×

Global uniqueness and reconstruction for the multi-channel Gel’fand-Calderón inverse problem in two dimensions. (English) Zbl 1228.35271

Let \(D\) be a bounded domain in \(\mathbb R^2\). The paper under review is concerned with the inverse boundary problem for the linear elliptic equation \(\Delta\psi =v(x)\psi\) in \(D\), where \(v\) is a smooth matrix-valued potential. The main result in this paper establishes an exact global reconstruction method for finding \(v\) from the associated boundary operator. A global uniqueness result is also provided. The results in the present paper are motivated by phenomena arising in quantum mechanics, acoustics or electrodynamics.

MSC:

35R30 Inverse problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J10 Schrödinger operator, Schrödinger equation
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Agranovich, Z. S.; Marchenko, V. A., The Inverse Problem of Scattering Theory (1963), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers New York, London, xiii+291 pp., translated from Russian by B.D. Seckler · Zbl 0117.06003
[2] Alessandrini, G., Stable determination of conductivity by boundary measurements, Appl. Anal., 27, 153-172 (1988) · Zbl 0616.35082
[3] S.V. Baykov, V.A. Burov, S.N. Sergeev, Mode tomography of moving ocean, in: Proc. of the 3rd European Conference on Underwater Acoustics, 1996, pp. 845-850.; S.V. Baykov, V.A. Burov, S.N. Sergeev, Mode tomography of moving ocean, in: Proc. of the 3rd European Conference on Underwater Acoustics, 1996, pp. 845-850.
[4] Beals, R.; Coifman, R. R., Multidimensional inverse scatterings and nonlinear partial differential equations, (Pseudodifferential Operators and Applications. Pseudodifferential Operators and Applications, Notre Dame, IN, 1984. Pseudodifferential Operators and Applications. Pseudodifferential Operators and Applications, Notre Dame, IN, 1984, Proc. Sympos. Pure Math., vol. 43 (1985), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 45-70 · Zbl 0575.35011
[5] Bikowski, J.; Knudsen, K.; Mueller, J. L., Direct numerical reconstruction of conductivities in three dimensions using scattering transforms, Inverse Problems, 27, 015002 (2011) · Zbl 1232.78007
[6] Bukhgeim, A. L., Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16, 1, 19-33 (2008) · Zbl 1142.30018
[7] Burov, V. A.; Rumyantseva, O. D.; Suchkova, T. V., Practical application possibilities of the functional approach to solving inverse scattering problems, Mosc. Phys. Soc., 3, 275-278 (1990), (in Russian)
[8] Calderón, A. P., On an inverse boundary problem, (Seminar on Numerical Analysis and Its Applications to Continuum Physics (1980), Soc. Brasileira de Matematica: Soc. Brasileira de Matematica Rio de Janeiro), 61-73
[9] Dubrovin, B. A.; Krichever, I. M.; Novikov, S. P., The Schrödinger equation in a periodic field and Riemann surfaces, Dokl. Akad. Nauk SSSR, 229, 1, 15-18 (1976) · Zbl 0441.35021
[10] Faddeev, L. D., Growing solutions of the Schrödinger equation, Dokl. Akad. Nauk SSSR, 165, 3, 514-517 (1965)
[11] I.M. Gelʼfand, Some problems of functional analysis and algebra, in: Proc. Int. Congr. Math., Amsterdam, 1954, pp. 253-276.; I.M. Gelʼfand, Some problems of functional analysis and algebra, in: Proc. Int. Congr. Math., Amsterdam, 1954, pp. 253-276.
[12] Grinevich, P. G., The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed nonzero energy, Uspekhi Mat. Nauk. Uspekhi Mat. Nauk, Russian Math. Surveys, 55, 6, 1015-1083 (2000), (in Russian), translation in: · Zbl 1022.81057
[13] Novikov, R. G., Multidimensional inverse spectral problem for the equation \(- \Delta \psi +(v(x) - E u(x)) \psi = 0\), Funktsional. Anal. i Prilozhen.. Funktsional. Anal. i Prilozhen., Funct. Anal. Appl., 22, 4, 263-272 (1988), (in Russian); English transl.: · Zbl 0689.35098
[14] Novikov, R. G., The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator, J. Funct. Anal., 103, 2, 409-463 (1992) · Zbl 0762.35077
[15] Novikov, R. G., Formulae and equations for finding scattering data from the Dirichlet-to-Neumann map with nonzero background potential, Inverse Problems, 21, 1, 257-270 (2005) · Zbl 1063.35152
[16] Novikov, R. G., New global stability estimates for the Gelʼfand-Calderon inverse problem, Inverse Problems, 27, 015001 (2011) · Zbl 1208.35173
[17] Novikov, R. G.; Santacesaria, M., A global stability estimate for the Gelʼfand-Calderón inverse problem in two dimensions, J. Inverse Ill-Posed Probl., 18, 765-785 (2010), e-print · Zbl 1279.35120
[18] Vekua, I. N., Generalized Analytic Functions (1962), Pergamon Press Ltd. · Zbl 0127.03505
[19] Xiaosheng, L., Inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials in two dimensions at fixed energy, Comm. Partial Differential Equations, 30, 4-6, 451-482 (2005) · Zbl 1076.35131
[20] Zakhariev, B. N.; Suzko, A. A., Direct and Inverse Problems. Potentials in Quantum Scattering (1990), Springer-Verlag: Springer-Verlag Berlin, xiv+223 pp., translated from Russian by G. Pontecorvo · Zbl 0636.35002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.