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Journal of Mathematics
Volume 2013 (2013), Article ID 318154, 10 pages
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.
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.
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 . Therefore, approaches of [2–12] 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, 14–24] 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  for Problem 1 in dimension . A principal improvement of estimates of  was given recently in : stability of  efficiently increases with increasing regularity of .
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  for and in  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.
2. Stability Estimates
2.1. Estimates for
In this subsection we assume for simplicity that where where Let
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 .
2.2. Estimates for
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 , , .
2.3. Concluding Remarks
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
Consider the operators , defined as follows Note that We recall that (see ) 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.
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 .
4. Faddeev Functions
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.
Let Let then we have that and, for any , where , ;
Results of the type (31), (32) go back to . For more information concerning (32) see estimate of . Results of the type (33), (34) (with less precise right-hand side in (34)) go back to . 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  for . Concerning proof of (35), see .
5. Proof of Theorem 1
Using the inverse Fourier transform formula we have that where
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).
This completes the proof of (12).
6. Buckhgeim-Type Analogs of the Faddeev Functions
Let us identify with and use coordinates , , where . Following [21–23, 37], we consider the functions , , , going back to Buckhgeim’s paper  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 . Estimates (70a) and (70b) were proved in . 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).
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., ): 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. ): Using (94) and (98), we get that for some Combining (97) and (99), we obtain that for
This work was partially supported by FCP Kadry no. 14.A18.21.0866.
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