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August  2011, 5(3): 715-730. doi: 10.3934/ipi.2011.5.715

Inverse boundary value problems for discrete Schrödinger operators on the multi-dimensional square lattice

 1 Graduate school of pure and applied sciences, University of Tsukuba, Tennnoudai 1-1-1, Tsukuba, Ibaraki, 305-0821, Japan

Received  August 2010 Revised  May 2011 Published  August 2011

We consider an inverse boundary value problem for a discrete Schrödinger operator $-\Delta + \hat{q}$ on a bounded domain in the square lattice. We define an analogue of the Dirichlet-to-Neumann map, and give a reconstruction procedure of the potential $\hat{q}$ from the D-to-N map for all energies.
Citation: Hisashi Morioka. Inverse boundary value problems for discrete Schrödinger operators on the multi-dimensional square lattice. Inverse Problems & Imaging, 2011, 5 (3) : 715-730. doi: 10.3934/ipi.2011.5.715
References:
 [1] K. Ando, Inverse scattering theory for discrete Schrödinger operators on the hexagonal lattice,, preprint., ().   Google Scholar [2] L. Borcea, V. Druskin and A. Mamonov, Circular resistor networks for electrical impedance tomography with partial boundary measurements,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/4/045010.  Google Scholar [3] L. Borcea, V. Druskin, A. Mamonov and F. Guevara Vasquez, Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/10/105009.  Google Scholar [4] F. R. Chung, "Spectral Graph Theory,", CBMS Regional Conference Series in Mathematics, 92 (1997).   Google Scholar [5] E. Curtis and J. Morrow, The Dirichlet to Neumann map for a resistor network,, SIAM J. Appl. Math., 51 (1991), 1011.  doi: 10.1137/0151051.  Google Scholar [6] E. Curtis, E. Mooers and J. Morrow, Finding the conductors in circular networks from boundary measurements,, RAIRO Modél. Math. Anal. Numér., 28 (1994), 781.   Google Scholar [7] R. Diestel, "Graph Theory,", 2nd edition, 173 (2000).   Google Scholar [8] H. Isozaki and E. Korotyaev, Inverse problems, trace formulae for discrete Schrödinger operators,, submitted., ().   Google Scholar [9] H. Isozaki, Some remarks on the multi-dimensional Borg-Levinson theorem,, J. Math. Kyoto Univ., 31 (1991), 743.   Google Scholar [10] R. G. Novikov and G. M. Khenkin, The $\overline\partial$-equation in the multidimensional inverse scattering problem,, (Russian) Uspekhi Mat. Nauk, 42 (1987), 93.   Google Scholar [11] A. I. Nachman, Reconstruction from boundary measurements,, Ann. Math. (2), 128 (1988), 531.  doi: 10.2307/1971435.  Google Scholar [12] A. I. Nachman, J. Sylvester and G. Uhlmann, An $n$-dimensional Borg-Levinson theorem,, Commun. Math. Phys., 115 (1988), 595.  doi: 10.1007/BF01224129.  Google Scholar [13] R. Oberlin, Discrete inverse problems for Schrödinger and resistor networks,, Research archive of Research Experiences for Undergraduates program at Univ. of Washington, (2000).   Google Scholar [14] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. Math. (2), 125 (1987), 153.  doi: 10.2307/1971291.  Google Scholar

show all references

References:
 [1] K. Ando, Inverse scattering theory for discrete Schrödinger operators on the hexagonal lattice,, preprint., ().   Google Scholar [2] L. Borcea, V. Druskin and A. Mamonov, Circular resistor networks for electrical impedance tomography with partial boundary measurements,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/4/045010.  Google Scholar [3] L. Borcea, V. Druskin, A. Mamonov and F. Guevara Vasquez, Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/10/105009.  Google Scholar [4] F. R. Chung, "Spectral Graph Theory,", CBMS Regional Conference Series in Mathematics, 92 (1997).   Google Scholar [5] E. Curtis and J. Morrow, The Dirichlet to Neumann map for a resistor network,, SIAM J. Appl. Math., 51 (1991), 1011.  doi: 10.1137/0151051.  Google Scholar [6] E. Curtis, E. Mooers and J. Morrow, Finding the conductors in circular networks from boundary measurements,, RAIRO Modél. Math. Anal. Numér., 28 (1994), 781.   Google Scholar [7] R. Diestel, "Graph Theory,", 2nd edition, 173 (2000).   Google Scholar [8] H. Isozaki and E. Korotyaev, Inverse problems, trace formulae for discrete Schrödinger operators,, submitted., ().   Google Scholar [9] H. Isozaki, Some remarks on the multi-dimensional Borg-Levinson theorem,, J. Math. Kyoto Univ., 31 (1991), 743.   Google Scholar [10] R. G. Novikov and G. M. Khenkin, The $\overline\partial$-equation in the multidimensional inverse scattering problem,, (Russian) Uspekhi Mat. Nauk, 42 (1987), 93.   Google Scholar [11] A. I. Nachman, Reconstruction from boundary measurements,, Ann. Math. (2), 128 (1988), 531.  doi: 10.2307/1971435.  Google Scholar [12] A. I. Nachman, J. Sylvester and G. Uhlmann, An $n$-dimensional Borg-Levinson theorem,, Commun. Math. Phys., 115 (1988), 595.  doi: 10.1007/BF01224129.  Google Scholar [13] R. Oberlin, Discrete inverse problems for Schrödinger and resistor networks,, Research archive of Research Experiences for Undergraduates program at Univ. of Washington, (2000).   Google Scholar [14] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. Math. (2), 125 (1987), 153.  doi: 10.2307/1971291.  Google Scholar
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