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Journal of Mathematics
Volume 2013, Article ID 318154, 10 pages
http://dx.doi.org/10.1155/2013/318154
Research Article

Energy- and Regularity-Dependent Stability Estimates for Near-Field Inverse Scattering in Multidimensions

Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau, France

Received 20 November 2012; Revised 17 December 2012; Accepted 17 December 2012

Academic Editor: Zindoga Mukandavire

Copyright © 2013 M. I. Isaev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove new global Hölder-logarithmic stability estimates for the near-field inverse scattering problem in dimension . Our estimates are given in uniform norm for coefficient difference and related stability efficiently increases with increasing energy and/or coefficient regularity. In addition, a global logarithmic stability estimate for this inverse problem in dimension is also given.

1. Introduction

We consider the Schrödinger equation: where We consider the resolvent of the Schrödinger operator in : where is the spectrum of in . We assume that denotes the Schwartz kernel of as of an integral operator. We consider also We recall that in the framework of (1) the function describes scattering of the spherical waves, generated by a source at (where is the Hankel function of the first kind of order ). We recall also that is the Green function for , , with the Sommerfeld radiation condition at infinity.

In addition, the function is considered as near-field scattering data for (1), where is the open ball of radius centered at .

We consider, in particular, the following near-field inverse scattering problem for (1).

Problem 1. Given on for some fixed , , find on .

This problem can be considered under the assumption that is a priori known on . Actually, in the present paper we consider Problem 1 under the assumption that on for some fixed . Below in this paper we always assume that this additional condition is fulfilled.

It is well known that the near-field scattering data of Problem 1 uniquely and efficiently determine the scattering amplitude for (1) at fixed energy , see [1]. Therefore, approaches of [212] can be applied to Problem 1 via this reduction.

In addition, it is also known that the near-field data of Problem 1 uniquely determine the Dirichlet-to-Neumann map in the case when is not a Dirichlet eigenvalue for operator in , see [8, 13]. Therefore, approaches of [3, 8, 1424] can be also applied to Problem 1 via this reduction.

However, in some case it is much more optimal to deal with Problem 1 directly, see, for example, logarithmic stability results of [25] for Problem 1 in dimension . A principal improvement of estimates of [25] was given recently in [26]: stability of [26] efficiently increases with increasing regularity of .

Problem 1 can be also considered as an example of ill-posed problem: see [27, 28] for an introduction to this theory.

In the present paper we continue studies of [25, 26]. We give new global Hölder-logarithmic stability estimates for Problem 1 in dimension , see Theorem 1. Our estimates are given in uniform norm for coefficient difference and related stability efficiently increases with increasing energy and/or coefficient regularity. Results of such a type for the Gel'fand inverse problem were obtained recently in [15] for and in [29] for .

The main feature of our new estimates is the explicit dependence on the energy . These estimates consist of two parts, the first is Hölder and the second is logarithmic; when increases, the logarithmic part decreases and the Hölder part becomes dominant.

In addition, we give also global logarithmic stability estimates for Problem 1 in dimension , see Theorem 2.

2. Stability Estimates

We recall that if satisfies (2) and for some , then where is the near-field scattering data of for (1) with , for more details see, for example, Section 2 of [25].

2.1. Estimates for

In this subsection we assume for simplicity that where where Let

Note that (8)(2).

Theorem 1. Let and be given constants. Let dimension and potentials , satisfy (8). Let , for some . Let and denote the near-field scattering data for and , respectively. Then for and any the following estimate holds: where , , and constants depend only on .

Proof of Theorem 1 is given in Section 5. This proof is based on results presented in Sections 3 and 4.

2.2. Estimates for

In this subsection we assume for simplicity that Note also that (13)(2).

Theorem 2. Let and be given constants. Let dimension and potentials , satisfy (13). Let , for some . Let and denote the near-field scattering data for and , respectively. Then where and constant depends only on , , .

Proof of Theorem 2 is given in Section 7. This proof is based on results presented in Sections 3 and 6.

2.3. Concluding Remarks

Remark 3. The logarithmic stability estimates for Problem 1.1 of [25, 26] follow from estimate (12) for and . Apparently, using the methods of [19, 20] it is possible to improve estimate (12) for .

Remark 4. In the same way as in [25, 26] for dimesnsion , using estimates (12) and (14), one can obtain logarithmic stability estimates for the reconstruction of a potential from the inverse scattering amplitude for any .

Remark 5. Actually, in the proof of Theorem 1 we obtain the following estimate (see formula (57)): where constants depend only on and the parameter is such that is sufficiently large: . Estimate of Theorem 1 follows from estimate (15).

3. Alessandrini-Type Identity for Near-Field Scattering

In this section we always assume that assumptions of Theorems 1 and 2 are fulfilled (in the cases of dimension and , resp.).

Consider the operators , defined as follows Note that We recall that (see [25]) for any functions , sufficiently regular in and satisfying with and , respectively, the following identity holds: where where and are the outward and inward normals to , respectively.

Remark 6. The identity (19) is similar to the Alessandrini identity (see Lemma 1 of [14]), where the Dirichlet-to-Neumann maps are considered instead of operators .

To apply identity (19) to our considerations, we use also the following lemma.

Lemma 7. Let and . Then, there is a positive constant (depending only on and ) such that for any satisfying the following inequality holds: where denotes the standard Sobolev space on .

The proof of Lemma 7 is given in Section 8.

4. Faddeev Functions

In dimension , we consider the Faddeev functions , , (see [6, 8, 30, 31]): where , , , where , , , ,

One can consider (22), (23) assuming that For example, in connection with Theorem 1, we consider (22), (23) assuming that

We recall that (see [6, 8, 30, 31]) formula (23) at fixed is considered as an equation for where is sought in ; as a corollary of (23), (24), and (27), satisfies (1) for ; of (22) is a generalized “scattering” amplitude.

In addition, , , in their zero energy restriction, that is for were considered for the first time in [32]. The Faddeev functions , , were, actually, rediscovered in [32].

Let Let then we have that and, for any , where , ;

Results of the type (31), (32) go back to [32]. For more information concerning (32) see estimate of [33]. Results of the type (33), (34) (with less precise right-hand side in (34)) go back to [6]. Estimate (34) follows, for example, from formulas (23), (22), and the estimate for , where denotes the integral operator with the Schwartz kernel and denotes the multiplication operator by the function . Estimate (35) was formulated, first, in [34] for . Concerning proof of (35), see [35].

In addition, we have that and, under assumptions of Theorem 1, where , denote and of (22) and (23) for .

Formula (36) was given in [36]. Estimate (37) was given for example in [15].

5. Proof of Theorem 1

Let

Note that where , are the spaces of (9), (38),

Using the inverse Fourier transform formula we have that where

Using (39), we obtain that Let Combining (43), (44), we find that, for any ,

Due to (37), we have that

Let Combining (17), (19), and (36), we get that where , , denotes the solution of (18) with , satisfying Using (21), (32), and the fact that , we find that Here and bellow in this section the constant is the same that in (32).

Combining (49) and (51), we obtain that Using (47), (52), we get that Let and be such that Using (43), (53), we get that Combining (42), (46), and (56) for and (55), we get that

Let , and be such that Then for the case when , due to (57), we have that Combining (59) and (60), we obtain that for and the following estimate holds: where and depend only on , , , , , , and .

Estimate (61) in the general case (with modified and ) follows from (61) for and the property that

This completes the proof of (12).

6. Buckhgeim-Type Analogs of the Faddeev Functions

Let us identify with and use coordinates , , where . Following [2123, 37], we consider the functions , , , going back to Buckhgeim’s paper [3] and being analogs of the Faddeev functions: where satisfies (13), where , satisfy (13) and , denote , of (63) for and , respectively.

We recall that (see [21, 22]): (i) the function satisfies the equations where , , and is the Dirac delta function;(ii) formulas (63) at fixed and are considered as equations for , in ;(iii) as a corollary of (63), (64), (66), the functions , satisfy (1) in for and ;(iv) the function is similar to the right side of (36).

Let potentials , and then we have that and, for any , where , , , , ; Formulas (68) can be considered as definitions of , . Formulas (69), (71) were given in [21, 22] and go back to [3]. Estimates (70a) and (70b) were proved in [33]. Estimate (72) was obtained in [21, 37].

7. Proof of Theorem 2

We suppose that   are defined as in Section 6 but with in place of , . Note that functions   satisfy (1) in with , , respectively. We also use the notation . Then, using (72), we have that Let Combining (17), (19), and (65), we get that where , , denotes the solution of (18) with , satisfying Using (21), (70a), and (70b) and the fact that , we find that Here and bellow in this section the constant is the same that in (70a) and (70b).

Combining (75), (77), we obtain that Using (73) and (78), we get that We fix some and let where is so small that . Then due to (79), we have that where , , and are the same as in (80).

Using (81), we obtain that for , where is a sufficiently small positive constant. Estimate (82) in the general case (with modified ) follows from (82) for and the property that .

This completes the proof of (14).

8. Proof of Lemma 7

In this section we assume for simplicity that and therefore .

We fix an orthonormal basis in : where is the dimension of the space of spherical harmonics of order , where The precise choice of is irrelevant for our purposes. Besides orthonormality, we only need to be the restriction of a homogeneous harmonic polynomial of degree to the sphere and so is harmonic pn . In the Sobolev spaces the norm is defined by The solution of the exterior Dirichlet problem can be expressed in the following form (see, e.g., [1, 38]): where are expansion coefficients of in the basis , and where is the Hankel function of the first kind. Let Note that is harmonic in and Using the Green formula and the radiation condition for , we get that Due to (89) and (90), we have that Using also the following property of the Hankel function of the first kind (see, e.g., [39]): we get that Combining (89), (90), (92), (93), and (95), we obtain that

Let consider the cases when .

Case 1 (). Using the property , we get that We recall that functions and have the following asymptotic forms (see, e.g. [39]): Using (94) and (98), we get that for some Combining (97) and (99), we obtain that for

Case 2 (). We have that Using (89) and (101), we get that for Combining (86)–(89), (96), (100), and (102), we get that for some constant : Using (86) and (103), we obtain (21).

Acknowledgment

This work was partially supported by FCP Kadry no. 14.A18.21.0866.

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