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Reconstruction of a potential from the impedance boundary map

We give formulas and equations for finding generalized scattering data for the Schr\"odinger equation in open bounded domain at fixed energy from the impedance boundary map (or Robin-to-Robin map). Combining these results with results of the inverse scattering theory we obtain efficient methods for reconstructing potential from the impedance boundary map.


Introduction
We consider the equation Following [19], [26], we consider the impedance boundary mapM α = M α,v (E) defined byM  [26] it was shown that there is not more than a countable number of α ∈ R such that E is an eigenvalue for the operator −∆ + v in D with the boundary condition cos α ψ| ∂D − sin α ∂ψ ∂ν | ∂D = 0.
(1.6) Therefore, for any fixed E we can assume that for some fixed α ∈ R E is not an eigenvalue for the operator −∆ + v in D with boundary condition (1.6) (1.7) and, as a corollary,M α can be defined correctly. We considerM α =M α,v (E) as an operator representation of all possible boundary measurements for the physical model described by (1.1). We recall that the impedance boundary mapM α is reduced to the Dirichlet-to-Neumann(DtN) map if α = 0 and is reduced to the Neumann-to-Dirichlet(NtD) map if α = π/2. The mapM α can be called also as the Robin-to-Robin map.
As in [26], we consider the following inverse boundary value problem for equation (1.1).
Problem 1.1. GivenM α for some fixed E and α, find v. This problem can be considered as the Gel'fand inverse boundary value problem for the Schrödinger equation at fixed energy (see [18], [36]). Note that in the initial Gel'fand formulation energy E was not yet fixed and boundary measurements were considered as an operator relating ψ| ∂D and ∂ψ ∂ν | ∂D for ψ satisfying (1.1). Problem 1.1 for E = 0 can be considered also as a generalization of the Calderon problem of the electrical impedance tomography (see [13], [36]).
Note also that Problem 1.1 can be considered as an example of ill-posed problem: see [4], [31] for an introduction to this theory. Problem 1.1 includes, in particular, the following questions: (a) uniqueness, (b) reconstruction, (c) stability.
Global uniqueness theorems and global reconstruction methods for Problem 1.1 with α = 0 (i.e. for the DtN case) were given for the first time in [36] in dimension d ≥ 3 and in [10] in dimension d = 2.
Global stability estimates for Problem 1.1 with α = 0 were given for the first time in [1] in dimension d ≥ 3 and in [45] in dimension d = 2. A principal improvement of the result of [1] was given recently in [44] (for E = 0). Due to [32] these logarithmic stability results are optimal (up to the value of the exponent). An extention of the instability estimates of [32] to the case of non-zero energy as well as to the case of Dirichlet-to-Neumann map given on the energy intervals was obtained in [24]. An extention of stability estimates of [44] to the energy dependent case was given recently in [27]. Instability estimates complementing stability results of [27] were obtained in [25].
Note also that for the Calderon problem (of the electrical impedance tomography) in its initial formulation the global uniqueness was firstly proved in [51] for d ≥ 3 and in [35] for d = 2. In addition, for the case of piecewise constant or piecewise real analytic conductivity the first uniqueness results for the Calderon problem in dimension d ≥ 2 were given in [15], [28]. Lipschitz stability estimate for the case of piecewise constant conductivity was proved in [2] and additional studies in this direction were fulfilled in [48].
It should be noted that in most of previous works on inverse boundary value problems for equation (1.1) at fixed E it was assumed in one way or another that E is not a Dirichlet eigenvalue for the operator −∆ + v in D, see [1], [32], [36], [44]- [49]. Nevertheless, the results of [10] can be considered as global uniqueness and reconstruction results for Problem 1.1 in dimension d = 2 with general α.
Global stability estimates for Problem 1.1 in dimension d ≥ 2 with general α were recently given in [26].
In the present work we give formulas and equations for finding (generalized) scattering data from the impedance boundary mapM α with general α. Combining these results with results of [21], [23], [35], [37]- [39], [41]- [43], we obtain efficient reconstruction methods for Problem 1.1 in multidimensions with general α. To our knowledge these results are new already for the NtD case.
In particular, in the present work we give first mathematically justified approach for reconstructing coefficient v from boundary measurements for (1.1) via inverse scattering without the assumption that E is not a Dirichlet eigenvalue for −∆ + v in D. In addition, numerical efficiency of related inverse scattering techniques was shown in [3], [9], [11], [12]; see also [8].
Definitions of (generalized) scattering data are recalled in Section 2. Our main results are presented in Section 3. Proofs of these results are given in Sections 4, 5 and 6.
We consider also the functions R, R γ , R + defined as follows: x, y ∈ R d , k ∈ C d , Im k = 0, (2.12) where G is defined by (2.7) and formula (2.12) at fixed y, k is considered as an equation for R(x, y, k) = e ik(x−y) r(x, y, k), (2.13) where r is sought with the properties In addition, the functions R(x, y, k), R γ (x, y, k) and R + (x, y, k) (for their domains of definition in k and γ) satisfy the following equations: (2.17) The function R + (x, y, k) (defined by means of (2.12) for k ∈ R d \ {0} with G replaced by G + of (2.4)) is well-known in the scattering theory for equations (2.1), (2.17) (see, for example, [7]). In particular, this function describes scattering of the spherical waves G + (x − y, k) generated by a source at y. In addition R + (x, y, k) is a radial function in k, i.e.
Apparently, the functions R and R γ were considered for the first time in [38]. In addition, under the assumption (2.2): equation (2.12) at fixed y and k is uniquely solvable for R with the properties (2.13), (2.14) if and only if at fixed y, ζ and γ is uniquely solvable for R γ if and only if ζ ∈ R d \({0}∪E γ ); equation (2.12) with k = ζ + i0ζ/|ζ|, ζ ∈ R d \ 0, at fixed y and ζ is uniquely solvable for R + if and only if ζ ∈ R d \ ({0} ∪ E + ).

Main results
Let v and v 0 satisfy (1.3), (1.7) for some fixed E and α. Let M α,v (x, y, E), M α,v 0 (x, y, E), x, y ∈ ∂D, denote the Schwartz kernels of the impedance boundary mapsM α,v ,M α,v 0 , for potentials v and v 0 , respectively, whereM α,v , M α,v 0 are considered as linear integral operators. In addition, we consider v 0 as some known background potential.

Proposition 3.1 is proved in Section 4. Note that (3.2) is considered as a linear integral equation for finding
In addition, we use similar interpretation for similar formulas for ψ + , f and for ψ γ , h γ , mentioned in Theorem 3.1 and Proposition 3.1.
Under the assumptions of Theorem 3.1, the following propositions are valid: is a Fredholm linear integral equation of the second kind in the space of bounded functions on ∂D. In addition, the same is also valid for the equation for Proposition 3.2 is proved in Section 4.
, are uniquely solvable in the space of bounded functions on ∂D if and only if k / ∈ E + and k / ∈ E γ , respectively. Proposition 3.3 is proved in Section 5.
Proposition 3.4. Let φ α (x, y) be the solution of the Dirichlet boundary value problem at fixed y ∈ ∂D, λ ∈ C: where we assume that λ is not a Dirichlet eigenvalue for −∆ in D. Then is the Dirichlet-to-Neumann map for (3.7). In addition, formulas completely similar to (3.8) are also valid for the kernels A + α (but with R + 0 in place of R 0 ) and A α,γ (but with R 0 γ in place of R 0 ), arising in the equations for [ψ + ] α and [ψ γ ] α , mentioned in Theorem 3.1 and Proposition 3.1.
In addition, under assumptions of Theorem 3.1, the following identity holds (see formula (3.9) of [26]): We will use also that: where Ω is such an open bounded domain in R d thatD ∈ Ω and C 1+δ denotes C 1 with the first derivatives belonging to the Hölder space C δ . We will use also the Green formula: where φ 1 and φ 2 are arbitrary sufficiently regular functions inD.

Proof of Theorem 3.1 and Proposition 3.1
For the case when sin α = 0, Theorem 3.1 and Proposition 3.1 were proved in [40]. In this subsection we generalize the proof of [40] to the case sin α = 0. We proceed from the following formulas and equations (being valid under assumption (2.2) on v 0 and v): where (4.10) at fixed k is considered as an equation where (4.12) at fixed k is an equation for ψ + ∈ L ∞ (R d ); where (4.14) at fixed γ and k is considered as an equation for ψ γ ∈ L ∞ (R d ).

Proof of Proposition 3.3
For the case when sin α = 0, Proposition 3.3 was proved in [40]. In this section we prove Proposition 3.3 for sin α = 0. We will prove Proposition 3.3 for the case of equation (3.2) for [ψ] α . The proofs for the cases of ψ + and ψ γ are similar. According to (4.15), to prove Proposition 3.3 (for the case of ψ) it is sufficient to show that equation (3.2) (at fixed k ∈ C d \ (R d ∪ E 0 )) is uniquely solvable in the space of bounded functions on ∂D if and only if equation (4.10) is uniquely solvable for ψ = e ikx µ(x, k) with µ ∈ L ∞ (R d ).
Let equation (4.10) have several solutions. Then, repeating the proof of Theorem 3.1 separately for each solution, we find that [ψ] α on ∂D for each of these solutions satisfies equation (3.2). Thus, using also (1.7) we obtain that equation (3.2) has at least as many solutions as equation (4.10).
To prove the converse (and thereby to prove Proposition 3.3) it remains to show that any solution [ψ] α of (3.2) can be continued to a continuos solution of (4.10).
6 Proofs of properties (4.4), (4.5) As it was mentioned in Subsection 4.1, properties (4.4), (4.5) are well-known for cos α = 0 (the case of the Neumann boundary condition). To extend these properties to the case of general α, v, E, we use the following schema: ,v by means of Lemma 6.1 given bellow (with sin α 1 = 0 and sin α 2 = 0), 2. G α,v 1 → G α,v 2 by means of Lemma 6.2 given bellow.