no. 3
Inverse scattering at a fixed energy for Discrete Schrödinger Operators on the square lattice
[Diffusion inverse à énergie fixée pour l’opérateur Schrödinger sur un réseau carré]
Isozaki, Hiroshi1 ; Morioka, Hisashi1
1 University of Tsukuba Division of Mathematics 1-1-1 Tennoudai, Tsukuba, Ibaraki, 305-8571 (Japan)
Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1153-1200.

Nous étudions un problème inverse de diffusion pour l’opérateur de Schrödinger discret sur un réseau carré ℤ d , d≥2, avec un potentiel à support compact. Nous montrons que le potentiel est uniquement determiné en utilisant la matrice de diffusion à énergie fixée.

We study an inverse scattering problem for the discrete Schrödinger operator on the square lattice ℤ d , d≥2, with compactly supported potential. We show that the potential is uniquely reconstructed from a scattering matrix for a fixed energy.

DOI : 10.5802/aif.2954
Classification : 81U40, 47A40, 39A12
Keywords: Schrödinger operator, Scattering theory, Inverse Problem
Mot clés : l’opérateur de Schrödinger, la théorie de diffusion, le problème inverse

Isozaki, Hiroshi 1 ; Morioka, Hisashi 1

1 University of Tsukuba Division of Mathematics 1-1-1 Tennoudai, Tsukuba, Ibaraki, 305-8571 (Japan) 
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     journal = {Annales de l'Institut Fourier},
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Isozaki, Hiroshi; Morioka, Hisashi. Inverse scattering at a fixed energy for Discrete Schrödinger Operators on the square lattice. Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1153-1200. doi : 10.5802/aif.2954. https://aif.centre-mersenne.org/articles/10.5802/aif.2954/

[1] Agmon, S.; Hörmander, L. Asymptotic properties of solutions of differential equations with simple characteristics, J. d’Anal. Math., Volume 30 (1976), pp. 1-38 | DOI | MR | Zbl

[2] Bukhgeim, A. Recovering the potential from Cauchy data in two dimensions, J. Inverse Ill-Posed Probl., Volume 16 (2008), pp. 19-34 | DOI | MR | Zbl

[3] Curtis, E.; Mooers, E.; Morrow, J. Finding the conductors in circular networks from boundary measurements, RAIRO Modél. Math. Anal. Numél., Volume 28 (1994), pp. 781-814 | Numdam | MR | Zbl

[4] Curtis, E.; Morrow, J. The Dirichlet to Neumann map for a resistor network, SIAM J. Appl. Math., Volume 51 (1991), pp. 1011-1029 | DOI | MR | Zbl

[5] Dodziuk, J. Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc., Volume 284 (1984), pp. 787-794 | DOI | MR | Zbl

[6] Eskina, M. S. The direct and the inverse scattering problem for a partial difference equation, Soviet Math. Doklady, Volume 7 (1966), pp. 193-197 | MR | Zbl

[7] Imanouilov, O. Y.; Uhlmann, G.; Yamamoto, M. The Calderón problem with partial Cauchy data in two dimensions, J. Amer. Math. Soc., Volume 23 (2010), pp. 655-691 | DOI | MR | Zbl

[8] Isakov, V.; Nachman, A. Global uniqueness for a two-dimensional semilinear elliptic inverse problem, Trans. Amer. Math. Soc., Volume 347 (1995), pp. 3375-3390 | DOI | MR | Zbl

[9] Isozaki, H.; Araki, H.; Ezawa, H. Inverse spectral theory, Topics In The Theory of Schrödinger Operators, World Scientific, 2003, pp. 93-143 | MR | Zbl

[10] Isozaki, H.; Korotyaev, E. Inverse problems, trace formulae for discrete Schrödinger operators, Ann. Henri Poincaré, Volume 13 (2012), pp. 751-788 | DOI | MR | Zbl

[11] Isozaki, H.; Morioka, H. A Rellich type theorem for discrete Schrödinger operators, Inverse Problems and Imaging, Volume 8 (2014), pp. 475-489 | DOI | MR

[12] Khenkin, G. M.; Novikov, R. G. The ∂ ¯-equation in the multi-dimensional inverse scattering problem, Russian Math. Surveys, Volume 42 (1987), pp. 109-180 | DOI | MR | Zbl

[13] Littman, W. Fourier transforms of surface-carried measures and differentiablity of surface averages, Bull, Amer. Math. Soc., Volume 69 (1963), pp. 766-770 | DOI | MR | Zbl

[14] Matsumura, M. Comportement des solutions de quelques problèmes mixtes pour certains systèmes huperboliques symétriques à coefficients constants, Publ. RIMS, Kyoto Univ. Ser. A, Volume 4 (1968), pp. 309-359 | DOI | MR | Zbl

[15] Nachman, A. Reconstruction from boundary measurements, Ann. of Math., Volume 128 (1988), pp. 531-576 | DOI | MR | Zbl

[16] Nachman, A. Global uniqueness for a two-dimensional inverse boundary value problem, Ann. Math., Volume 143 (1996), pp. 71-96 | DOI | MR | Zbl

[17] Nachman, A.; Päivärinta, L.; Teirilä, A. On imaging obstacles inside inhomogeneous media, J. Funct. Anal., Volume 252 (2007), pp. 490-516 | DOI | MR | Zbl

[18] Novikov, R. G. A multidimensional inverse spectral problem for the equation -Δψ+(v(x)-E)ψ=0, Funct. Anal. Appl., Volume 22 (1988), pp. 263-272 | DOI | MR | Zbl

[19] Oberlin, R. Discrete inverse problems for Schrödinger and resistor networks, Research archive of Research Experiences for Undergraduates program at Univ. of Washington, (2000) (www.math.washington.edu/~reu/papers/2000/oberlin/oberlin_schrodinger.pdf)

[20] Rellich, F. Über das asymptotische Verhalten der Lösungen von Δu+λu=0 in unendlichen Gebieten, Jahresber. Deitch. Math. Verein., Volume 53 (1943), pp. 57-65 | MR | Zbl

[21] Shaban, W.; Vainberg, B. Radiation conditions for the difference Schrödinger operators, Applicable Analysis, Volume 80 (2001), pp. 525-556 | DOI | MR | Zbl

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

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