Stability of the inverse problem in potential scattering at fixed energy

Stefanov, Plamen

doi:10.5802/aif.1239

Stefanov, Plamen

Annales de l'Institut Fourier, Tome 40 (1990) no. 4, pp. 867-884.

Résumé
Abstract

Nous prouvons une estimation du type $∥ q_{1} - q_{2} ∥_{L^{\infty}} \leq C φ (∥ A_{q_{1}} - A_{q_{2}} ∥_{R, 3 / 2 - 1 / 2})$ , où $A_{q_{i}} (ω, θ)$ , $i = 1, 2$ est l’amplitude de “scattering” relative au potentiel à support compact $q_{i} (x)$ à un niveau d’énergie fixée $k =$ const., où $φ (t) = (- ln t)^{- δ}$ , $0 < δ < 1$ et $∥ \cdot ∥_{R, 3 / 2 - 1 / 2}$ est une norme définie.

We prove an estimate of the kind $∥ q_{1} - q_{2} ∥_{L^{\infty}} \leq C φ (∥ A_{q_{1}} - A_{q_{2}} ∥_{R, 3 / 2 - 1 / 2})$ , where $A_{q_{i}} (ω, θ)$ , $i = 1, 2$ is the scattering amplitude related to the compactly supported potential $q_{i} (x)$ at a fixed energy level $k =$ const., $φ (t) = (- ln t)^{- δ}$ , $0 < δ < 1$ and $∥ \cdot ∥_{R, 3 / 2 - 1 / 2}$ is a suitably defined norm.

MR Zbl | 2 citations dans Numdam

DOI : 10.5802/aif.1239

@article{AIF_1990__40_4_867_0,
     author = {Stefanov, Plamen},
     title = {Stability of the inverse problem in potential scattering at fixed energy},
     journal = {Annales de l'Institut Fourier},
     pages = {867--884},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {40},
     number = {4},
     year = {1990},
     doi = {10.5802/aif.1239},
     mrnumber = {92d:35217},
     zbl = {0715.35082},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1239/}
}

TY  - JOUR
AU  - Stefanov, Plamen
TI  - Stability of the inverse problem in potential scattering at fixed energy
JO  - Annales de l'Institut Fourier
PY  - 1990
SP  - 867
EP  - 884
VL  - 40
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1239/
DO  - 10.5802/aif.1239
LA  - en
ID  - AIF_1990__40_4_867_0
ER  -

%0 Journal Article
%A Stefanov, Plamen
%T Stability of the inverse problem in potential scattering at fixed energy
%J Annales de l'Institut Fourier
%D 1990
%P 867-884
%V 40
%N 4
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1239/
%R 10.5802/aif.1239
%G en
%F AIF_1990__40_4_867_0

Stefanov, Plamen. Stability of the inverse problem in potential scattering at fixed energy. Annales de l'Institut Fourier, Tome 40 (1990) no. 4, pp. 867-884. doi : 10.5802/aif.1239. http://www.numdam.org/articles/10.5802/aif.1239/

Bibliographie
Cité par

[A] S. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172. | MR | Zbl

[EMOT] A. Erdélyi, W. Magnus, F. Oberhettinger and F. C. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw Hill, New York, 1953. | MR | Zbl

[L] R. Leis, Zur Monotonie der Eigenwerte selbstadjungierter elliptischer Differentialgleichungen, Math. Z., 96 (1967), 26-32. | MR | Zbl

[M] C. Müller, Grundprobleme der Mathematischen Theorie Elektromagnetischer Schwingungen, Springer, 1957. | Zbl

[Na1] A. Nachman, Reconstruction from boundary measurements, Ann. Math., 128 (1988), 531-576. | MR | Zbl

[Na2] A. Nachman, Exact reconstruction procedures for some inverse scattering and inverse boundary problems, Meeting on inverse problems, Montpellier, 1989.

[NSU] A. Nachman, J. Sylvester and G. Uhlmann, An n-dimensional Borg-Levison theorem, Comm. Math. Physics, 115 (1988), 595-605. | MR | Zbl

[NO] R. G. Novikov, Multidimensional inverse spectral problems for the equation - Δѱ + (v(x) - Eu(x))ѱ = 0, Funkt. Analizi i Ego Prilozheniya, 22 (4) (1988), 11-12; Translation in Funct. Anal. and its Appl., 22 (4) (1988), 263-272. | MR | Zbl

[R] A. G. Ramm, Recovery of the potential from fixed energy scattering data, Inverse Probl., 4 (1988), 877-886. | MR | Zbl

[RS] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV, Accademic Press, New York, 1978. | Zbl

[S] B. Simon, Schrödinger semigroups, Bull. A.M.S., 7 (3) (1982), 447-526. | MR | Zbl

[SU1] J. Sylvester and G. Uhlmann, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure Appl. Math., 39 (1986), 91-112. | MR | Zbl

[SU2] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem. Ann. Math., 125 (1987), 153-169. | MR | Zbl

[SU3] J. Sylvester and G. Uhlmann, Inverse boundary value problem at the boundary - continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197-221. | MR | Zbl

[SU4] J. Sylvester and G. Uhlmann, The Dirichlet to Neumann map and its applications, Proceedings of meeting in Arcata, California on inverse problems, July 29-August 4, 1989. | Zbl

Cité par Sources :