Journal of Mathematics

Volume 2013 (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

M. I. Isaev

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 [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 [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 [21–23, 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.

References

Ju. M. Berezanskiĭ, “The uniqueness theorem in the inverse problem of spectral analysis for the Schrödinger equation,” Trudy Moskovskogo Matematičeskogo Obščestva, vol. 7, pp. 1–62, 1958 (Russian).

N. V. Alekseenko, V. A. Burov, and O. D. Rumyantseva, “Solution of the three-dimensional acoustic inverse scattering problem. the modified Novikov algorithm,” Akusticheskiĭ Zhurnal, vol. 54, no. 3, pp. 469–482, 2008 (Russian), English translation in Acoustical Physics, vol. 54, no. 3, pp. 407–419, 2008

A. L. Bukhgeim, “Recovering a potential from Cauchy data in the two-dimensional case,” Journal of Inverse and Ill-Posed Problems, vol. 16, no. 1, pp. 19–33, 2008.

V. A. Burov, N. V. Alekseenko, and O. D. Rumyantseva, “Multifrequency generalization of the Novikov algorithm for the two-dimensional inverse scattering problem,” Acoustical Physics, vol. 55, no. 6, pp. 843–856, 2009.

G. Eskin and J. Ralston, “Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy,” Communications in Mathematical Physics, vol. 173, no. 1, pp. 199–224, 1995.

R. G. Novikov and G. M. Khenkin, “The -equation in the multidimensional inverse scattering problem,” Uspekhi Matematicheskikh Nauk, vol. 42, no. 3(255), pp. 93–152, 1987 (Russian), English translation in Russian Mathematical Surveys, vol. 42, no. 3, pp.109–180, 1987

M. I. Isaev, “Exponential instability in the inverse scattering problem on the energy interval,” to appear in Funktsional'nyi Analiz i ego Prilozheniya, http://arxiv.org/abs/1012.5526

R. G. Novikov, “A multidimensional inverse spectral problem for the equation ,” Funktsional'nyĭ Analiz i ego Prilozheniya, vol. 22, no. 4, pp. 11–22, 1988, English translation in Functional Analysis and Its Applications, vol. 22, pp. 263–272, 1988

R. G. Novikov, “The inverse scattering problem at fixed energy for the three-dimensional Schrödinger equation with an exponentially decreasing potential,” Communications in Mathematical Physics, vol. 161, no. 3, pp. 569–595, 1994.

R. G. Novikov, “Approximate solution of the inverse problem of quantum scattering theory with fixed energy in dimension ,” Trudy Matematicheskogo Instituta Imeni V. A. Steklova, vol. 225, pp. 301–318, 1999 (Russian), English translation in Proceedings of the Steklov Institute of Mathematic, no. 2(225), pp. 285–302, 1999

R. G. Novikov, “The -approach to approximate inverse scattering at fixed energy in three dimensions,” IMRP. International Mathematics Research Papers, no. 6, pp. 287–349, 2005.

P. Stefanov, “Stability of the inverse problem in potential scattering at fixed energy,” Annales de l'Institut Fourier, vol. 40, no. 4, pp. 867–884, 1990.

A. I. Nachman, “Reconstructions from boundary measurements,” Annals of Mathematics, vol. 128, no. 3, pp. 531–576, 1988.

G. Alessandrini, “Stable determination of conductivity by boundary measurements,” Applicable Analysis, vol. 27, no. 1–3, pp. 153–172, 1988.

M. I. Isaev and R. G. Novikov, “Energy and regularity dependent stability estimates for the Gel'fand inverse problem in multidimensions,” Journal of Inverse and III-posed Problems, vol. 20, no. 3, pp. 313–325, 2012.

V. Isakov, “Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map,” Discrete and Continuous Dynamical Systems. Series S, vol. 4, no. 3, pp. 631–640, 2011.

N. Mandache, “Exponential instability in an inverse problem for the Schrödinger equation,” Inverse Problems, vol. 17, no. 5, pp. 1435–1444, 2001.

R. G. Novikov, “Formulae and equations for finding scattering data from the Dirichlet-to-Neumann map with nonzero background potential,” Inverse Problems, vol. 21, no. 1, pp. 257–270, 2005.

R. G. Novikov, “An effectivization of the global reconstruction in the Gel'fand-Calderón inverse problem in three dimensions,” in Imaging Microstructures, vol. 494 of Contemp. Math., pp. 161–184, American Mathematical Society, Providence, RI, USA, 2009.

R. G. Novikov, “New global stability estimates for the Gel'fand-Calderon inverse problem,” Inverse Problems, vol. 27, no. 1, Article ID 015001, 21 pages, 2011.

R. Novikov and M. Santacesaria, “A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions,” Journal of Inverse and Ill-Posed Problems, vol. 18, no. 7, pp. 765–785, 2010.

R. G. Novikov and M. Santacesaria, “Global uniqueness and reconstruction for the multi-channel Gel'fand-Calderón inverse problem in two dimensions,” Bulletin des Sciences Mathématiques, vol. 135, no. 5, pp. 421–434, 2011.

R. Novikov and M. Santacesaria, “Monochromatic reconstruction algorithms for two-dimensional multi-channel inverse problems,” International Mathematics Research Notes, 2012.

J. Sylvester and G. Uhlmann, “A global uniqueness theorem for an inverse boundary value problem,” Annals of Mathematics, vol. 125, no. 1, pp. 153–169, 1987.

P. Hähner and T. Hohage, “New stability estimates for the inverse acoustic inhomogeneous medium problem and applications,” SIAM Journal on Mathematical Analysis, vol. 33, no. 3, pp. 670–685, 2001.

M. I. Isaev and R. G. Novikov, “New global stability estimates for monochromatic inverse acoustic scattering,” http://arxiv.org/abs/1210.3821

L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, NY, USA, 2012.

M. M. Lavrent'ev, V. G. Romanov, and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis, vol. 64 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1986.

M. Santacesaria, “Stability estimates for an inverse problem for the Schrödinger equation at negative energy in two dimensions,” Applicable Analysis, 2012.

L. D. Faddeev, “Growing solutions of the Schrödinger equation,” Doklady Akademii Nauk SSSR, vol. 165, no. 3, pp. 514–517, 1965 (Russian), English translation in Soviet Physics. Doklady, vol. 10, pp. 1033–1035, 1966

L. D. Faddeev, “The inverse problem in the quantum theory of scattering. II,” Current Problems in Mathematics, vol. 3, pp. 93–180, 259, 1974 (Russian), English translation in Journal of Soviet Mathematics, vol. 5, pp. 334–396, 1976

R. Beals and R. R. Coifman, “Multidimensional inverse scatterings and nonlinear partial differential equations,” in Pseudodifferential Operators and Applications (Notre Dame, Ind., 1984), vol. 43 of Proc. Sympos. Pure Math., pp. 45–70, American Mathematical Society, Providence, RI, USA, 1985.

M. I. Isaev and R. G. Novikov, “Stability estimates for determination of potential from the impedance boundary map,” Algebra and Analysis, 2013, http://128.84.158.119/abs/1112.3728v2

R. B. Lavine and A. I. Nachman, “On the inverse scattering transform for the -dimensional Schrödinger operator,” in Topics in Soliton Theory and Exactly Solvable Nonlinear Equations (Oberwolfach, 1986), pp. 33–44, World Scientific, Singapore, 1987.

R. Weder, “Generalized limiting absorption method and multidimensional inverse scattering theory,” Mathematical Methods in the Applied Sciences, vol. 14, no. 7, pp. 509–524, 1991.

R. G. Novikov, “-method with nonzero background potential. Application to inverse scattering for the two-dimensional acoustic equation,” Communications in Partial Differential Equations, vol. 21, no. 3-4, pp. 597–618, 1996.

M. Santacesaria, “Global stability for the multi-channel Gel'fandCalderon inverse problem in two dimensions,” Bulletin des Sciences Mathématiques, vol. 136, no. 7, pp. 731–744, 2012.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, Springer, Berlin, Germany, 2nd edition, 1998.

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, UK, 1944.