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An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions

By developing the d-bar approach to global"inverse scattering"at zero energy we give a principal effectivization of the global reconstruction method for the Gel'fand-Calderon inverse boundary value problem in three dimensions. This work goes back to results published by the author in 1987, 1988 and proceeds from recent progress in the d-bar approach to inverse scattering in 3D published by the author in 2005, 2006.


Introduction
The global reconstruction method for the Gel'fand-Calderon inverse boundary value problem in 3D was proposed for the first time by the author in 1987, 1988, see [HN] (Note added in proof) and [No1]. The sheme of this global reconstruction can be presented as follows: where Φ denotes boundary measurements, h is the Faddeev generalized scattering amplitude (at zero energy for simplicity) defined in complex domain Θ,Θ ρ is the subset of Θ with the imaginary part not greater than ρ, v ρ is an approximate inverse scattering reconstruction from h Θ ρ , v ρ → v for ρ → +∞, where v is the unknown potential to be reconstructed from Φ. The global uniqueness in the Gel'fand-Calderon problem in 3D follows from this global reconstruction as a corollary, see [No1]. Slightly earlier with respect to [No1] the global uniqueness in the Calderon problem in 3D was proved in [SU]. Note that [SU] gives no reconstruction method.
Slightly later with respect to [No1] the global reconstruction from boundary measurements in 3D similar to the global reconstruction of [No1] was also published in [Na1]. Note that [Na1] contains a reference to the preprint of [No1].
The Gel'fand-Calderon problem is formulated as Problem 1.2 of Subsection 1.2 of this introduction.
Reconstruction problems from the Faddeev generalized scattering amplitude h in the complex domain at zero energy are formulated as Problems 1.1a, 1.1b and 1.1c of Subsection 1.1 of this introduction.
The reduction of Problem 1.2 to Problems 1.1 is given by formulas and equations (1.23)-(1.25) mentioned in Subsection 1.2 of this introduction, see also [No2] for more advanced version of these formulas and equations. For a long time the main disadvantage of the global reconstruction (*) in 3D was related with the following two facts: (1) The determination of h Θ ρ from Φ via formulas and equations of the type (1.23)- (1.25) is stable for relatively small ρ, but is very unstable for ρ → +∞ in the points ofΘ ρ with sufficiently great imaginary part, see Subsection 1.2 of this introduction.
(2) The decay of the error v − v ρ for ρ → +∞ was very slow (not faster than O(ρ −1 ) even for infinitely smooth compactly supported v) in existing global results for stable construction of v ρ from h Θ ρ in 3D, see Remarks 1.1, 1.2, 1.3 of Subsection 1.1 of this introduction.
As a corollary, the global reconstruction (*) in 3D was not efficient with respect to its stability properties.
The key point is that in the present work we give a global and stable construction of v ρ from h bΘ ρ in 3D, where bΘ ρ denotes the boundary ofΘ ρ , with rapid decay of the error v − v ρ for ρ → +∞ (in particular, with v − v ρ = O(ρ −∞ ) for v of the Schwartz class). This gives a principal effectivization of the global reconstruction (*) with respect to its stability properties.
Our new results are presented in detail below in Subsections 1.1, 1.2, 1.3 (of the introduction) and in Sections 2 and 6. These results were obtained proceeding from [No3], [No4].

Inverse scattering at zero energy. Consider the equation
(1.1) where v is a sufficiently regular function on R d with sufficient decay at infinity (1.2) (precise assumptions on v are specified below in this introduction and in Section 2). Equation (1.1) arises, in particular, in quantum mechanics, acoustics, electrodynamics. Formally, (1.1) looks as the Schrödinger equation with potential v at fixed energy E = 0.
For equation (1.1), under assumptions (1.2), we consider the Faddeev generalized scattering amplitude h(k, l), where (k, l) ∈ Θ, Θ = {k ∈ C d , l ∈ C d : k 2 = l 2 = 0, Im k = Im l}. (1.3) Given v, to determine h on Θ one can use, in particular, the formula (1.4) and the linear integral equation Actually, h on Θ is a zero energy restriction of a function h introduced by Faddeev as an extension to the complex domain of the classical scattering amplitude for the Schrödinger equation at positive energies (see [F2], [HN]). Note that the restriction h Θ was not considered in Faddeev's works and that h in its zero energy restriction was considered for the first time in [BC] for d = 3 in the framework of Problem 1.1a formulated below. The Faddeev function h was, actually, rediscovered in [BC]. The fact that∂-scattering data of [BC] coincide with the Faddeev function h was observed, in particular, in [HN].
In the present work, in addition to h on Θ, we consider, in particular, (1.9) Using (1.4), (1.5) one can see that (1.10) in the Born approximation (that is in the linear approximation near zero potential). In addition, one can see also that In the present work we consider, in particular, the following inverse scattering problems for equation (1.1) under assumptions (1.2).
Problem 1.1. (a) Given h on Θ, find v on R d ; (b) Given h onΘ ρ for some (sufficiently great) ρ > 0, find v on R d , at least, approximately; (c) Given h on bΘ ρ for some (sufficiently great) ρ > 0, find v on R d , at least, approximately.
Note that Problems 1.1a, 1.1b make sense for any d ≥ 2, whereas Problem 1.1c is reasonable for d ≥ 3 only: dim bΘ ρ < dim R d for d = 2, see (1.9).
Note that: (1) any reconstruction method for Problem 1.1b with decaying error as ρ → +∞ gives also a reconstruction method for Problem 1.1a and (2) for d ≥ 3, any reconstruction method for Problem 1.1c gives also a reconstruction method for Problem 1.1b.
Note that in the Born approximation (1.10): (a) Problem 1.1a is reduced to finding v on R d fromv on R d , (b) Problem 1.1b is reduced to (approximate) finding v on R d from v on B 2ρ , (c) Problem 1.1c for d ≥ 3 is reduced to (approximate) finding v on R d fromv on B 2ρ , wherev is defined by (1.6). Thus, in the Born approximation, Problem 1.1c for d ≥ 3 (as well as Problem 1.1b for d ≥ 2) can be solved by the formula In addition, if, for example, v ∈ W n,1 (R d ) for some n ∈ N, (1.14) and v n,1 ≤ C, where W n,1 (R d ) denotes the space of n-times smooth functions on R d in L 1 -sense and · n,1 denotes some fixed standard norm in W n,1 (R d ), then (1.15) and, therefore, for n > d, where c 1 (n, d), c 2 (n, d) are some fixed positive constants and v lin err (x, ρ) is the error term of (1.13).
Thus, in the Born approximation (1.10) (that is in the linear approximation near zero potential) we have that: (1) h on bΘ ρ for d ≥ 3 (as well as h onΘ ρ for d ≥ 2) stably determines v lin appr (x, ρ) of (1.13) and (2) the error v lin ) in the uniform norm as ρ → +∞ for n-times smooth v in the sense (1.14), where n > d. In particular, v lin err = O(ρ −∞ ) in the uniform norm as ρ → +∞ for v of the Schwartz class on R d .
The main results of the present work consist in global analogs for the nonlinearized case for d = 3 of the aforementioned Born-approximation results for Problem 1.1c, see Theorem 2.1 and Corollary 2.1 of Section 2. In particular, we give a stable approximate solution of nonlinearized Problem 1.1c for d = 3 and v satisfying (1.14), n > d = 3, with the error term decaying as O(ρ −(n−d) ln ρ) in the uniform norm as ρ → +∞ (that is with almost the same decay rate of the error for ρ → +∞ as in the linearized case near zero potential, see (1.13), (1.16)). The results of the present work were obtained in the framework of a development of the∂-approach to inverse scattering at fixed energy in dimension d ≥ 3 of [BC], [HN], [No3], [No4], with applications to the Gel'fand-Calderon inverse boundary value problem via the reduction going back to [No1]. See Subsections 1.2, 1.3 and Sections 2, 3, 4, 5, 6 for details.
Remark 1.2. On the other hand (in comparison with the result mentioned in Remark 1.1), for sufficiently small potentials v, in [No4] we succeeded, in particular, to give a stable method for solving Problem 1.1b for d = 3 with the same type rapid decay of the error term for ρ → +∞ as in formulas (1.13), (1.16) for the linearized case near zero potential. Moreover, in this result of [No4], v is approximately reconstructed already from nonoverdetermined restriction h Θ ρ ∩Γ , where Γ ⊂ Θ, dim Γ = d = 3. However, this result of [No4] is local: the smallness of v is used essentially. Remark 1.3. We emphasize that before the present work no results were given, in general, in the literature on solving Problems 1.1c and 1.1b for d ≥ 3 with the error term decaying more rapidly than O(ρ −1 ) as ρ → +∞ even for v of the Schwartz class on R d (and even for the infinitely smooth compactly supported case in the framework of sufficiently stable rigorous algorithms). In addition, rapid decay of this error term is a property of principal importance in the framework of applications of methods for solving Problems 1.1 to the Gel'fand-Calderon inverse boundary value problem (Problem 1.2) via the reduction of [No1], see the next part of introduction.
Note that Problem 1.1a was considered for the first time in [BC] for d = 3 from pure mathematical point of view without any physical applications. No possibility to measure h on Θ\{(0)} directly in some physical experiment is known at present (here {0} = {(k, l) ∈ Θ : |k| = |l| = 0}). However, as it was shown in [No1] (see also [HN] (Note added in proof), [Na1], [No2]), Problems 1.1 naturally arise in the electrical impedance tomography and, more generally, in connection with Problem 1.2 formulated in the next subsection.
v is a sufficiently regular function onD = D ∪ ∂D. (1.20) For simplicity we assume also that 0 is not a Dirichlet eigenvalue for for all sufficiently regular solutions ψ of (1.1) inD, where ν is the outward normal to ∂D. The map Φ is called the Dirichlet-to-Neumann map for equation (1.1) in D. We consider the following inverse boundary value problem for equation (1.1) in D: This problem can be considered as the Gel'fand inverse boundary value problem for the Schrödinger equation at zero energy (see [G], [No1]). This problem can be also considered as a generalization of the Calderon problem of the electrical impedance tomography (see [C], [SU], [No1]).
One can see that the Faddeev function h of Problems 1.1 does not appear in Problem 1.2. However, as it was shown in [No1] (see also [HN] (where this result of [No1] was announced in Note added in proof), [Na1], [No2]), if h corresponds to equation (1.1) on R d , where v is the potential of Problem 1.2 on D and v ≡ 0 on R d \D, then h on Θ can be determined from the Dirichlet-to-Neumann map Φ via the following formulas and equation:  .24)); see, for example, [BRS], [No2], [No4]. This explains the principal importance (mentioned in Remark 1.3) of the error term rapid decay as ρ → +∞ in methods for solving Problems 1.1b and 1.1c.

Final remarks.
In the present work we consider, mainly, Problems 1.1 and 1.2 for d = 3. The main results of the present work are presented in Sections 2 and 6. Some of these results were already mentioned above. Note that only restrictions in time prevent us from generalizing all main results of the present work to the case of the Schrödinger equation at arbitrary (not necessarily zero) fixed energy E for d ≥ 3.
Note that results of the present work permit to complete (at least for d = 3) the proof of new stability estimates for Problem 1.2, d ≥ 3, announced as Theorem 2.2 of [NN]. We plan to return to this proof in a separate article.
Our new global reconstruction for Problem 1.2 in 3D is summarized in schemes (6.1), (6.2) of Section 6. We expect that this reconstruction can be implemented numerically in a similar way with implementations developed in [ABR], where the parameter ρ of (6.1), (6.2) will play in some sense the role of the wave number k 0 of [ABR].

2.Main new results
In the present work we consider, mainly, the three dimensional case d = 3. In addition, in the main considerations of the present work for d = 3 our basic assumption on v consists in the following condition on its Fourier transform: wherev is defined by (1.6), 2) and C denotes the space of continuous functions. Actually, (2.1) is a specification of (1.2).
An outline of the reconstruction formalized in Theorem 2.1 is given also at the end of the present section. Let wherev ± (p, τ, ρ) is the approximation of Theorem 2.1. Formula (2.3) and Theorem 2.1 imply the following Corollary 2.1. Let v satisfy (1.14), n > d = 3, and v n,1 ≤ D. Let τ and ρ satisfy (2.6) for µ = n, µ 0 = 3, C = c 4 (n, 3)D and fixed δ ∈]0, 1[. Then h on bΘ ρ,τ stably determines v ± (·, τ, ρ) via (6.2), (2.8) and where the constants c 6 , c 7 are simply related with c 2 and c 5 . In addition, in particular, (2.9) We consider Theorem 2.1 as a global nonlinear analog of the result that in the Born approximation h bΘ ρ is reduced to av on B 2ρ for d ≥ 3. One can see, in particular, that in Theorem 2.1 we even do not try to reconstructv on R 3 \B 2ρ from h on bΘ ρ . We consider Corollary 2.1 as a global nonlinear analog of the Born approximation result (for Problem 1.1c) consisting in formulas (1.13), (1.16).
As it was already mentioned in introduction, the results of the present work were obtained in the framework of a development of the∂-approach to inverse scattering at fixed energy in dimension d ≥ 3 of [BC], [HN], [No3], [No4]. In particular, there is a considerable similarity between the reconstruction scheme of the present work for the case of Problem 1.1c and the reconstruction scheme of [No3] for the case of inverse scattering at fixed positive energy E. Actually, in the present work the parameter ρ of Problem 1.1 plays the role of E 1/2 of [No3]. Some results of [BC], [HN], [No4] we use in the present work are recalled in the next section.

Background results
For simplicity always in this section we assume that d = 3.

Estimates for
where a(µ) is the constant c 2 (µ) of [No4]. Then (according to [No4]): where |k| = ((Re k) 2 + (Im k) 2 ) 1/2 . (These and some additional estimates on H are given in Proposition 3.2 of [No4].) Note that, for sufficiently regular v on R 3 with sufficient decay at infinity, formula (3.4) was obtained for the first time in [HN].

The∂-equation for H on Ω\Ω ρ .
Let v satisfy (2.1), v µ ≤ C, and (3.1) hold. Then (see [No4]): ds is arc-length measure on the circle S k in R 3 . Actually, under some stronger assumptions on v than in the present subsection, the∂-equation (3.5) was obtained for the first time in [BC]. An important property of the∂-equation (3.5) is that (3.5) can be considered for H on Ω\Ω ρ only, see, in particular, formulas (3.24) of Subsection 3.4.
Lemma 4.1 is proved in Section 7.
One can see that on its steps 1 and 3 reconstruction (6.1) is reduced to results of [No1], [No2] and to the inverse Fourier transform, whereas (6.2) is developed in the present work. Some rigorous results concerning (2.8), (6.2), were already summarized as Theorem 2.1 and Corollary 2.1 of Section 2.
Proof of Lemma 4.5. We have that where I ρ,τ (U, V ) is defined by (7.5).