 (r, ν) ∼ e ±ir , r → +∞, are given by

F + 0 (r, ν) = e i(νR+ 1 2 ) π 2 πr 2 H (1) νR (r), ∀r ≥ r 0 (2.18)
and

F - 0 (r, ν) = e -i(νR+ 1 2 ) π 2 πr 2 H (2) νR (r), ∀r ≥ r 0 . (2.19)
Lemma 2.1. For every fixed r ≥ r 0 , the free Jost solutions ν → F ± 0 (r, ν) are holomorphic functions of order 1 with infinite type with respect to ν ∈ C. Moreover, these functions are even with respect to ν R = νγ(R).

Remark 2.2. As it has been recalled in [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF], for any fixed r, there exist positive constants A and B such that for all Re(ν R ) ≥ 0, |F + 0 (r, ν)| ≤ Ae B|νR| |Γ(ν R + 1)|. But, since we work for r 0 ≤ r ≤ R we can actually find uniform positive constants A and B such that for all such r and for all Re(ν R ) ≥ 0

|F + 0 (r, ν)| ≤ Ae B|νR| |Γ(ν R + 1)|.
Moreover, we also know that the free regular solution, which is the solution of (2.17) satisfying a Dirichlet condition at r = r 0 , is given, up to a multiplicative constant, by

Φ 0 (r, ν) = i F - 0 (r 0 , ν)F + 0 (r, ν) -F + 0 (r 0 , ν)F - 0 (r, ν) = i π √ rr 0 2 H (2)
νR (r 0 )H (1) νR (r) -H (1) νR (r 0 )H (2) νR (r) .

Proposition 2.3. There exists a positive constant C such that for all r 0 ≤ r ≤ R and for all Re(ν R ) ≥ 0

|Φ 0 (r, ν)| ≤ C |ν R | + 1 r r 0 Re(νR) .
Proof. We note that for all r ≥ r 0

Φ 0 (r, ν) = 1 2 N (r, r 0 , ν),
and we then conclude thanks to Proposition A.6.

Finally, we obtain that the free Jost functions, which are defined by Φ 0 (r, ν) = α 0 (ν)F + 0 (r, ν) + β 0 (ν)F - 0 (r, ν), since the Jost solutions are a Fundamental System of Solutions of (2.17), are given by

α 0 (ν) = iF - 0 (r 0 , ν) = ie -i(νR+ 1 2 ) π 2 πr 2 H (2) νR (r 0 )
and

β 0 (ν) = -iF + 0 (r 0 , ν) = -ie i(νR+ 1 2 ) π 2 πr 2 H (1) νR (r 0 ). (2.20)
Hence, the free Regge interpolation function is

σ 0 (ν) = e iπ(ν+ 1 2 ) α 0 (ν) β 0 (ν) = -e iπ(ν-νR) H (2)
νR (r 0 ) H

νR (r 0 )

.

In particular, for l ∈ Z we obtain the result given in [START_REF] Ruisjenaars | The Aharonov-Bohm effect and scattering theory[END_REF], Eq. (4.22),

σ 0 (l) = -e iπγ(R) H (2) l-γ(R) (r 0 ) H (1)
l-γ(R) (r 0 ) .

Therefore, using the asymptotics of the Bessel functions (A.37) and (A.38) and the symmetry identities (A.34) and (A.35), we obtain that σ 0 (l) → e iπγ(R) as l → +∞ and σ 0 (l) → e -iπγ(R) as l → -∞.

Definition and properties of the Jost solutions

In this Section we study the Jost solutions. First, let us recall the definition of these functions.

Definition 2.1 (Jost solutions). The Jost solutions are the solutions of the stationary equation (1.8) satisfying the asymptotics F ± (r, ν) ∼ e ±ir , r → +∞.

Remark 2.4. These Jost solutions are in fact closely related to the ones introduced in [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF]. Precisely,

F ± (r, ν) = f ± (r, ν R ), ∀r ≥ r 0 ,
where f ± (r, ν R ) are the Jost functions of [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF].

We easily obtain the following Lemma which gives us useful properties on the Jost solutions.

Lemma 2.5. For r ≥ r 0 fixed, the Jost solutions are holomorphic functions with respect to ν satisfying the following identity:

F ± (r, ν) = F ∓ (r, ν).
Moreover, F ± (r, ν) = 0, ∀ν ∈ R, ∀r ≥ r 0 .

(2.21)

Proof. Let us give the proof of the last point. Assume for instance that F + (r, ν) = 0 for some r ≥ r 0 . Since, F + (r, ν) = F -(r, ν) we also have F -(r, ν) = 0 which contradicts

W (F + (r, ν), F -(r, ν)) = -2i.
Because of the presence of ν in the potential q ν the Jost solutions are not even functions with respect to ν contrary to the ones introduced in the non-magnetic case studied in [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF]. However, there is an important symmetry identity that will be usefull in the following.

Lemma 2.6. Let F ± γ (r, ν) be the Jost solutions associated with the couple (V, B) ∈ C. F ± -γ (r, ν) are then the Jost solutions associated with the couple (V, -B) ∈ C and

F ± γ (r, ν) = F ± -γ (r, -ν), ∀r ≥ r 0 , ∀ν ∈ C. (2.22)
Proof. We first note that the Hamiltonian defined in (1.5) satisfies

H ν,γ = H -ν,-γ . (2.23)
We recall that the Jost solutions F ± γ (r, ν) are the unique solutions of

H ν,γ F ± γ (r, ν) = F ± γ (r, ν)
satisying suitable asymptotics at infinity. From (2.23), we then deduce that

H -ν,-γ F ± γ (r, ν) = F ± γ (r, ν)
and by uniqueness we can then conclude that

F ± γ (r, ν) = F ± -γ (r, -ν).
Remark 2.7. We will use this equality to extend the results we can obtain on the right half-plane to the left half-plane. Indeed, if (V, B) ∈ C then (V, -B) ∈ C. So, if we can prove a property for the Jost solutions F ± γ (r, ν) on the right half-plane, this property is also true for F ± -γ (r, ν) and thanks to previous Lemma we then obtain that this property is also true for F ± γ (r, -ν), i.e. on the left half-plane.

The aim of this Section is to prove the following Proposition on the Jost solutions.

Proposition 2.8. For r ≥ r 0 fixed, the Jost solutions satisfy the following properties.

1. They are of order 1 with infinite type on the whole complex plane C.

2. There exists a positive constant C such that

F ± (r, ν) ∼ CF ± 0 (r, ν), as ν → +∞.
3. They are bounded on iR + γ(R).

We use the method of variation of constants which tells us that for every r ≥ r 0 the Jost solutions satisfy

F ± (r, ν) = F ± 0 (r, ν) + +∞ r N (r, s, ν)q ν (s)F ± (s, ν)ds, (2.24)
where the Green kernel N (r, s, ν) is defined by

N (r, s, ν) = u(r)v(s) -u(s)v(r),
where (u, v) is a Fundamental System of Solutions of (1.8) when q ν = 0 defined by

u(r) = πr 2 J νR (r) and v(r) = -i πr 2 H (1) νR (r).
First, we give an estimate of the kernel N (r, s, ν) which will be useful in the following.

Lemma 2.9. There exists a constant C > 0 such that for all ν ∈ {Re(ν

R ) ≥ 0} = {Re(ν) ≥ γ(R)} and for all r 0 ≤ r ≤ s ≤ R |N (r, s, ν)| ≤ C |ν R | + 1 s r Re(νR) .
Proof. See Proposition A.6.

Remark 2.10. In [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF] the authors introduced the Green kernel

M (r, s, ν) = F + 0 (s, ν) F + 0 (r, ν) N (r, s, ν),
but it forces to work only on particular regions of the complex plane because of the zeros of the Jost solution F + 0 (r, ν). Since in our case we work only for r 0 ≤ r ≤ s ≤ R we can introduce a new kernel which is slightly different and permits us to avoid this problem.

We introduce a new Green kernel

M (r, s, ν) = r s νR N (r, s, ν).
Thanks to Lemma 2.9 we immediately obtain the following Lemma.

Lemma 2.11. There exists a constant C > 0 such that for all ν ∈ {Re(ν R ) ≥ 0} = {Re(ν) ≥ γ(R)} and for all r 0 ≤ r ≤ s ≤ R:

|M (r, s, ν)| ≤ C |ν R | + 1
.

We now want to prove the following Lemma.

Lemma 2.12. The Jost solutions F ± (r, ν) are holomorphic functions of order 1 with infinite type for

ν ∈ {Re(ν R ) ≥ 0}, i.e. ν ∈ {Re(ν) ≥ γ(R)}.
Proof. We just give the proof for the Jost solution F + (r, ν). We use an iterative method to prove this Lemma. First, we multiply the integral equation ( We then use an iterative method by putting

g 0 (r, ν) = r R νR F + 0 (r, ν)
and

g k+1 (r, ν) = +∞ r M (r, s, ν)q ν (s)g k (s, ν)ds, ∀k ≥ 0.
We then show using Lemma 2.11 and Remark 2.2 that, for all k ≥ 0, for all Re(ν R ) ≥ 0

|g k (r, ν)| ≤ Ae B|νR| |Γ(ν R + 1)| 1 k! C |ν R | + 1 +∞ r |q ν (s)|ds k .
Therefore, for all Re(ν R ) ≥ 0,

|g(r, ν)| ≤ k≥0 |g k (s, ν)| ≤ Ae B|νR| |Γ(ν R + 1)| exp C |ν R | +∞ r |q ν (s)|ds
and finally, for all Re(ν R ) ≥ 0,

|F + (r, ν)| ≤ Ae B|νR| |Γ(ν R + 1)| exp C |ν R | +∞ r |q ν (s)|ds R r 0 Re(νR)
.

Remark 2.13. We note that the term

Ae B|νR| |Γ(ν R + 1)|
corresponds to the contribution of F + 0 (r, ν) (see Remark 2.2). From this inequality we deduce that for any fixed r ∈ [r 0 , R], the Jost solution ν → F + (r, ν) is a function of order 1 with infinite type for all Re(ν R ) ≥ 0 thanks to the Stirling's formula (see [START_REF] Lebedev | Special functions and their applications[END_REF], Eq (1.4.24)).

Corollary 2.14. The Jost solutions F ± (r, ν) are holomorphic functions of order 1 with infinite type for ν ∈ C.

Proof. Thanks to the previous Lemma we already know that the Jost solutions are holomorphic functions of order 1 with infinite type for {Re(ν R ) ≥ 0}, i.e. {Re(ν) ≥ γ(R)}. We now want to extend this property to the region {Re(ν) ≤ γ(R)}. In this aim, we use Lemma 2.6 which tells us that

F ± γ (r, ν) = F -γ (r, -ν), ∀r ≥ r 0 , ∀ν ∈ C. (2.25)
Indeed, since if (V, B) ∈ C then (V, -B) ∈ C, we thus also know that the corresponding Jost solutions F -γ (r, ν) are holomorphic functions of order 1 with infinite type for {Re(ν) ≥ -γ(R)}. Moreover, if Re(ν) ≤ γ(R), then Re(-ν) ≥ -γ(R). Therefore, thanks to (2.25), the Jost solutions F ± γ (r, ν) are also holomorphic functions of order 1 with infinite type in the region {Re(ν) ≤ γ(R)}. Proposition 2.15. For any fixed r ∈ [r 0 , R], there exists a positive constant C such that F + (r, ν) ∼ CF + 0 (r, ν), as ν → +∞. Proof. Since F + 0 (r, ν) = 0 for all ν ∈ R (see Lemma 2.5), we can here strictly follow the idea of [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF], Lemma 4.4 and Proposition 4.5. We first rewrite the integral equation (2.24) as

h(r, ν) = 1 + +∞ r K(r, s, ν)q ν (s)h(s, ν)ds, where h(r, ν) = F + (r, ν) F + 0 (r, ν) and K(r, s, ν) = F + 0 (s, ν) F 0 (r, ν) N (r, s, ν).
We note that thanks to Lemma A.7, for all

r 0 ≤ r ≤ s ≤ R, for ν R ≥ 0, K(r, s, ν) = s 2ν R + 1 (1 + o(1)).
We then use an iterative method to obtain, using Lemma A.7, that for all ν R ≥ 0,

h(r, ν) = exp -1 2ν R + 1 +∞ r sq ν (s)ds (1 + o(1)).
We now recall that

q ν (r) := - 2ν(γ(r) -γ(R)) r 2 + γ(r) 2 -γ(R) 2 r 2 + V (r)
and we thus obtain that

h(r, ν) = exp 2ν 2ν R + 1 +∞ r γ(s) -γ(R) s ds - 1 2ν R + 1 +∞ r γ(s) 2 -γ(R) 2 s + V (s)ds (1 + o(1)) = exp 2ν 2ν R + 1 +∞ r γ(s) -γ(R) s ds exp O 1 ν R (1 + o(1)).
Therefore,

h(r, ν) ∼ exp +∞ r γ(s) -γ(R) s ds , as ν → +∞.
By definition, we can thus conclude that

F + (r, ν) ∼ C r F + 0 (r, ν), as ν → +∞,
where

C r := exp +∞ r γ(s) -γ(R) s ds . Remark 2.16. C r = 1, ∀r ≥ R.
Proposition 2.17. The Jost solutions ν → F ± (r, ν) are bounded on iR + γ(R). In particular, there exists a positive constant C such that for all r 0 ≤ r ≤ R and for all y ∈ R,

|F ± (r, iy + γ(R))| ≤ C exp C |y| + 1 +∞ r |q ν (s)|ds .
Proof. We just give the proof for the Jost solution F + (r, ν). We recall the integral equation on this Jost solution

F + (r, ν) = F + 0 (r, ν) + +∞ r N (r, s, ν)q ν (s)F + (s, ν)ds.
We then use an iterative method, for ν = iy + γ(R), i.e. ν R = iy, to obtain, using (A.39) and Proposition A.5, that

|F + (r, iy + γ(R))| ≤ C exp C |y| + 1 +∞ r |q ν (s)|ds .

Study of the regular solution

In this Section we study the regular solution. First, let us recall the definition of this function. By uniqueness and using the fact that the Jost solutions F ± (r, ν) are a Fundamental System of Solutions of (1.8) we obtain the following Lemma.

Lemma 2.18. The regular solution Φ(r, ν) is given, up to a multiplicative constant, by

Φ(r, ν) = i F -(r 0 , ν)F + (r, ν) -F + (r 0 , ν)F -(r, ν) .
We then easily obtain the following Lemma thanks to Lemma 2.5.

Lemma 2.19. For r ≥ r 0 fixed, the regular solution is a holomorphic function with respect to ν on C satisfying the following identity:

Φ(r, ν) = Φ(r, ν).
The aim of this section is then to prove the following Proposition on the regular solution.

Proposition 2.20. There exists a positive constant C such that for all r 0 ≤ r ≤ R and for all Re(ν R ) ≥ 0, i.e. for all Re(ν) ≥ γ(R),

|Φ(r, ν)| ≤ C 1 + |ν R | r r 0 Re(νR) .
Proof. We use the following integral equation for the regular solution for all r ≥ r 0

Φ(r, ν) = Φ 0 (r, ν) + r r0 N (r, s, ν)q ν (s)Φ(s, ν)ds.
We then use an iterative method to obtain, using Propositions 2.3 and A.6 in the case where s ≤ r, that for all Re(ν R ) ≥ 0,

|Φ(r, ν)| ≤ C |ν R | + 1 r r 0 Re(νR) exp C |ν R | + 1 r r0 |q ν (s)|ds .
We thus finally obtain that for all r ≥ r 0 and for all Re(ν R ) ≥ 0

|Φ(r, ν)| ≤ C 1 + |ν R | r r 0 Re(νR) .

Study of the Jost functions

The aim of this Section is to study the Jost functions and to obtain an algebraic identity on these functions that will be useful is the resolution of our inverse problem. Let us first recall the definition of these functions. Definition 2.3 (Jost functions). The Jost functions are defined as the coefficients of the expansion of the regular solution on the Fundamental System of Solutions given by the Jost solutions, i.e.

Φ(r, ν) = α(ν)F + (r, ν) + β(ν)F -(r, ν).
By uniqueness and using Lemma 2.18 we can actually show the following Lemma.

Therefore, thanks to (2.18-2.19), when ν → +∞,

σ(ν) = -e iπ(ν+ 1 2 ) F -(r 0 , ν) F + (r 0 , ν) ∼ -e iπ(ν+ 1 2 ) CF - 0 (r 0 , ν) CF + 0 (r 0 , ν) = -e -iπγ(R) H (2)
νR (r) H

= e -iπγ(R) , thanks to the identity (A.34).

Remark 2.27. We obtain that the eigenvalues of the scattering matrix σ(l) = e 2iδ l concentrate to e -iπγ(R) when l → +∞. Actually, we can also show that σ(l) concentrate to e iπγ(R) when l → -∞. We thus obtain in our framework the result of [START_REF] Ph | On the mathematical theory of the Aharonov-Bohm effect[END_REF], Theorem 2.1 and Corollary 2.2 where the authors show that the essential spectrum of the scattering matrix consists in two complex conjugate points of the unit circle.

Inverse scattering

In this Section we want to prove Theorem 1.6 and we thus introduce (V, B) ∈ C and ( Ṽ , B) ∈ C as radial and compactly supported electric potentials and magnetic fields respectively. Let L ⊂ N ⋆ a subset satisfying the Müntz condition

n∈L 1 n = +∞.
We assume that δ l = δl , ∀l ∈ L.

We then want to show that V (x) = Ṽ (x) and B(x) = B(x), ∀|x| ≥ r 0 . Proof. First, we recall that if δ l = δl then σ(l) = e 2iδ l = e 2i δl = σ(l).

Reconstruction of the magnetic flux

Moreover, we know thanks to Proposition 2.26 that σ(l) → e -iπγ(R) , l → +∞.

Therefore, we obtain that e -iπγ(R) = e -iπγ(R) , so γ(R) = γ(R) mod 2.

At this point we only reconstruct the magnetic flux up to an even integer 2k. Let us explain why we can actually assume that 2k = 0 with no loss of generality. First, note that

H ν,γ+2k = H ν-2k,γ .
In particular,

α γ+2k (ν) = α γ (ν -2k) and β γ+2k (ν) = β γ (ν -2k).
Hence, by Definition 2.4, we obtain that

σ γ+2k (ν) = σ γ (ν -2k).
Therefore, even if it means to change our main assumption in

δ l+2k = δl , ∀l ∈ L,
where L ⊂ N ⋆ satisfies the Müntz condition and 2k, k ∈ Z, is the difference of the magnetic flux we can assume with no loss of generality that 2k = 0.

First part of the proof: extension of the assumption

In this Section we follow an idea introduced by Ramm in [START_REF] Ramm | An inverse scattering problem with part of the fixed-energy phase shifts[END_REF] to obtain the uniqueness of the Regge interpolation function on almost the whole complex plane. In this aim, we consider the function

F (ν) = 2i(α(ν) β(ν) -α(ν)β(ν)). (3.27)
As we proved previously in Proposition 2.25 (using Theorem 3.1), we know that

F (ν) = +∞ r0 p ν (r)Φ(r, ν) Φ(r, ν)dr,
where p ν (r) = q ν (r) -qν (r) = 0, ∀r ≥ R.

We now prove that the function F belongs to a particular class of holomorphic functions when restricted to the half plane

Π + = {z ∈ C, Re(z) > γ(R)}.
Recall first the definition of the Nevanlinna class N (Π + ).

Definition 3.1 (Nevanlinna class, see [START_REF] Rudin | Real and Complex Analysis[END_REF] p.311). The Nevanlinna class N (Π + ) is defined as the set of all analytic functions f on Π + that satisfy the estimate

sup 0<r<1 π -π ln + f 1 -re iϕ 1 + re iϕ dϕ < ∞,
where ln + (x) = ln(x) if ln(x) ≥ 0 and 0 if ln(x) < 0.

We shall use the following result proved in [START_REF] Ramm | An inverse scattering problem with part of the fixed-energy phase shifts[END_REF]. As a consequence of Lemma 3.2, we get the following result.

Corollary 3.3. The map ν → F (ν) belong to N (Π + ).

Proof. We know thanks to Proposition 2.20 that

|Φ(r, ν)| ≤ C |ν R | + 1 r r 0 Re(νR)
, ∀Re(ν) ≥ γ(R).

Using this inequality, we obtain that

|F (ν)| ≤ R r0 |p ν (r)||Φ(r, ν)|| Φ(r, ν)|dr ≤ C |ν R | + 1 2 R r0 |p ν (r)| r r 0 2Re(νR)
dr.

Therefore, since r ∈ [r 0 , R], we obtain

|F (ν)| ≤ C 2 |ν| (|ν R | + 1) 2 R r 0 2Re(νR)
.

(3.28)

From this last inequality we deduce that there exist two positive constants A and C such that νR) , ∀Re(ν) ≥ γ(R).

|F (ν)| ≤ Ce ARe(
We now recall the following result proved in [START_REF] Ramm | An inverse scattering problem with part of the fixed-energy phase shifts[END_REF], Theorem 1.3. Proof. We recall that our main assumption is

δ l = δl , ∀l ∈ L, (3.29)
where L ⊂ N ⋆ satisfies l∈L 1 l = +∞.

From (3.29) we easily deduce that σ(l) = e 2iδ l = e 2i δl = σ(l).

Therefore, since σ(l) = e iπ(l+γ(R)+ 1 2 ) α(l) β(l) and σ(l) = e iπ(l+γ(R)+ 1 2 ) α(l) β(l) , using Theorem 3.1 and the definition of the function F given by (3.27) we obtain that

F (l) = 0, ∀l ∈ L.
Using Corollary 3.3 and Theorem 3.4 we thus obtain that the map ν → F (ν) is identically zero on Π + . Finally F is identically zero on the whole complex plane C since it is a holomorphic function.

Theorem 3.6. Assume that δ l = δl , ∀l ∈ L,

where L ⊂ N ⋆ satisfies l∈L 1 l = +∞.
Then,

σ(ν) = σ(ν), ∀ν ∈ C \ {β(ν) = 0}.
Proof. Thanks to Corollary 3.5 we know that for all ν ∈ C

F (ν) = 2i(α(ν) β(ν) -α(ν)β(ν)) = 0.
Thus, using the definition of the Regge interpolation function σ given by (2.26), we obtain the result.

Second part of the proof: conclusion

We will follow the elegant, simple and self-contained proof given in [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF]. This proof follows an idea close to the local Börg-Marchenko uniqueness Theorem (see [START_REF] Bennewitz | A proof of the local Borg-Marchenko Theorem[END_REF][START_REF] Gesztesy | On local Borg-Marchenko uniqueness results[END_REF][START_REF] Simon | A new approach to inverse spectral theory, I. fundamental formalism[END_REF][START_REF] Teschl | Mathematical Methods in Quantum Mechanics[END_REF]). Let r ≥ r 0 fixed, we introduce for ν ∈ C the function

F (r, ν) = F + (r, ν) F -(r, ν) -F -(r, ν) F + (r, ν). (3.30)
As we have seen in Section 2.2, Lemma 2.5, the function ν → F (r, ν) is holomorphic on C and of order 1 with infinite type. The aim is now to prove that this function is identically zero using the Phragmén-Lindelöf Theorem (see for instance [START_REF] Boas | Entire Functions[END_REF], Theorem 1.4.2).

Step 1: F (r, ν) is bounded on iR + γ(R). We proved in Section 2.2, Proposition 2.17, that the Jost solutions ν → F ± (r, ν) are bounded on iR + γ(R). Therefore, F (r, ν) is, by definition, also bounded on iR + γ(R).

Step 2: F (r, ν) → 0 when ν → +∞. We strictly follow the proof given in [START_REF] Daudé | Local inverse scattering at a fixed energy for radial Schrödinger operators and localization of the Regge poles[END_REF]. Indeed, for ν ∈ R, we know that β(ν) = 0 (see Lemma 2.21) and we can then set Ψ(r, ν) = Φ(r, ν) β(ν) .

We thus obtain, using (1.10), that

F -(r, ν) = Ψ(r, ν) - α(ν) β(ν) F + (r, ν).
Therefore,

F (r, ν) = Ψ(r, ν)F + (r, ν) -Ψ(r, ν) F + (r, ν) + α(ν) β(ν) - α(ν) β(ν) F + (r, ν) F + (r, ν).
So, using (2.26), we deduce

F (r, ν) = Ψ(r, ν)F + (r, ν) -Ψ(r, ν) F + (r, ν) + e -iπ(ν+ 1 2 ) (σ(ν) -σ(ν)) F + (r, ν) F + (r, ν).
Hence, by Theorem 3.6, we obtain that for ν ∈ R,

F (r, ν) = Ψ(r, ν)F + (r, ν) -Ψ(r, ν) F + (r, ν).
For instance, let us examine Ψ(r, ν) F + (r, ν). Propositions 2.15 and 2.23 imply that for ν ≥ γ(R)

|Ψ(r, ν) F + (r, ν)| = Φ(r, ν) β(ν) | F + (r, ν)| ≤ C Φ(r, ν) β 0 (ν) | F + 0 (r, ν)|.
Moreover, using Proposition 2.20, we know that there exists a positive constant C such that for a fixed r ≥ r 0 and for all ν ≥ γ(R)

|Φ(r, ν)| ≤ C |ν R | + 1 r r 0 νR ,
thanks to (2.18) and (A.37)

|F + 0 (r, ν)| ≤ C r 2 -νR+ 1 2 Γ(ν R ),
and thanks to (2.20)

|β 0 (ν)| ∼ C r 0 2 -νR+ 1 2 Γ(ν R ).
Therefore,

|Ψ(r, ν) F + (r, ν)| ≤ C |ν R | + 1 → 0, ν → +∞. Similarly, we obtain | Ψ(r, ν)F + (r, ν)| → 0, ν → +∞.
Therefore, F (r, ν) → 0, ν → +∞.

Step 3: F (r, ν) → 0 when ν → -∞. Thanks to the uniquenesss of the Regge interpolation function on the real line we know that for all ν ∈ R

F (r, ν) = Ψ(r, ν)F + (r, ν) -Ψ(r, ν) F + (r, ν).
We then use Lemma 2.6 to come back to the limit as ν → +∞. Indeed, thanks to this result we obtain that, for large positive ν F

+ γ (r, -ν) = F + -γ (r, ν), β γ (-ν) = β -γ (ν), and Φ γ (r, -ν) = Φ -γ (r, ν).
Moreover, since if (V, B) ∈ C then (V, -B) ∈ C, Propositions 2.15, 2.23 and 2.20 are also true for (V, -B), i.e. for -γ. Therefore, we can follow the end of the previous step to obtain that

F (r, ν) → 0, ν → -∞.
Step 4: F (r, ν) is identically zero on C. At this point, we know that F (r, ν) → 0 when ν → ±∞. In particular, ν → F (r, ν) is bounded on the real axis. Since F (r, ν) is also bounded in iR + γ(R), applying the Phragmén-Lindelöf Theorem in each quadrant of the complex plane, we see that F (r, ν) is bounded on the whole complex place C. Finally, F (r, ν) is constant by Liouville's Theorem. As the limit is 0 when ν → +∞ we have

F (r, ν) = 0, ∀ν ∈ C. (3.31)
Step 5: Conclusion of the proof. Using (3.30) and (3.31) we have

F + (r, ν) F -(r, ν) = F -(r, ν) F + (r, ν), ∀r ≥ r 0 , ∀ν ∈ C. (3.32)
For ν ∈ R fixed, we note that for all r ≥ r 0 , F ± (r, ν) = 0 (see Lemma 2.21). We can then rewrite (3.32) as

F + (r, ν) F -(r, ν) = F + (r, ν) F -(r, ν) , ∀r ≥ r 0 , ∀ν ∈ R.
Differentiating and using that W

(F + (r, ν), F -(r, ν)) = -2i, it follows that F -(r, ν) 2 = F -(r, ν) 2 .
We take the logarithmic derivative of this and we differentiate once more. We then obtain

F -′′ (r, ν) F -(r, ν) = F -′′ (r, ν) F -(r, ν) , ∀r ≥ r 0 , ∀ν ∈ R.
Using (1.8) we then obtain that q ν (r) = qν (r), ∀r ≥ r 0 , ∀ν ∈ R.

We now recall that for all r ≥ r 0 ,

q ν (r) = - 2ν(γ(r) -γ(R)) r 2 + γ(r) 2 -γ(R) 2 r 2 + V (r) and qν (r) = - 2ν(γ(r) -γ(R)) r 2 + γ(r) 2 -γ(R) 2 r 2 + Ṽ (r).
Since the equality (3.33) is satisfied for all ν ∈ R, we can decouple it and we thus obtain the identities

γ(r) -γ(R) = γ(r) -γ(R), ∀r ≥ r 0 and γ(r) 2 -γ(R) 2 r 2 + V (r) = γ(r) 2 -γ(R) 2 r 2 + Ṽ (r), ∀r ≥ r 0 . So, |J iy (r)| = 1 |Γ(1 + iy)| 1 + O 1 |y| = 1 |y||Γ(iy)| 1 + O 1 |y| , |y| → +∞.
We know (see [START_REF] Lebedev | Special functions and their applications[END_REF], p.14) that for real y |Γ(iy)| 2 = π y sinh(πy) .

Thus,

|J iy (r)| = | sinh(πy)| |y|π 1 + O 1 |y| , |y| → +∞.
Moreover, using (A.34) we obtain that

|J -iy (r)| = |J -iy (r)| = |J iy (r)| = | sinh(πy)| |y|π 1 + O 1 |y| , |y| → +∞.
We now should split our study into two cases.

Case 1: y ≥ 0. In this case J -iy (z) = o(e πy J iy (z)), y → +∞.

So, when y → +∞, |H Using these two asymptotics we then obtain (A.39). Finally, we prove (A.

A.3 Estimates on the Green kernels

The aim of this Section is to obtain useful estimates on several Green kernels. We recall that the Green kernel N (r, s, ν) is defined by N (r, s, ν) = u(r)v(s)u(s)v(r),

where (u, v) is a Fondamental System of Solutions of the free stationary equation where ν R = ν -γ(R). The main idea to obtain an estimate on N (r, s, ν) on the half plane {Re(ν) ≥ γ(R)} is to use the Phragmén-Lindelöf Theorem because u and v (and so N (r, s, ν)) are entire functions of order 1 with infinite type. We thus roughly speaking just need to estimate N (r, s, ν) for ν ≥ γ(R) and for ν ∈ iR + γ(R). We first study the behaviour of N (r, s, ν) when ν → +∞. Thanks to Proposition A.1, we can first prove the following Lemma. We now need to obtain an estimate of N (r, s, ν) for ν ∈ iR + γ(R) which is given by the following Proposition. Proof. We recall that N (r, s, ν) = u(r)v(s)u(s)v(r) = iπ 2 √ rs J νR (s)H (1) νR (r) -J νR (r)H (1) νR (s) = iπ 4 √ rs H (1) νR (r)H (2) νR (s) -H (1) νR (s)H

Proof. We recall that N (r, s, ν) = u(r)v(s)u(s)v(r).

Since u and v are entire functions of order 1 with infinite type, N (r, s, ν) is also an entire functions of order 1 with infinite type and thanks to the Phragmén-Lindelöf Theorem we thus just need to estimate N (r, s, ν) for ν ≥ γ(R) and for ν ∈ iR + γ(R).

Estimate of N (r, s, ν) on iR + γ(R). We use Proposition A.

1

  Introduction and statement of the main result 1.1 Description of our framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Statement of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Overview of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Direct scattering 2.1 The free case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Definition and properties of the Jost solutions . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Study of the regular solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Study of the Jost functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Inverse scattering 3.1 Reconstruction of the magnetic flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 First part of the proof: extension of the assumption . . . . . . . . . . . . . . . . . . . . . 3.3 Second part of the proof: conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

M

  (r, s, ν)q ν (s)g(s, ν)ds, where g(r, ν) = r R νR F + (r, ν).

Definition 2 . 2 (

 22 Regular solution). The regular solution is the solution of the stationary equation (1.8) satisfying the Dirichlet condition Φ(r, ν) |r=r0 = 0.

Theorem 3 . 1 . 1 l

 311 Assume that δ l = δl , ∀l ∈ L, where L ⊂ N ⋆ satisfies the Müntz condition l∈L = +∞. Then, γ(R) = γ(R).

Lemma 3 . 2 .

 32 Let h ∈ H(Π + ) be a holomorphic function in Π + satisfying |h(z)| ≤ Ce ARe(z) , ∀z ∈ Π + ,where A and C are two constants. Then h ∈ N (Π + ).

Theorem 3 . 4 ( 1 n

 341 [START_REF] Ramm | An inverse scattering problem with part of the fixed-energy phase shifts[END_REF], Thm.1.3). Let h ∈ N (Π + ) satisfying h(n) = 0 for all n ∈ L where L ⊂ N ⋆ with n∈L = ∞. Then h ≡ 0 in Π + . Corollary 3.5. The map ν → F (ν) is identically zero on C.

2 y

 2 (1 + o(1)), where we used for the last step sinh(πy) = e πye -πy 2 ∼ e πy 2 , y → +∞.Case 2: y ≤ 0. In this case e πy J iy (z) = o(J -iy (z)), y → -∞.So, when y → -

2 )

 2 40) by using (A.35) as follows |H (iy (r)| = |H

2 y

 2 (1 + o(1)), |y| → +∞.

πr 2 J 2 H ( 1 )

 221 νR (r) and v(r) = -i πr νR (r),

Lemma A. 3 . 4 .

 34 For r, s ≥ r 0 belonging to a compact set, the following uniform asymptotics are satisfied when ν R → +∞ u(r)v(s) For r 0 ≤ r ≤ s ≤ R belonging to a compact set, we have the uniform asymptotics when ν → +∞ N (r, s, ν) For r ≤ s, we know that when ν → +∞, )v(s) = o(u(s)v(r))and then using thatN (r, s, ν) = u(r)v(s)u(s)v(r) = -u(s)v(r)(1 + o(1)), ν → +∞,we obtain the result we want thanks to Lemma A.3.

Proposition A. 5 .

 5 There exists a positive constant C such that for r, s > 0 belonging to a compact set, we have the following estimate when ν = iy + γ(R) and |y| → +∞|N (r, s, iy + γ(R))| ≤ C |y| .

( 2 )

 2 νR (r) and we use Proposition A.2 since ν R = νγ(R) = iy.Proposition A.6. There exists a constant C > 0 such that for all ν ∈ {Re(ν) ≥ γ(R)} and for allr 0 ≤ r ≤ s ≤ R |N (r, s, ν)| ≤ C |ν R | + 1

1 . 7 .) π 2 πr 2 H

 1722 5 which gives us that |N (r, s, iy + γ(R))| ≤ C |y| , |y| → +∞. Therefore, there exists a positive constant C such that for all y ∈ R, |N (r, s, iy + γ(R))| ≤ C |y| + Estimate of N (r, s, ν) on [γ(R), +∞). We use Proposition A.4 which gives us that|N (r, s, ν)| ≤ C ν R + 1 s r νR , ν → +∞.Therefore, there exists a positive constant C such that for all ν ≥ γ(R),|N (r, s, ν)| ≤ C ν + 1)N (r, s, ν).By definition, f is a holomorphic function of order 1 and with infinite type on {Re(ν) ≥ γ(R)}. Moreover, thanks to the previous estimates f is bounded on [γ(R), +∞) and iR + γ(R). Therefore, thanks to the Phragmén-Lindelöf Theorem we can conclude that f is bounded on {Re(ν) ≥ γ(R)}. By definition, it provides us that for all ν ∈ {Re(ν) ≥ γ(R)} |N (r, s, ν)| ≤ C ν For all ν ≥ γ(R) and for all r 0 ≤ r ≤ s ≤ R |K(r, s, ν)| = s 2ν R (1 + o(1)).whereK(r, s, ν) = F + 0 (s, ν) F + 0 (r, ν)N (r, s, ν).Proof. First, we use Proposition A.4 which provides us that when ν → +∞ |N (r, s, ν)recall thatF + 0 (r, ν) = e i(νR+1 2 (1) νR (r), ∀r ≥ r 0 . Hence, when ν → +∞, thanks to (A.37) thus obtain when ν → +∞ |K(r, s, ν)| = |F + 0 (s, ν)| |F + 0 (r, ν)| |N (r, s, ν)|

Acknowledgments: This paper was initiated by Thierry Daudé and François Nicoleau during the PhD of the author. The author wants to deeply thank Thierry Daudé and François Nicoleau for their help and their support and Niky Kamran for his encouragement.

Lemma 2.21. The Jost solutions are given, up to a multiplicative constant, by α(ν) = iF -(r 0 , ν) and β(ν) = -iF + (r 0 , ν).

Thanks to this Lemma we easily obtain the following Proposition.

Proposition 2.22. The Jost functions α(ν) and β(ν) are holomorphic functions of order 1 with infinite type on the whole complex plane C.

Proof. We know that this is true for the Jost solutions (see Proposition 2.8) and we recall that

and β(ν) = -iF + (r 0 , ν).

Proposition 2.23. There exists a positive constant C such that

Proof. This is an easy consequence of Proposition 2.15 and the fact that

To solve our inverse problem we will need an algebraic identity on the Jost functions. Assume that (α, β) and (α, β) are the Jost functions corresponding to two magnetic Schrödinger equations having the same magnetic flux, that is to say γ(R) = γ(R).

Lemma 2.24. If γ(R) = γ(R) then, for all Re(ν R ) ≥ 0, i.e. for all Re(ν) ≥ γ(R),

Proof. We just give the proof of the first identity. First, using Equation (1.8), we know that for all r ≥ r 0

since γ(R) = γ(R) and so ν R = νγ(R) = νR . We integrate this identity on [r 0 , +∞) and we obtain

We thus obtain, since

(q ν (r) -qν (r))Φ(r, ν) Φ(r, ν)dr.

Definition 2.4 (Regge interpolation function). The Regge interpolation function is defined by

Proof. Thanks to the definition of the Jost functions

we know that

We recall that thanks to Proposition 2.8

Moreover, we already shown in Theorem 3.1 that

Therefore, we can conclude that γ(r) = γ(r), ∀r ≥ r 0 , which implies by differentiation (see (1.1)) that

Remark 3.7. The equality γ(r) = γ(r), ∀r ≥ r 0 , actually allows us to conclude that A(x) = Ã(x), ∀|x| ≥ r 0 , by (1.1). This is not surprising that we can actually recover the magnetic potential A because of the gauge choice we made in Definition 1.1.

Finally, we also obtain that V (r) = Ṽ (r), ∀r ≥ r 0 .

A Estimates on Bessel functions A.1 Definition and symmetry properties of the Bessel functions

In this Section we recall the definition of the Bessel functions and we give some basic properties of these functions. We refer the reader to [START_REF] Lebedev | Special functions and their applications[END_REF], Chapter 5, or to the Watson's treatise [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF] that we will use in the following.

The Bessel function of the first kind J ν (z) is defined for ν ∈ C and |Arg(z)| < π by

.

The Bessel function of the second kind are then defined for ν ∈ C \ Z and |Arg(z)| < π by

Finally, the Bessel functions of the third kind, or Hankel functions, denoted by H

ν (z) and H

(2) ν (z) are defined in terms of the Bessel functions of the first and the second kind by

and H (2) ν (z) = J ν (z) -iY ν (z). These functions can be written as

and

The Bessel functions J ν (z) and the Hankel functions

ν (z) are entire functions with respect to ν. Moreover, they satisfy the following identities (see [START_REF] Oberhettinger | Formulas and Theorems for the Special Functions of Mathematical Physics[END_REF], p.66)

-ν (z) = e iπν H (1) ν (z) and H

-ν (z) = e -iπν H (2) ν (z). (A.35)

A.2 Review of estimates on Bessel functions on a compact set

In this Section, we will recall some uniform asymptotics for the Bessel function J ν (r) and the Hankel functions

ν (r) with respect to r when r belongs to a real compact set. We emphasize that these uniform asymptotics fail if r ∈ (0, +∞).

Proposition A. 1 ([11], Proposition A.7). Let δ > 0 be small enough. For r > 0 belonging to a compact set, we have the following uniform asymptotics when ν → ∞

On the imaginary axis the asymptotics of the Hankel functions for a parameter r belonging to a compact set are given by the following Proposition.


Bill Allombert
email: bill.allombert@math.u-bordeaux.fr

Nicolas Brisebarre
email: nicolas.brisebarre@ens-lyon.fr

Alain Lasjaunias
email: alain.lasjaunias@math.u-bordeaux.fr

On a two-valued sequence and related continued fractions in power series fields

Keywords: formal power series, power series over a finite field, continued fractions, finite automata, automatic sequences, words, finite alphabet Mathematics Subject Classification (2000) Primary 11J70, 11T55; Secondary 11B85

We explicitly describe a noteworthy transcendental continued fraction in the field of power series over Q, having irrationality measure equal to 3. This continued fraction is a generating function of a particular sequence in the set {1, 2}. The origin of this sequence, whose study was initiated in a recent paper, is to be found in another continued fraction, in the field of power series over F 3 , which satisfies a simple algebraic equation of degree 4, introduced thirty years ago by D. Robbins.

Introduction

In this paper, we present a remarkable continued fraction, built from an infinite word containing only the letters 1 and 2. The partial study of the structure of this word shows that this sequence is complex enough to imply surprising arithmetical properties for the resulting generating function. This generating function is indeed transcendental and it has a very peculiar continued fraction expansion that we will fully describe. We first start by describing our framework and we recall the origin of this strange sequence.

Here, K is a field which is either the finite field F q , containing q elements and of characteristic p, or the field Q of the rational numbers. Let T be a formal indeterminate. As usual K[T ] and K(T ) are, respectively, the ring of polynomials and the field of rational functions in T with coefficients in K. We consider the field of power series in 1/T , with coefficients in K, denoted by F(K) or K((T -1 )). Hence a non-zero element of F(K) can be written as

α = k k0 a k T k
where k 0 ∈ Z, a k ∈ K and a k0 = 0.

An ultrametric absolute value is defined over these fields of power series. For α as above we have |α| = |T | k0 where |T | is a fixed real number greater than 1. Note that F(K) is the completion of the field K(T ) for this absolute value.

If K = F q , we simply write F(q) for F(K). The case K = F q is particularly important for several reasons. The first one is the link between these fields of power series and certain sequences taking a finite number of values. The fields F(q) are analogues of the field of real numbers and, as in the case of real numbers, if α ∈ F(q) then the sequence of coefficients (or digits) (a k ) k k0 for α is ultimately periodic if and only if α is rational, that is, if α belongs to F q (T ) (note that this fails if K = Q). However, and this is a distinguishing aspect of the formal case, this sequence of digits can also be characterized for all elements in F(q) which are algebraic over F q (T ); see Theorem 1. Indeed, a large class of sequences, taking values in a finite set, were introduced around the 1960's by computer scientists. These sequences tend to generalize the particularly simple periodic sequences and are called automatic sequences. A full account on this topic and a very complete list of references are to be found in the book of Allouche and Shallit [START_REF] Allouche | Automatic sequences. Theory, applications, generalizations[END_REF]. Automatic sequences are classified by an integer parameter k 2, and consequently we have k-automatic sequences for all k 2. The link between these sequences and power series over a finite field appears in the following theorem found in the work of Christol [START_REF] Christol | Ensembles presque périodiques k-reconnaissables[END_REF] (see also the article of Christol, Kamae, Mendès France, and Rauzy [START_REF] Christol | Suites algébriques, automates et substitutions[END_REF]).

Theorem 1 (Christol) Let α in F(q) with q = p s . Let (a k ) k k0 be the sequence of digits of α and u(n) = a -n for all integers n 0. Then α is algebraic over F q (T ) if and only if the sequence (u(n)) n 0 is p-automatic.

In the fields of power series over K, as for real numbers, there is a continued fraction algorithm. The integers are replaced by elements of K[T ]. The reader may consult [START_REF] Lasjaunias | A survey of Diophantine approximation in fields of power series[END_REF] for a general account concerning continued fractions and Diophantine approximation in power series fields. We know that any irrational element α in F(K) can be expanded as an infinite continued fraction where the partial quotients a n are polynomials in K[T ], all of positive degree, except perhaps for the first one. Traditionally, this expansion is denoted by α = [a 0 , a 1 , a 2 , . . . , a n , . . .]. As for real numbers, this expansion is fundamental to measure the quality of the rational approximation of formal power series. The irrationality measure of an irrational power series α ∈ F(K) is defined by

ν(α) = -lim sup |Q|→∞ log(|α -P/Q|)/ log(|Q|),
where P, Q ∈ K[T ]. This irrationality measure is directly related to the growth of the sequence of the degrees of the partial quotients a n in the continued fraction expansion of α. Indeed we have (see, for instance, [5, p. 214])

ν(α) = 2 + lim sup n 1 (deg(a n+1 )/ 1 i n deg(a i )).
Here again the difference between K finite and K = Q is striking. We recall that, by an adaptation of a classical theorem of Liouville, due to Mahler, if α ∈ F(K) is algebraic over K(T ), such that [K(T, α) : K(T )] = n then we have ν(α) ∈ [2, n]. For algebraic real numbers this irrationality measure is known to be equal to 2 by the celebrated Roth theorem [START_REF] Roth | Rational approximation to algebraic numbers[END_REF]. It was proved by Uchiyama [START_REF] Uchiyama | Rational approximation to algebraic functions[END_REF], shortly after Roth theorem, that the same is true for fields of power series over K, if K has characteristic zero. In the case K = F q , the situation is more complex, and Mahler [START_REF] Mahler | On a theorem of Liouville in fields of positive characteristic[END_REF] observed that we may have ν(α) = n for certain algebraic elements of degree n. This observation was the beginning of the study of Diophantine approximation in positive characteristic. At the same time the study of continued fractions for certain algebraic power series over finite fields was developed.

The starting point of this note is a particular continued fraction in the field F(3). It was introduced by Mills and Robbins [START_REF] Mills | Continued fractions for certain algebraic power series[END_REF], and it is the root of the following quartic equation: x 4 + x 2 -T x + 1 = 0. This equation has a unique root in F(p) for all p 2. The continued fraction expansion for this root α in F(3), calculated by computer, was conjectured in [START_REF] Mills | Continued fractions for certain algebraic power series[END_REF]. Some ten years later this conjecture was proved in [START_REF] Buck | The continued fraction expansion of an algebraic power series satisfying a quartic equation[END_REF] and also, shortly afterwards, with a different method in [START_REF] Lasjaunias | Diophantine approximation and continued fraction expansions of algebraic power series in positive characteristic[END_REF]. Here we have α = [0, a 1 , a 2 , . . . , a n , . . . ] where

a n = λ n T un , (λ n ) n 1 is a sequence in F *
3 and (u n ) n 1 another sequence in N * . The knowledge of this last sequence in N * implies that here we have ν(α) = 2, even though the sequence of the degrees of the partial quotients (u n ) is unbounded. This implies a remarkable property of this element (see [6, p. 209] and also [7, p. 58]). In this note, we only are interested in the first sequence (λ n ) n 1 in F * 3 , i.e., in {1, 2}. This sequence is described in the following theorem.

Theorem 2 Let (W n ) n 0 be the sequence of finite words over the alphabet {1, 2}, defined recursively as follows:

W 0 = ǫ, W 1 = 1,
and

W n = W n-1 , 2, W n-2 , 2, W n-1 , for n 2,
where commas indicate concatenation of words and ǫ denotes the empty word. Let W = (w(n)) n 1 be the infinite sequence beginning with W n for all n 0. Then the sequence W = (w(n)) n 1 is substitutive but it is not k-automatic for any integer k 2.

This theorem was proved in [START_REF] Lasjaunias | Hyperquadratic continued fractions and automatic sequences[END_REF]Theorem 4]. The reader can consult [START_REF] Lasjaunias | Hyperquadratic continued fractions and automatic sequences[END_REF] for a brief account of automatic and substitutive sequences, and also naturally [START_REF] Allouche | Automatic sequences. Theory, applications, generalizations[END_REF] for a full exposition on these matters.

We set θ = n 1 w(n)T -n . Then θ can be considered in F(K), where K = F p , for any prime number p 2 or K = Q, and θ is transcendental over K(T ) in all cases. In the case K = F p , this is simply a straightforward consequence of Theorem 1. Indeed if θ were algebraic over F p (T ), then the sequence (w(n)) n 1 would be p-automatic, in contradiction with Theorem 2. In the case, K = Q, by Uchiyama's adaptation of Roth's theorem, it will simply follow from the fact that the irrationality measure for θ is equal to 3. This will be established in the next section, as a consequence of our main result Theorem 3, in which we explicitly give the degrees of the partial quotients of the continued fraction expansion for θ when considered in the field Q((T -1 )). We will end our text with a conjectural description of this continued fraction expansion.

2 A transcendental continued fraction in Q((T -1 ))
We start from the sequence W = (w(n)) n 1 , defined above in Theorem 2, and we consider the formal power series θ = n 1 w(n)T -n ∈ Q((T -1 )). We shall see that this power series is not rational and we shall describe partially the infinite sequence of the partial quotients in the theorem below.

First we define some notions needed for a deep study of the sequence (W n ) n 0 of finite words over the alphabet {1, 2}. If M is a finite word, we let |M | denote the length of this word. This notation on words should not be confused with the use of the absolute value for elements in the power series fields. Hence we have |W 0 | = 0 and |W 1 | = 1. We set ℓ n = |W n |. By the recursive definition of (W n ) n 0 , for n 2, we have ℓ n = 2ℓ n-1 + ℓ n-2 + 2. We will also use the following: if M = m 1 , m 2 , . . . , m n is a finite word we set

φ(M ) = m 1 T n-1 +m 2 T n-2 +• • •+m n ∈ Q[T ] and also Φ(M ) = φ(M )T -|M| ∈ Q(T )
. By extension, for an infinite word, with our definitions of W and θ, we can simply write θ = Φ(W ). We have the following theorem which is our main result.

Theorem 3 Let θ = n 1 w(n)T -n ∈ Q((T -1
)), then θ is irrational and we have the infinite continued fraction expansion θ = [0, a 1 , a 2 , . . . , a n , . . . ] with a n ∈ Q[T ] for n 1. Let (ℓ n ) n 0 be the sequence of integers defined by

ℓ 0 = 0, ℓ 1 = 1 and ℓ n+1 = 2ℓ n + ℓ n-1 + 2 for n 1. Indeed, we have ℓ n = ((2 + √ 2)(1 + √ 2) n + (2 - √ 2)(1 - √ 2) n )/4 -1. Let (d n ) n 1 be the sequence of integers defined by d n = deg(a n ) for n 1.
Then this sequence is described as follows:

d 1 = d 2 = d 3 = d 4 =
1 and, for n 1, we have

d 4n+1 = (3ℓ n + ℓ n-1 + 1)/2, d 4n+3 = (ℓ n + ℓ n-1 + 1)/2 and d 4n+2 = 1, d 4n+4 = 1.
Corollary 1 The irrationality measure of θ is equal to 3.

Proof We know that the irrationality measure of θ satisfies

ν(θ) = 2 + lim sup n 1 (d n+1 / 1 i n d i ).
We observe that, for n 1, we have

d 4n+1 > max(d 4n , d 4n-1 , d 4n-2 , d 4n-3
) and consequently, we have ν(θ) = 2 + lim n 1 (d 4n+1 / 1 i 4n d i ). We set t n = 1 i 4n d i . We have t 1 = 4 and by induction, we obtain easily

t n = 2 + d 4n+1 for n 1. Hence, we have ν(θ) = 2 + lim n 1 (d 4n+1 /(2 + d 4n+1 )) = 3. ⊓ ⊔
The proof of Theorem 3 is obtained using three lemmas. Each of the first two lemmas gives a sequence of good rational approximations to θ.

In the sequel, we shall avoid the comma for the concatenation of words.

Lemma 1 For n 1, we set S n = T (ℓn+ℓn-1+3)/2 (T ℓn+1 -1). Then, for n 1, there exists

R n ∈ Q[T ] such that |θ -R n /S n | = |S n | -ωn with ω n = 3 -4/(3ℓ n + ℓ n-1 + 5).
Proof For n 1, we introduce the following words

U n = W n 2W n-1 and V n = 2W n .
Note that W n+1 = U n V n . We consider the infinite word

X 1 = U n V n V n • • • V n • • • = U n (V n ) ∞ .
Now let us consider the element X 2 ∈ Q((T -1 )) such that Φ(X 1 ) = X 2 . Indeed, we can write

X 2 = φ(U n ) T |Un| + φ(V n ) T |Un|+|Vn| + φ(V n ) T |Un|+2|Vn| + • • • .
Consequently X 2 ∈ Q(T ) and we have

X 2 = φ(U n ) T |Un| + φ(V n ) T |Un|+|Vn| . T |Vn| T |Vn| -1 = (T |Vn| -1)φ(U n ) + φ(V n ) T |Un| (T |Vn| -1) . We set N 2 = (T |Vn| -1)φ(U n ) + φ(V n ) and D 2 = T |Un| (T |Vn| -1) so that we have X 2 = N 2 /D 2 .
In order to simplify the rational X 2 , we need to introduce the following relation on finite words. For two finite words A and B , we write A • B if and only if the letters ending A and B respectively are different. Note that A • B if and only if T does not divide φ(A) -φ(B). For n = 1, we have

U 1 = 12, V 1 = 21 and U 1 • V 1 .
For n = 2 we can write

U 2 = W 2 2W 1 = G 2 2W 1 and V 2 = 2W 2 = 2W 1 2W 0 2W 1 = H 2 2W 1 with G 2 • H 2 . We set (F 1 , G 1 , H 1 ) = (ǫ, U 1 , V 1 ) and (F 2 , G 2 , H 2 ) = (2W 1 , W 2 , 2W 1 2). So we have U 1 = G 1 F 1 , V 1 = H 1 F 1 , U 2 = G 2 F 2 and V 2 = H 2 F 2 .
By induction, we can define three finite words F n , G n and H n such that, for n 1,

U n = G n F n V n = H n F n with G n • H n .
The sequence (F n , G n , H n ) satisfies, for n 2, the following formulas:

F n = 2W 1 2W 2 • • • 2W n-1 , H n = 2G n-1 and G n = U n-1 H n-1 .
Now we will compute the length of the words. We have

|U n | = ℓ n + ℓ n-1 + 1 and |V n | = ℓ n + 1.
But we also have |F 1 | = 0 and, for n 2, by using a simple induction, we get

|F n | = n -1 + ℓ 1 + ℓ 2 + • • • + ℓ n-1 = (ℓ n + ℓ n-1 -1)/2.
Consequently we have, for n 1,

|U n | -|F n | = |G n | = (ℓ n + ℓ n-1 + 3)/2.
It is easily verified that for two finite words A and B we have φ(AB) = T |B| φ(A) + φ(B). Therefore, for three finite words A, B and C we also have

φ(AC) -φ(BC) = T |C| (φ(A) -φ(B)). Now we define R n ∈ Q[T ] by R n = φ(G n+1 ) -φ(G n ).
Hence, we can write

N 2 = T |Vn| φ(U n ) + φ(V n ) -φ(U n ) = φ(U n V n ) -φ(U n )
and

N 2 = φ(U n H n F n ) -φ(G n F n ) = T |Fn| (φ(U n H n ) -φ(G n )) = T |Fn| R n .
Since |V n | = ℓ n + 1, we can also write, in agreement with the formula given in the lemma for S n ,

D 2 = T |Fn| (T |Un|-|Fn| (T |Vn| -1)) = T |Fn| S n .
Consequently we have X 2 = R n /S n . Note that we have N 2 (1) = φ(V n )(1) = 0 and consequently R n (1) = 0. Now we need to study the approximation of θ by X 2 . First, for n 1, we observe the following equalities

W n+2 = W n+1 2W n 2W n+1 = W n 2W n-1 2W n 2W n 2W n 2W n-1 2W n = U n V 3 n V n-1 V n .
Hence for all n 1, the infinite word W begins with

U n V 3 n V n-1 V n whereas we have X 1 = U n V ∞ n . With
-R n /S n | = |Φ(W ) -Φ(X 1 )| = |T | -tn , if t n is the rank of the first letter differing in the words U n V 3 n V n-1 V n and U n V ∞ n . From this we deduce |θ -R n /S n | = |T | -tn = |S n | -tn/ deg(Sn) = |S n | -ωn .
To compute this value t n , we introduce another relation between finite words. For two finite words A and B , we write A • •B if and only if the letters beginning A and B respectively are different. We define the following sequence (J n ) n 1 of finite words:

J 1 = ǫ and J n = W n-1 2W n-2 2 • • • W 1 2 for n 2.
Then, by induction, we show that there exists a sequence (I n ) n 1 of finite words

I 0 = 1, I 1 = 21, I 2 = 1221, . . . and I n • •I n-1 for n 1 such that V n-1 V n = 2J n I n and V n = 2J n I n-1 for n 1.

Hence we have

W = U n V 3 n 2J n I n • • • and X 1 = U n V 3 n 2J n I n-1 • • • and this implies t n = |U n | + 3|V n | + |J n | + 2. We observe that, for n 1, we have |J n | = |F n | = (ℓ n + ℓ n-1 -1)/2.
Consequently we get

t n = ℓ n + ℓ n-1 + 1 + 3(ℓ n + 1) + (ℓ n + ℓ n-1 -1)/2 + 2 = (9ℓ n + 3ℓ n-1 + 11)/2.
Since deg(S n ) = (ℓ n + ℓ n-1 + 3)/2 + ℓ n + 1 = (3ℓ n + ℓ n-1 + 5)/2, we obtain

ω n = t n / deg(S n ) = 3 -4/(3ℓ n + ℓ n-1 + 5),
as stated in the lemma.

⊓ ⊔ Lemma 2 For n 1, we set S ′ n = T 3ℓn+ℓn-1+4 -1. Then, for n 1, there exists R ′ n ∈ Q[T ] such that |θ -R ′ n /S ′ n | = |S ′ n | -ω ′ n with ω ′ n = 2 + (ℓ n + ℓ n-1 + 1)/(6ℓ n + 2ℓ n-1 + 8).
Proof For n 1, we introduce the following word

U ′ n = U n+1 2 = W n+1 2W n 2.
We consider the infinite word

X 3 = U ′ n U ′ n • • • U ′ n • • • = (U ′ n ) ∞ .
Lemma 3 Let R n , S n , R ′ n and S ′ n be the polynomials defined in the two preceding lemmas. For n 1, we have the equality

R n S ′ n -R ′ n S n = (-1) n (T -1). Consequently, for n 1, we have gcd(R n , S n ) = gcd(R ′ n , S ′ n ) = 1. Proof We shall use the finite words U n , V n , F n , G n , H n and U ′
n , introduced in the previous lemmas. For n 1, let us introduce in Q[T ] the polynomials:

P n = T |Fn+1|+1 (T |Vn+1| + 1) and Q n = T |Gn| .

Note that we have

|F n+1 | + 1 (3ℓ n + ℓ n-1 + 3)/2, |V n+1 | = ℓ n+1 + 1 and |G n | = (ℓ n +ℓ n-1 +3)/2.
Consequently, using the recurrence relation on ℓ n and the definitions of S n , S ′ n , S n+1 and S ′ n+1 , a direct and elementary computation shows that, for n 1, we have

S ′ n+1 = P n S n+1 + S ′ n , (1)
S n+1 = Q n S ′ n -S n . (2)
Now we shall see that, for n 1, the very same equalities as above hold, if S is replaced by R. Indeed, for n 1, we have

R n+1 = φ(G n+2 ) -φ(G n+1 ) = φ(U n+1 H n+1 ) -φ(G n+1 ) = T |Hn+1| φ(U n+1 ) + φ(H n+1 ) -φ(G n+1 ) = T |Hn+1| φ(U n+1 ) + φ(2G n ) -φ(G n+1 ) = T |Hn+1| φ(U n+1 ) + 2T |Gn| + φ(G n ) -φ(G n+1 ) = T |Gn| (T |Hn+1|-|Gn| φ(U n+1 ) + 2) -R n = T |Gn| (T φ(U n+1 ) + 2) -R n = T |Gn| φ(U ′ n ) -R n = Q n R ′ n -R n , since H n+1 = 2G n implies |H n+1 | -|G n | = 1.
To establish the next formula, we will use the following observation: for n 2 we have

U n = W n 2W n-1 = W n-1 2W n-2 2W n-1 2W n-1 = U n-1 V n-1 V n-1 .
Then, we also have

R ′ n+1 -R ′ n = φ(U ′ n+2 ) -φ(U ′ n+1 ) = T φ(U n+2 ) + 2 -(T φ(U n+1 ) + 2) = T (φ(U n+2 ) -φ(U n+1 )) = T (φ(U n+1 V n+1 V n+1 ) -φ(U n+1 )) = T (T 2|Vn+1| φ(U n+1 ) + φ(V n+1 V n+1 ) -φ(U n+1 )) = T (T 2|Vn+1| φ(U n+1 ) + T |Vn+1| φ(V n+1 ) + φ(V n+1 ) -φ(U n+1 )) = T ((T 2|Vn+1| -1)φ(U n+1 ) + (T |Vn+1| + 1)φ(V n+1 )) = T (T |Vn+1| + 1)((T |Vn+1| -1)φ(U n+1 ) + φ(V n+1 )) = T (T |Vn+1| + 1)(T |Fn+1| R n+1 ) = P n R n+1 .
From the approximations given in Lemma 1 and Lemma 2, since we clearly have ω n > 2 and ω ′ n > 2 for n 1, we can conclude that the rational functions R n /S n and R ′ n /S ′ n are convergent to θ. Hence, for n 1, there are three integers N (n), M (n) and L(n) such that we have

R n /S n = x N (n) /y N (n) , R ′ n /S ′ n = x M(n) /y M(n) and R n+1 /S n+1 = x L(n) /y L(n) .
Since the rational functions R n /S n and R ′ n /S ′ n are in their lowest terms, we have

|S n | = |y N (n) | and |S ′ n | = |y M(n) |.
On the other hand if we set D n = deg(S n ) and D ′ n = deg(S ′ n ), we observe the following

D n = (3ℓ n + ℓ n-1 + 5)/2 < D ′ n = 3ℓ n + ℓ n-1 + 4 < D n+1 for n 1.
Consequently, for n 1, we have

N (n) < M (n) < L(n). Since we have deg(y n ) = 1 i n deg(a i ), we obtain D ′ n = D n + d N (n)+1 + • • • + d M(n) ,
and

D n+1 = D ′ n + d M(n)+1 + • • • + d L(n) .
Due to the rational approximation of the convergents, for n 1, we also have

ω n = 2 + d N (n)+1 /D n and ω ′ n = 2 + d M(n)+1 /D ′ n .
From these two equalities, with the values for ω n and ω ′ n given in the lemmas, for n 1, we get

d N (n)+1 = (3ℓ n + ℓ n-1 + 1)/2 and d M(n)+1 = (ℓ n + ℓ n-1 + 1)/2.
At last, a straightforward computation shows that

D ′ n -D n = d N (n)+1 + 1 and D n+1 -D ′ n = d M(n)+1 + 1.
Since d i 1 for i 1, comparing with the above formulas, we conclude that for instance, we get S 1 = T 2 (T 2 -1) and R 1 = (T 2 -2)(T -1), and the property of coprimality for the pair (R 1 , S 1 ) fails. This coprimality is necessary to get the regularity of the sequence of the degrees as stated in Theorem 3, and when it fails this brings a slight perturbation in the continued fraction expansion of the corresponding generating function. Of course, we are well aware of the many possible generalizations and extensions, concerning the word W itself and the continued fraction expansion which is derived from it, in the formal case, in characteristic zero and also in positive characteristic, but even in the real case by specializing the indeterminate. However, the aim of this note is simply to describe the continued fraction presented here below, thinking that it has the accidental beauty of some singular mathematical objects.

M (n) = N (n) + 2 and d M(n) = 1, but also L(n) = M (n) + 2 and d L(n) = 1. We observe that R 1 /S 1 = (T 3 + 2T 2 + T -1)/(T 4 -T 2 ) = [0, a 1 , a 2 ,
Finally, we state a conjecture giving the precise form, and not only the degree, of the partial quotients in the continued fraction expansion for θ. However, in order to avoid long and sophisticated arguments, we have not tried to prove what is stated below. The first partial quotients are given by expanding the convergent R 1 /S 1 . We have R 1 /S 1 = [0, a 1 , a 2 , a 3 , a 4 ] and a direct computation gives For the partial quotients from the fifth on, we have the following conjecture.

Conjecture 1 Let (ℓ n ) n 0 be the sequence of integers defined in Theorem 3. For n 1, there exists (λ 1,n , λ 2,n , λ 3,n , λ 4,n ) ∈ Q 4 such that a 4n+1 = λ 1,n (T (3ℓn+ℓn-1+3)/2 + T (ℓn+ℓn-1+1)/2 -2)/(T -1), a 4n+3 = λ 3,n (T (ℓn+ℓn-1+3)/2 -1)/(T -1) and a 4n+2 = λ 2,n (T -1), a 4n+4 = λ 4,n (T -1).

Moreover the sequences (λ i,n ) n 1 in Q, for i = 1, 2, 3 and 4, are as follows:

λ 1,n = (-1) n+1 r 2 n , λ 2,n = (-1) n+1 (r 2 n + r n r n+1 ) -1 , λ 3,n = (-1) n+1 (r n + r n+1 ) 2 , λ 4,n = (-1) n+1 (r 2 n+1 + r n r n+1 ) -1

where the sequence (r n ) n 1 in Q is defined by r n = 4(2ℓ n -ℓ n-1 + 1)/25 for n 1.

  our notation, if M 1 and M 2 are two different words, then we have |Φ(M 1 ) -Φ(M 2 )| = |T | -t , where t is the first position where M 1 and M 2 differ. Therefore we have |θ

Remark 1

 1 a 3 , a 4 ]. Consequently, we have N (1) = 4 and by induction, N (n) = 4n, M (n) = 4n+ 2 and L(n) = 4n + 4, for n 1. So the proof of the theorem is complete. The reader will observe that the choice of the pair (1, 2) in the definition of the infinite word W is arbitrary. Clearly, the structure of W only depends on a pair of symbols (a, b) with a = b. To introduce the generating function associated to W , we could have taken an arbitrary pair (a, b) in Q 2 with a = b. However, deciding to keep close to the original sequence, we take (a, b) = (1, 2). The structure of the infinite world for an arbitrary (a, b) implies the existence of the rational approximations given in the first lemmas and the pseudo-coprimality obtained in the last one. Note that, with (a, b) = (1, -1)

a 1 =

 1 T -2, a 2 = T /2 + 1/4, a 3 = 8T /5 + 76/25, a 4 = -125T /48 + 25/24.

Let us consider now the element X 4 ∈ Q((T -1 )) such that Φ(X 3 ) = X 4 . Indeed, we can write

Consequently X 4 ∈ Q(T ) and we have

We shall now study the approximation of θ by X 4 = R ′ n /S ′ n . First, for n 1, we observe the following equalities

To compute t ′ n , we have to compare the finite words W n-1 2W n 2W n+2 and U ′ n . For n 1, let us consider the word J n , introduced in the proof of Lemma 1.

For n 1, by induction, we prove that there are two finite words A n and B n such that we have

Hence we have

This gives us the desired value for ω ′ n stated in the lemma.

⊓ ⊔

Before proving Theorem 3, we need one last lemma to establish the irreducibility of the rational functions R n /S n and R ′ n /S ′ n .

Consequently, we have obtained

Combining ( 1), ( 2), ( 3) and ( 4), we have

and

For n 1, we set

and, in the same way, we have from ( 6)

From ( 7) and ( 8), we obtain

Finally, for n = 1, we have

From this we get directly ∆ 1 = -T + 1 and therefore ∆ n = (-1) n (T -1), for n 1, as desired. The statement on the greatest common divisor follows, since R n (1) = 0 and R ′ n (1) = 0. ⊓ ⊔ Proof of Theorem 3: We first observe that, due to the very good rational approximations for θ stated in Lemma 2 and Lemma 3, it is clear that θ is irrational. Let us start by recalling classical results concerning the continued fraction algorithm in the formal case. Let α be an irrational element in F(K) and α = [a 0 , a 1 , . . . , a n , . . . ] its continued fraction expansion. Let (x n /y n ) n 0 be the sequence of convergents to α. The polynomials x n and y n are defined recursively by the same relation z n = a n z n-1 + z n-2 , with initial conditions (x 0 , x 1 ) = (a 0 , a 0 a 1 + 1) and (y 0 , y 1 ) = (1, a 1 ). We have gcd(x n , y n ) = 1 and deg(y