THE BORN APPROXIMATION IN THE THREE-DIMENSIONAL CALDER ´ON PROBLEM

. Uniqueness and reconstruction in the three-dimensional Calder´on inverse conductivity problem can be reduced to the study of the inverse boundary problem for Schr¨odinger operators − ∆ + q . We study the Born approximation of q in the ball, which amounts to studying the linearization of the inverse problem. We ﬁrst analyze this approximation for real and radial potentials in any dimension d ≥ 3. We show that this approximation is well-deﬁned and obtain a closed formula that involves the spectrum of the Dirichlet-to-Neumann map associated to − ∆ + q . We then turn to general real and essentially bounded potentials in three dimensions and introduce the notion of averaged Born approximation, which captures the exact invariance properties of the inverse problem. We obtain explicit formulas for the averaged Born approximation in terms of the matrix elements of the Dirichlet to Neumann map in the basis spherical harmonics. To show that the averaged Born approximation does not destroy information on the potential, we also study the high-energy behavior of the matrix elements of the Dirichlet to Neumann map.


Introduction and main results
1.1.Motivations and setting of the results.Let d ≥ 2 and consider a bounded open set Ω ⊂ R d and a function γ ∈ L ∞ (Ω, R) such that γ(x) ≥ γ 0 > 0 for almost every x in Ω.The solution u ∈ H 1 (Ω) of the problem (1.1) −∇ • (γ∇u) = 0 in Ω, u| ∂Ω = g, models the electrostatic potential generated in the conductor Ω, having conductivity γ, by a voltage g at the boundary.If ∂Ω is smooth enough, the exterior normal vector field ν to ∂Ω is well-defined and the quantity γ(x)∂ ν u(x) = γ(x)ν • ∇u(x) represents the electric current density flowing through the boundary at x ∈ ∂Ω.
The Calderón inverse problem (which dates back to 1980, see [Cal06]) consists in determining whether or not one can recover the values of γ on Ω from measurements of γ∂ ν u made at the boundary ∂Ω for different choices of g in (1.1).In order to give a more mathematically precise statement, it is convenient to think of γ∂ ν u as the value at g ∈ C ∞ (∂Ω) of the linear operator Λ[γ]g := γ∂ ν u, where u is the unique solution of (1.1) corresponding to this choice if g.This operator is known as the Dirichlet to Neumann map, or D-N map for short, since it maps the Dirichlet data to the Neumann data of the boundary value problem (1.1).It was shown by Sylvester and Uhlmann [SU88] that, at least when γ is smooth, that Λ[γ] is a first order pseudodifferential operator on ∂Ω, and more generally, it always maps H 1/2 (∂Ω) into H −1/2 (∂Ω).The Calderón problem can be seen as a model of Electrical Impedance Tomography (EIT), which is related to important applications in medical imaging [ALMM + 01, CIN99, IMNS06], or geophysics.
The Calderón problem can be reformulated as determining whether or not the mapping γ −→ Λ[γ] is injective (at least when γ varies in some subspace of L ∞ (Ω, R)) (uniqueness), and in the affirmative case, in providing appropriate algorithms to recover γ explicitly from Λ[γ] (reconstruction).The uniqueness and the related question of stability have driven a considerable amount of research since Calderón's seminal contribution and the early works [KV84,SU87].The reconstruction aspect has also been the subject of intensive research since the work [Nac88].When d = 3 we mention among many others [Ale90, Bro01, CR16, GLU03, Hab15], see for example [GKLU09,Uhl14] for more references.We will give more references on the numerical aspects of this problem later on.
A well known simplification, particularly relevant when it comes to designing algorithms for solving the reconstruction problem, is that one can assume that Ω is a ball in euclidean space without loss of generality.In fact one can also assume that the conductivity is equal to 1 in a neighborhood of the boundary.This is a consequence of the boundary determination results of [Ale90,Bro01], which make possible the construction of an appropriate extension of γ to a ball containing Ω.One can recover the D-N map in ∂Ω from the D-N map in the boundary of the ball.See [CR16, Section 3] for more comments and references on this procedure.
From this observation, it is possible to show that the uniqueness question can be reduced to the unique determination of a potential q ∈ L ∞ (Ω, R) from the Dirichlet to Neumann map Λ q associated to the Schrödinger operator −∆ + q on Ω.In particular, one has that The approach used since [SU87] to prove to uniqueness and reconstruction of q from Λ q in d ≥ 3 involves certain exponentially growing solutions on Ω of the equation −∆u + qu = 0 known as Complex Geometrical Optic solutions or CGOs.To describe these solutions, we define the following algebraic variety of C d : (1.4) for the exponential harmonic functions introduced in this context by Calderón in [Cal06].Given h > 0 and ζ ∈ V(d), a CGO is a family of functions ψ h ζ ∈ H 1 (Ω) that solve Notice that when ζ ∈ V(d), the exponential e ζ/h is a harmonic function in R d whose L 2 (Ω)-norm diverges as h → 0 + .
The origins of CGOs date back to Faddeev [Fad65], who introduced similar objects in the context of Scattering Theory.They have been shown to exist under different regularity assumptions on the potential/conductivity.Their importance stems from the fact that, as first proved by Sylverster and Uhlmann [SU87], if one extends q by zero to the whole of R d then its Fourier transform satisfies: (1.5) q(ξ) = lim , We remark that (1.5) also holds when (1.6) provided that r h ∈ R d and |r h | = o(h) as h → 0 + .Here we are following the convention for the Fourier transform of a function f ∈ L 1 (R d ).
In order to use (1.5) to reconstruct q in d = 3, one needs first to know the boundary values of ψ h ζ .This can be obtained by solving a boundary integral equation using only the Λ q map, as shown by Nachman in [Nac88].From the numerical point of view, a number of challenges appear when trying to implement this scheme, not least the exponentially growing and oscillatory nature of the CGO solutions as h → 0 + .See for example [DHK11,DHK12,DK14] for the case of d = 3, and [MS20,MS12] for more references on the numerical aspects of EIT.
A possible strategy to avoid those difficulties is to proceed as it is customary in Scattering Theory, and introduce the so-called Born approximation q exp of the potential: the linearization of the inverse problem.We can formally define q exp by the expression Here the use of the duality pairing , H 1/2 (∂Ω)×H −1/2 (∂Ω) is not necessary due to the smoothness of e ζ .
The Born approximation q exp has been considered mainly as a tool to obtain numerical results in the reconstruction problem, see [BKM11,KM11].Note that a notion of Born approximation can be introduced also in the case of conductivities.In particular, one can obtain a Born approximation γ exp for the conductivity problem from q exp by linearizing (1.2) (see [BKM11,KM11] for this and other approaches).Therefore, obtaining explicit formulas for (1.7) in terms of the D-N map also yields to interesting consequences for the inverse conductivity problem.Note also that the linearization of the conductivity problem was already studied by Calderón in [Cal06].The effectiveness of the Born approximation has been recently compared to other numerical reconstruction methods in [HIK + 21] using synthetic data to simulate real discrete data from electrodes.
As far as we know, even the existence of the limit h → 0 + in (1.7) is a priori not clear.Also, the limit could depend on the choice of ζ 1 , ζ 2 ∈ V(d) satisfying (1.6).In this work we will address these questions and provide explicit formulas to compute q exp from the matrix elements of the D-N map when Ω is a ball.As we mentioned before, working in the ball does not involve any loss of generality when it comes to solve the reconstruction problem.
With these considerations in mind, in the rest of this article we shall assume that Ω is the ball B = {x ∈ R d : |x| ≤ 1}.We will denote the boundary of B, the unit sphere, by S d−1 and consider the family (1.9) The purpose of this work is to investigate the structure of the operators Λ q , partly motivated by the reconstruction problem.In order to do so, we will first show that one can obtain explicit formulas for the Born approximation in terms of matrix elements of the D-N map.From the numerical point of view, this avoids the difficulties of dealing with the exponentially growing and oscillatory e ζ/h functions and the limit h → 0 + .We then examine the structure of the matrix elements of the operator Λ q in the basis of spherical harmonics.In the particular case of radial potentials, they coincide with the eigenvalues of the D-N map; therefore we will examine this case in detail in order to get some insight and motivate our results for the general case.
1.2.The Calderón problem on the ball.The radial case.Recall that spherical harmonics are restrictions to the sphere S d−1 of the complex homogeneous polynomials of d variables that are harmonic.We denote the subspace of spherical harmonics of degree k by H k .Spherical harmonics of different degrees are orthogonal in L 2 (S d ).Moreover, if We denote by SO(d) the special orthogonal group in dimension d.For every Q ∈ SO(d) write: (1.12) In particular, if This means that q is a radial function and that Λ q | H k = λ k [q] Id H k for every k ∈ N 0 , where N 0 stands for the set of non-negative integers (see Section 2.2).Note that, since Λ q is self-adjoint over L 2 (S d−1 ), the eigenvalues λ k [q] are real and Λ q = χ q (−∆ S d−1 ) for some measurable function χ q defined on [0, ∞).
When q = 0 one has Spec Λ 0 = N 0 and, in view of (1.11), In fact, if u solves (1.10) with q = 0 and g = ϕ k for any ϕ k ∈ H k , then u must coincide with the solid spherical harmonic u one obtains by differentiation: Note that when d = 2, Λ 0 coincides with the fractional Laplacian (−∆ S d−1 ) 1/2 .
In the following theorem we prove that the limit (1.7) converges in the radial case by providing a closed formula.Moreover, we compute explicitly the difference between the Born approximation and the Fourier transform of the potential.Define the moments of the potential as: (1.16) where |S d−1 | is the volume of the sphere.
Theorem 1.Let d ≥ 3 and q ∈ Q d be of the form q = q 0 (| • |), where Then the Born approximation q exp (ξ) given by (1.7) is well-defined.Moreover, if (λ k [q]) k∈N 0 denote the eigenvalues of the D-N map then: (1.17) If, in addition, r h = 0 then no limit needs to be taken in (1.7): Identity (1.19) was conjectured by [BKM11] based on numerical evidence.Note also that an analogue of (1.17) has been given in [KLMS07] and [SMI00,equation (41)] in the two-dimensional case.
We remark that the series (1.17) converges absolutely, and thus q exp (ξ) is well defined, provided that λ k [q] − k grows as a power of k.This is a well known fact, see however our next result Theorem 2 below for more precise spectral asymptotics.On the other hand, note that (1.17) is a formal definition of q exp in the following sense: even if the series on the right hand side it is absolutely convergent for all ξ ∈ R d , there is not an a priori control of the growth in the ξ variable necessary to define the inverse Fourier transform of (1.17).
Identity (1.18) follows immediately from the fact that whenever where the series on the right is absolutely convergent for all ξ ∈ R d , see Section 3 for a short proof of this formula.The fact that q has compact support is essential here: the above formula is false for general Schwartz class functions.The similarity between identities (1.20) and (1.17) implies that the Born approximation satisfies formally that σ k [q exp ] = λ k [q] − k, and hence it follows that the linearization of the Calderón problem for the Schrödinger operator −∆ + q is a Hausdorff moment problem.This theorem, whose proof is presented in Section 3, motivates studying the spectrum of the D-N map in the radial case.Our next results gives a perturbation series for the eigenvalues of Λ q for a radial potential q ∈ Q d .
Theorem 2. Let d ≥ 2 and q ∈ Q d be of the form q = q 0 (| • |), where q 0 ∈ L ∞ ([0, 1], R).Let α := max ess supp q 0 .Then, for all k > α q 0 1/2 2 the eigenvalues λ k [q] of Λ q can be written as a series where the real numbers σ k,n [q] can be explicitly computed and satisfy that Remark 1.1.The explicit formula to compute σ k,n [q] is identity (5.18) in Section 5.1.For instance, the case n = 2 corresponds to: which is of lower order in k than: Remark 1.3.It is well known (see [SU88]) that whenever q is smooth and compactly supported in the interior of B, Λ q is a pseudodifferential operator that is smoothing to all orders.Theorem 2 quantifies the structure of the exponentially small remainder in the radial case.The fact that the exponential rate depends on the distance of the essential support to the boundary of B was already noticed in [Man01].
Let us note that the spectral theory of D-N operators is a very active subject, see for instance the survey article [GP17] and the references therein.See also [DKN20,DKN21] for recent results in a context closer to that of Theorem 2.
Theorem 1 shows in particular that from the knowledge of the moments σ k,1 [q] for all k ∈ N 0 one can recover completely the Fourier transform of q.On the the other hand, (1.24) shows that the moments σ k,1 [q] can be approximated by λ k [q] − k modulo an error of order k −2 σ k,1 [q].Heuristically this gives evidence that the Born approximation captures the high frequency information of the potential, as it is well known in scattering problems (see Section 3 for more references).In a similar spirit, the recovery of jumps of discontinuous conductivities from the linearization of the inverse problem has been recently studied in the two dimensional case in [GLS + 18] .
1.3.The Calderón problem on the ball.General Schrödinger operators.We now address the more complex non-radial case when d = 3.If one tries to follow the strategy we implemented in the radial case, the resulting expression in (1.7) involves all matrix elements of Λ q , some of them with divergent coefficients as h → 0 + .To avoid this problem we perform an averaging procedure that leads to a convergent expression.This average also has the advantage of reducing the number of matrix elements involved in the final result, yielding a formula that could be used in practical applications.The averaging procedure is motivated by the following observation.Given ω ∈ S 2 the Fourier transform of q which has the following invariance property where, if ω ∈ S d−1 , the set SO(d) ω is the stabilizer of ω: On the other hand, one expects the Born approximation to enjoy the same symmetry properties that the Fourier transform has.Recalling (1.7) however, for every where the pair This implies that, prior to taking the limit h → 0 + , the Born approximation (or more precisely, the scattering transform) does not necessarily share the invariance property (1.25).In fact, in general, the right hand side of (1.27) differs from This motivates the introduction of an averaged version of the Born approximation.In order to do so, let: and, for ω ∈ S 2 define the angular momentum with respect to ω as: This is a self-adjoint operator on L 2 (S 2 ) whose spectrum is equal to Z and whose definition is independent of the particular choice of P .Therefore, it generates a unitary group e −itLω , t ∈ R, that is 2πZ-periodic; in fact it generates the stabilizer of ω: see Section 2.3 for more details.We define the averaged operator: (1.29) Λ q ω := 1 2π 2π 0 e itLω Λ q e −itLω dt.
Note that, by construction, [ Λ q ω , L ω ] = 0.The operator Λ q ω is known as the quantum average of Λ q along the flow e −itLω .This construction is the starting point for implementing the averaging method in quantum mechanics that goes back to [Wei77] and has been used extensively in the spectral theory, see among many others, [Gui78, Uri84, Uri85, OVVB12, GUW12, MR16].
We are now in position to define formally the averaged Born approximation q exp by the formula (1.30) As in the case of the Born approximation, the existence of this limit (and the possible dependence on the choice of ζ 1 , ζ 2 ) is not clear a priori.The motivation behind this definition follows from the next remark.
The previous identities also show that for all Q ∈ SO(3) ω and s > 0 This is the analogue of (1.25).Moreover, Remark 1.4 implies that, as soon as q is invariant by SO(3) ω q exp (sω) = q exp (sω), for all s > 0.
As we will later justify, this averaging procedure does not destroy any relevant information on the potential q.In order to state the main result of this section, we specify an orthonormal basis of spherical harmonics, namely the one that diagonalizes the angular momentum operators L ω .Since L ω commutes with ∆ S 2 , it turns out that the self-adjoint operator L ω leaves H invariant.It can be proved that the spectrum of L ω | H is simple and is given by the integers m such that − ≤ m ≤ .We will denote Y ω ,m , with − ≤ m ≤ , the corresponding basis of eigenfunctions:

More details on the angular momentum and the specific choice of the Y ω
,m eigenfunctions are provided in Section 2.3.We start by defining the following matrix elements of the D-N map for any ω ∈ S 2 , and we now define the following moments of q Notice that m k, ;ω [q] is a matrix element in the solid harmonics of the multiplication operator associated to the potential q.
Take ξ ∈ R 3 /{0}.Let {η 1 , η 2 , ω} be a positively oriented orthonormal basis of R 3 , where For the sake of simplicity we also introduce the constants Our last main result is a closed formula for the averaged Born approximation.
Theorem 3. Consider a potential q ∈ Q 3 .Take ξ ∈ R 3 \ {0} and let ω := ξ/|ξ|.Assume q exp (ξ) is given by (1.30) with ζ 1 and ζ 2 as in (1.34).If λ k, ;ω [q] are given by (1.32) then where δ k, denotes the Kronecker delta.Also, the m k, ;ω [q] coefficients defined in (1.33) are the linearization of the matrix elements λ k, ;ω [q] − kδ k, , and one has that 1 where α := max x∈ess supp q |x|.Moreover, for all functions q ∈ L 1 (R 3 ) with compact support the following holds This theorem shows that the limit (1.30) is well defined, and (1.37) and (1.38) show that the average used to define q exp (ξ) does not destroy any relevant information on the potential.As expected, in the radial case one can recover (1.17) from formula (1.36), since Also, notice that in the case k = the moment m k,k;ω [q] behaves sharply as α 2k (2k) −1 -to see this just assume q is the indicator function of the ball B α with α ≤ 1-so estimate (1.37) implies an extra decay of k −1/2 in the right hand side.
Again, we remark that the series (1.38) is absolutely convergent provided q has compact support and this formula fails for general functions in the Schwartz class.The right hand side of (1.36) is also an absolutely convergent series, but there is no control on the growth of q exp (ξ) as |ξ| → ∞ to guarantee that it is a tempered distribution.Therefore we need to consider (1.36) just as a formal definition of q exp .Let us note that the results in Theorems 1 -3 can be used as the basis of efficient numerical algorithms to reconstruct the Born approximations γ exp and q exp .This can be used to improve and simplify the algorithms to reconstruct the potential and the conductivity from the D-N map.This is the subject of the article [BCMM22].
1.4.Structure of the article.Section 2 presents some well-known facts on spherical harmonics, the action of the group of rotations on the sphere and angular momentum operators that will be used throughout the paper.In Section 3 we prove Theorem 1, whereas Section 4 is devoted to the proof of identity (1.36).In Section 5 we study the spectrum and matrix elements of the D-N map.There we prove Theorem 2 and formula (1.37).Finally, Section 6 presents the proof of the identity (1.38) for the Fourier transform and completes the proof of Theorem 3.
1 Throughout the paper we write a b when a and b are positive constants and there exists C > 0 so that a ≤ Cb.We refer to C as the implicit constant in the estimate.

A few preliminary facts on spherical harmonics
In this section we summarize some properties of the angular momentum operators and of the spherical harmonics which will be useful later on.All the properties stated here without proofs are well known (see for example [AH12, SW71, Tay86]).
2.1.Spherical harmonics.The eigenfunctions of ∆ S d−1 acting on L 2 (S d−1 ) are the restriction to the sphere of the complex homogeneous polynomials of d variables that are harmonic.These functions are called the spherical harmonics; the set of spherical harmonics of degree k is denoted by H k .Spherical harmonics of different degrees are orthogonal in L 2 (S d−1 ).If ϕ ∈ H k then: One also has that: is dense in L 2 (S d−1 ), and that H k is spanned by the restriction to the sphere of the polynomials: where V(d) was defined in (1.4).This particular class of spherical harmonics will play an important role in this work, as we will later see in Sections 3 and 4.
Then, for all k ∈ N 0 we have that (2.1) where Proof.Let k ∈ N 0 and define Then, by (1.15) and Green formula we have that Therefore one gets, for k > 0, which implies that shows that the identity (2.1) holds with .
This finishes the proof since If we now define the spherical harmonics as a consequence of the previous lemma we have that In the particular case |ζ| = √ 2 -for example if ζ = e 1 + ie 2 with e 1 and e 2 orthonormal-it follows that 2.2.Representation of SO(d) on L 2 (S d−1 ) and angular momentum.We now briefly recall some properties of the representation of SO(d) on L 2 (S d−1 ) introduced in (1.12) (see for example [Tay86]).Recall that so(d), the Lie algebra of SO(d), can be identified to the set of skew-symmetric d × d matrices.Given ω ∈ so(d), the family (e tω ) t∈R defines a one parameter subgroup on SO(d) and SO(d for some self-adjoint first-order differential operator L ω on L 2 (S d−1 ) that we will call the angular momentum associated to ω.Then (2.5) Let A denote an operator on L 2 (S d−1 ) such that the space C ∞ (S d−1 ) is dense in D(A) and mapped into itself by A. Then for every ω ∈ so(d) one has that This is the case for A = ∆ S d−1 , and hence it follows that e −itLω , L ω : In addition, using the fact that ∆ S d−1 can be expressed in terms of the angular momentum operators, one can prove that an operator A as above satisfies Since the representation Q → R Q is irreducible when restricted to H k , Shur's lemma implies that every operator A that maps H k onto itself and commutes with rotations satisfies 2.3.Spherical harmonics and angular momentum when d = 3.When d = 3 the Lie algebra so(3) is isomorphic to R 3 equipped with the Lie bracket given by the cross product.
Because of this, the expression of the angular momentum operators is particularly simple.Any angular momentum operator is a linear combination of the operators corresponding to the vectors {e 1 , e 2 , e 3 } of the canonical basis of R 3 : (2.6) These operators act both on L 2 (S 2 ) and L 2 (R 3 ).For simplicity we will denote the operators by the same symbol independently of the space in which they are acting.If ω ∈ R 3 then where ω 1 , . . ., ω 3 , are the components of ω with respect to the canonical base.In addition, when |ω| = 1 then for every P ∈ SO(3) such that P (ω) = e 3 one has: (2.7) Therefore, all angular momentum operators associated to unitary vectors are unitarily equivalent.
In these coordinates the angular momentum operator L 3 is given by (2.9) Recall that the self-adjoint operatos L ω map, for any k ∈ N 0 , the space of spherical harmonics of degree k into itself.We now introduce the specific basis of eigenfunctions of L 3 that will be useful for our purposes.
To that aim, let us introduce a few preliminary facts on the associated Legendre polynomials.Let , m ∈ N 0 with 0 ≤ m ≤ , and let P m be the associated Legendre polynomial (2.10) where P is the Legendre polynomial of degree , given by: 2 (2.11) see for example [Leb65,p. 44].For later use we remark that P (1) = 1.From (2.10) it follows that (2.12) for an appropriate constant c > 0. Definition (2.10) can be extended to − ≤ m ≤ 0 as follows: Using the parametrization defined in (2.8), for ∈ N 0 , m ∈ Z such that − ≤ m ≤ the spherical harmonic Y ,m : S 2 → C of degree , order m and north pole e 3 is given by the formula With this convention the family (Y ,m ) ∈N 0 ,− ≤m≤ is an orthonormal basis of L 2 (S 2 ).Note that since the associated Legendre polynomials are real. 3It turns out that the functions in this basis are eigenfunctions of L 3 .In fact, it follows from (2.9) and (2.13) that: (2.15) This shows that the spectrum of L 3 is Z, and that the spectrum of The same holds, by virtue of (2.7), for any other angular momentum operator L ω with |ω| = 1.For any given P ∈ SO(3) such that P ω = e 3 the functions 3 The sign convention used here may seem cumbersome, but it is one of the usual conventions in quantum mechanics since it simplifies working with the ladder operators.When m > 0, the (−1) m factor appearing in (2.13) is known as Condon-Shortley phase.

The Born approximation: the radial case
In this section we prove formula (1.20) and Theorem 1.
We start by showing that the Fourier transform of a radial function q ∈ L 1 (R d ) with compact support can be expressed in terms of the moments σ k,1 [q] defined in (1.16).If q(x) = q 0 (|x|) we have that , where the Bessel function J ν (t) is given by the series with R + = (0, ∞).Using this with ν = (d − 2)/2 in the formula of the Fourier transform, and moving the summation outside the integral -this can be justified provided q has compact support-yields that from which (1.20) follows directly.As mentioned, the compact support of q is essential, since one can easily find a Schwartz class function f in R d such that f vanishes in a neighborhood of ξ = 0, so cannot satisfy (1.20).
The identity (1.20) provides a nice interpretation of the formula we obtain for the Born approximation (1.17): from the point of view of reconstruction, q exp (ξ) can bee seen as the function that one gets when substituting the unknown moments σ k,1 [q] by the a priori known quantity λ k [q] − k in (1.20).
Proof of Theorem 1.Let ζ ∈ V(d).One of the key facts that allows to obtain the formula (1.17) is the following simple property.Using Taylor's expansion of the exponential one gets where we recall that (ζ • x) k is a spherical harmonic of degree k (see Section 2.1).For simplicity we write On the other hand, as a consequence of (3.2) it follows that and since in the radial case Λ q is diagonal in the spherical harmonics, we have that Hence, Lemma 2.1 and (3.3) yield Note that when r h = 0 one has l h = 0, the above quantity does not depend on h and (1.19) follows.Otherwise, since l h = o(h 2 ), taking limits as h → 0 + and using (1.7) and (2.2) gives directly (1.17).Combining (1.17) with (1.20) gives the second identity in the statement.This finishes the proof of the theorem.
Let us mention a further interesting consequence of Theorem 1.The coefficients of the series (1.17) satisfy for k > 1 that which means, roughly, that |a k | increases until k ∼ |ξ| and then decreases to 0 as k → ∞.Hence, |a k | attains its (very large) maximum values when k ∼ |ξ|.This implies that the eigenvalues making a greater contribution to the value of q exp (ξ) in (1.17) for a fixed ξ, are the as k → ∞, this suggest that q exp (ξ) will improve as an approximation for q(ξ) as |ξ| → ∞.Thus, the Born approximation should recover the leading discontinuities of the potential.This property is well known in backscattering and in the fixed angle scattering problems, see among others

The Born approximation: the general case
Here we consider the Born approximation in the general case.We will prove formula (1.36) of Theorem 3. We restate this result in the following proposition.
The proof will follow from a series of intermediate results and is given at the end of this section.We start with the following lemma.
As we have seen in Section 2.2, the identity (4.2) holds if and only if Then, as in the proof of (4.1), we have where the last identity follows from the 2π-periodicity of the flow e itLω .This finishes the proof of (4.2).
We now prove (4.3).Using the definition (1.29) we have using (2.14) and (2.18).This finishes the proof of the lemma.
One of the advantages of the previous lemma is that it allows to reduce the arguments to the case of ω = e 3 , as we will show later on.The identity (4.2) states that Λ q e 3 commutes with the angular momentum operator L 3 defined in (2.6).For convenience we fix (4.4) By (3.4), to compute the right hand side of (1.30) it is enough to know explicitly how Λ q e 3 − Λ 0 acts on the spherical harmonics k for all k ∈ N 0 .These functions can be expressed in terms of spherical harmonics with north pole e 3 : To see this, check that ).An explicit computation in the spherical coordinates given in (2.8) to find the complex phase yields which by (2.12) and (2.13) implies (4.5).
As we will see, the right-hand side of (1.7) can be computed for any self-adjoint operator A that commutes with L 3 .Thus, here we will work with A in place of Λ q e 3 − Λ 0 .This extra generality will come handy later on.
Lemma 4.3.Let A be an operator on L 2 (S 2 ) such that D(A) = H 1 (S 2 ) and [A, L 3 ] = 0.Then, if ζ 1 , ζ 2 are given by (4.4), we have where µ k, satisfies (1.35) and Proof.For convenience, in this proof we write ζ i = ζ i , i = 1, 2. Since L 3 and A commute, the eigenspaces of L 3 are invariant subspaces of A. This implies that By (2.3), (4.5) and (4.7) it follows that Using (2.3) and (3.4), we obtain that (4.8) lim The key to compute the last limit is to understand the behavior of 3) is given by (4.4).Then, for all k, ∈ N 0 , ≥ k we have that where c 1/2 was defined in (2.2).
We postpone the proof of this lemma to proceed with the proof of Lemma 4.3.Combining Lemma 4.4 with (4.8) we get lim To finish the proof we need to show that µ k, satisfies (1.35).Combining (2.2) and the identity (4.9) we get which yields (1.35).This finishes the proof the lemma.
The proof of Lemma 4.4 uses the so-called ladder operators L + and L − : (4.10) -see for example [Tes14, Section 8.2].The term ladder operators is motivated by the fact that these operators act as right/left shift operators in the index m: L 1 and L 2 have the following expressions in terms of ladder operators (4.12) Proof of Lemma 4.4.Let ζ = e 1 + ie 2 .We start by noticing that , where O h|ξ| belongs to SO(3) and, in the {e 1 , e 2 , e 3 } basis, is given by the matrix where sin = h|ξ| for h small.Therefore where to get the last equality we have used (4.5).We must be careful now, since we recall that , is not the L 2 product -there is not complex conjugation, see (1.8).Since we want to use the L 2 (S 2 ) structure of certain integrals, it is convenient to use (2.14) to write the quantity that we want to compute as follows Observe that the O h|ξ| matrices fix e 1 , and the generator of rotations that fix the e 1 vector is exactly the angular momentum operator L 1 defined in (2.6).This means that the operator R O h|ξ| can be generated from L 1 through the exponential map, as shown in (2.5).Thus, the Taylor's expansion of the exponential The operator L 1 is not diagonal on the basis of spherical harmonics used here, so we recall the Ladder operators introduced in (4.10) and (4.12).We have which makes computing Y ,−k , L p 1 (Y , ) straightforward using (4.11).On the one hand, in the case 0 ≤ p < + k for appropriate coefficients a j we have that On the other hand, when p = + k, notice that This means that there is exactly one term in the expansion of L p 1 Y , which does not vanish after taking the L 2 (S 2 ) product with Y ,−k .Hence one gets Substituting this in (4.14), and using that sin = h|ξ|, and hence = h|ξ| + O(h 2 ), we obtain that Using this in (4.13) finishes the proof of the lemma.
We now prove the main result in this section.

The structure of D-N map
This section is devoted to studying the spectrum and matrix elements of the Dirichlet to Neumann map.We will make extensive use of the following version of Alessandrini's identity [Ale88]: for every f, g ∈ H 1/2 (S d−1 ) the following holds: where u solves (1.10) and v solves ∆v = 0 in B, v| S d−1 = f.5.1.The radial case.In this section we prove Theorem 2. Let d ≥ 2 and consider a radial potential q(x) = q 0 (|x|) such that q∈Q d .If g ∈ H 1/2 (∂Ω) the direct problem (1.10) (5.2) −∆u(x) + q 0 (|x|)u(x) = 0 for x ∈ B, u| ∂B = g, and the Dirichlet to Neumann map is well-defined.By (1.13) this map commutes with all rotations in SO(d) and therefore Note that there exist a unique solution b k with these properties, otherwise there would be more than one function u satisfying (5.2).Thus, . From now on we will write λ k = λ k [q].The solid spherical harmonic associated to ϕ k ∈ H k is the harmonic function ϕ k (x) = |x| k ϕ k (x/|x|) with x ∈ B. Thus, (5.1) yields that (5.4) transforms equation (5.3) into its Liouville normal form: (5.5) subject to the boundary conditions: and with (5.6) V (− log r) = r 2 q 0 (r), V (t) = e −2t q 0 (e −t ).
Write e κ (t) := e −κt ; then (5.10) is equivalent to the following integral equation Multiplying by V one gets which can be solved with a Neumann series: (5.14) Assuming this holds, (5.11) becomes (5.15) where for n ≥ 2 the numbers σ k,n satisfy (5.16) To prove Theorem 2 first notice that undoing the previous change of variables and using (5.8), we have that (5.17) which is exactly the first order term in (1.21).Then the proof of Theorem 2 follows from the next lemma.
On the other hand, (5.16) and Cauchy-Schwarz inequality imply that Using Lemma 5.1, (5.7) and that This finishes the proof of the theorem.
We can now invert the change of variables to obtain an explicit formula for σ k,n [q] in terms of q(x) = q 0 (|x|).If n ≥ 2 and k + (d − 2)/2 > 0 we have that where we have that, for β, r, s > 0, 5.2.The general 3-dimensional case.In this section we will assume that d = 3.Our goal is to understand the asymptotic behavior of the matrix elements of the operator Λ q − Λ 0 that appear in the formula (1.36).
Let u k,m be the solution of (5.19) where q ∈ Q 3 , and Y k,m is a spherical harmonic with north pole e 3 , as defined in (2.13).Equation (5.19) has a unique solution u k,m , and the Dirichlet to Neumann map Λ q Y k,m = ∂ ν u k,m is well-defined.The solid spherical harmonic associated to Y k,m is the function Let k, ∈ N 0 and m, n ∈ N 0 such that −k ≤ m ≤ k and − ≤ n ≤ .The Allessandrini identity (5.1) yields that As in radial case, we introduce a function v k,m such that We now introduce the resolvent operator R 0 : L 2 (B) → H 1 0 (B) given by the solution operator of the Poisson problem in the ball.If d ≥ 3 this operator is given by Applying the resolvent R 0 in (5.23), we obtain the integral equation Inserting this and (5.22) in (5.21) gives In general, if α := max x∈ess supp q |x|, the first term in (5.25) satisfies the bound Notice that the resolvent operator is independent of k, in contrast with the radial case.This prevents us from using the Neumann series approach unless one imposes smallness assumptions on the potential.Nonetheless, since we are mostly interested to know if the first term in the right hand side of (5.25) is the dominant one as k, → ∞, we can still use the regularization effect provided by the resolvent operator to obtain extra decay for the second term in (5.25).
As mentioned in the introduction, (5.27) is sharp for k = , as shown by taking the indicator function of the ball as q, so (5.28) gives in an improvement in decay of order k −1/2 .
First, notice that (5.27) is the same as estimate (5.26) for m = n = k, so we have just to prove (5.28).By (1.32), we have that Therefore we need to estimate the second term in (5.25) in the special case m = n = k: where u k,k solves (5.19) with m = k (recall B α stands for of radius α centered at the origin where q is supported).We denote the integral term on the right by I. Hence We now use that in R 3 (5.30) This yields To finish, we use that u k,k = v k,k + Y k,k together with the identity which follows from (5.23).Note, that, since 0 is not a Dirichlet eigenvalue of (−∆ + q) (recall (1.9)), R q is well defined and is bounded in L 2 (B).Using this we get Since for all k ∈ N 0 and m ∈ Z with |m| ≤ k, direct integration of the solid harmonic yields which proves (5.28).

Proof of the Fourier transform formula
On this section we prove representation (1.38) for the Fourier transform.This finishes the proof of Theorem 3, since (1.36) has been proved by Proposition 4.1 and (1.37) by Proposition 5.2.Let us state again the result given by (1.38).
Proof of Theorem 3. It follows directly putting together Propositions 4.1, 5.2 and 6.1.
The key to prove Proposition 6.1 is to express products of spherical harmonics as linear combinations of terms with one spherical harmonic.This is provided by the following lemma.The proof of this lemma involves computations with the Clebsch-Gordan coefficients, so we leave the details for the appendix A.
Proof of P roposition 6.1.The absolute convergence follows easily by the estimate (5.27).The µ k, ;ω [q] grow exponentially with k -here we don't assume q supported in B, so α can be larger that 1-but this is compensated by the factors (2k)!( − k)!(k + )! in the denominator.
Let x = |x|θ and ξ = |ξ|ω, where ω, θ ∈ S 2 .By (5.29) and the invariance properties of the Fourier tranform under rotations, it is enough to prove the case ω = e 3 .We recall that In the right hand side of (6.1), let us commute the integral of q that appears in the previous expression for m k, ;e 3 with the summation in k and .This is justified by the absolute convergence of the series and the absolute value estimate (5.27) for the integrals that define the moments.Thus, the identity (6.1) can be written as q(ξ) = As a consequence, to prove that (6.1) holds true with ξ = |ξ|e 3 , it is enough to show that a(x, ξ) = e −iξ•x = e −i|ξ||x|e 3 •θ .
On the other hand, the expansion of a plane wave in terms of spherical waves is given by the classical formula since Y n,0 (x) = 2n+1 4π P n (x • e 3 ) by (2.13).The set A(k, ), defined in the statement of the lemma, appears when using that C( k n; 0 0) = 0 if + k + n is odd, as stated previously.Therefore, inserting (A.2) and (A.3) in the previous identity proves the lemma.
Notice that ζ ∈ V(d) if and only if | Re(ζ)| = | Im(ζ)| and Re(ζ) • Im(ζ) = 0; write e ζ (x) = e ζ•x , x ∈ R d , form an orthonormal basis of eigenfunctions of L ω .Different choices for the rotation P give different basis of spherical harmonics.However, any two possible basis (Y ω ,m ) and 2 We follow one of the standard conventions used in quantum mechanics for the associated Legendre polynomials and the spherical harmonics, see for example [AW01, Chapter 12].