# American Institute of Mathematical Sciences

May  2014, 8(2): 475-489. doi: 10.3934/ipi.2014.8.475

## A Rellich type theorem for discrete Schrödinger operators

 1 Division of Mathematics, University of Tsukuba, Tennoudai 1-1-1, Tsukuba, Ibaraki, 305-8571, Japan, Japan

Received  July 2013 Revised  November 2013 Published  May 2014

An analogue of Rellich's theorem is proved for discrete Laplacians on square lattices, and applied to show unique continuation properties on certain domains as well as non-existence of embedded eigenvalues for discrete Schrödinger operators.
Citation: Hiroshi Isozaki, Hisashi Morioka. A Rellich type theorem for discrete Schrödinger operators. Inverse Problems & Imaging, 2014, 8 (2) : 475-489. doi: 10.3934/ipi.2014.8.475
##### References:
 [1] S. Agmon, Lower bounds for solutions of Schrödinger equations,, J. d'Anal. Math., 23 (1970), 1.  doi: 10.1007/BF02795485.  Google Scholar [2] S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics,, J. d'Anal. Math., 30 (1976), 1.  doi: 10.1007/BF02786703.  Google Scholar [3] E. M. Chirka, Complex Analytic Sets,, Mathematics and Its applications, (1989).   Google Scholar [4] J. Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks,, Trans. Amer. Math. Soc., 284 (1984), 787.  doi: 10.1090/S0002-9947-1984-0743744-X.  Google Scholar [5] D. M. Eidus, The principle of limiting absorption,, Amer. Math. Soc. Transl. Set. 2, 47 (1962), 157.   Google Scholar [6] M. S. Eskina, The direct and the inverse scattering problem for a partial difference equation,, Soviet Math. Doklady, 7 (1966), 193.   Google Scholar [7] C. Gérard and F. Nier, The Mourre theory for analytically fibered operators,, J. Funct. Anal., 152 (1989), 202.  doi: 10.1006/jfan.1997.3154.  Google Scholar [8] F. Hiroshima, I. Sasaki, T. Shirai and A. Suzuki, Note on the spectrum of discrete Schrödinger operators,, J. Math-for-Industry, 4 (2012), 105.   Google Scholar [9] L. Hörmander, Lower bounds at infinity for solutions of differential equations with constant coefficients,, Israel J. Math., 16 (1973), 103.  doi: 10.1007/BF02761975.  Google Scholar [10] L. Hörmander, The Analysis of Linear Partial Differential Operators III, Pseudo-Differential Operators,, Classics in Mathematics. Springer, (2007).   Google Scholar [11] H. Isozaki and E. Korotyaev, Inverse problems, trace formulae for discrete Schrödinger operators,, Ann. Henri Poincaré, 13 (2012), 751.  doi: 10.1007/s00023-011-0141-0.  Google Scholar [12] H. Isozaki and H. Morioka, Inverse scattering at a fixed energy for discrete Schrödinger operators on the square lattice,, preprint, ().   Google Scholar [13] T. Kato, Growth properties of solutions of the reduced wave equation with a variable coefficient,, CPAM, 12 (1959), 403.  doi: 10.1002/cpa.3160120302.  Google Scholar [14] S. G. Krantz, Function Theory of Several Complex Variables,, John Wiley and Sons Inc., (1982).   Google Scholar [15] P. Kuchment and B. Vainberg, On absence of embedded eigenvalues for Schrödinger operators with perturbed periodic potentials,, Comm. PDE, 25 (2000), 1809.  doi: 10.1080/03605300008821568.  Google Scholar [16] W. Littman, Decay at infinity of solutions to partial differential equations with constant coefficients,, Trans. Amer. Math. Soc., 123 (1966), 449.  doi: 10.1090/S0002-9947-1966-0197951-7.  Google Scholar [17] W. Littman, Decay at infinity of solutions to higher order partial differential equations: Removal of the curvature assumption,, Israel J. Math., 8 (1970), 403.  doi: 10.1007/BF02798687.  Google Scholar [18] M. Murata, Asymptotic behaviors at infinity of solutions to certain linear partial differential equations,, J. Fac. Sci. Univ. Tokyo Sec. IA, 23 (1976), 107.   Google Scholar [19] F. Rellich, Über das asymptotische Verhalten der Lösungen von $\Delta u + \lambda u = 0$ in unendlichen Gebieten,, Jahresber. Deitch. Math. Verein., 53 (1943), 57.   Google Scholar [20] S. N. Roze, The spectrum of a second-order elliptic operator,, Math. Sb., 80 (1969), 195.   Google Scholar [21] W. Shaban and B. Vainberg, Radiation conditions for the difference Schrödinger operators,, Applicable Analysis, 80 (2001), 525.  doi: 10.1080/00036810108841007.  Google Scholar [22] I. R. Shafarevich, Basic Algebraic Geometry 1,, $2^{nd}$ edition, (1994).   Google Scholar [23] F. Treves, Differential polynomials and decay at infinity,, Bull. Amer. Math. Soc., 66 (1960), 184.  doi: 10.1090/S0002-9904-1960-10423-5.  Google Scholar [24] E. Vekua, On metaharmonic functions,, Trudy Tbiliss. Mat. Inst., 12 (1943), 105.   Google Scholar [25] M. Zworski, Semiclassical Analysis,, Graduate studies in Mathematics, (2012).   Google Scholar

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##### References:
 [1] S. Agmon, Lower bounds for solutions of Schrödinger equations,, J. d'Anal. Math., 23 (1970), 1.  doi: 10.1007/BF02795485.  Google Scholar [2] S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics,, J. d'Anal. Math., 30 (1976), 1.  doi: 10.1007/BF02786703.  Google Scholar [3] E. M. Chirka, Complex Analytic Sets,, Mathematics and Its applications, (1989).   Google Scholar [4] J. Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks,, Trans. Amer. Math. Soc., 284 (1984), 787.  doi: 10.1090/S0002-9947-1984-0743744-X.  Google Scholar [5] D. M. Eidus, The principle of limiting absorption,, Amer. Math. Soc. Transl. Set. 2, 47 (1962), 157.   Google Scholar [6] M. S. Eskina, The direct and the inverse scattering problem for a partial difference equation,, Soviet Math. Doklady, 7 (1966), 193.   Google Scholar [7] C. Gérard and F. Nier, The Mourre theory for analytically fibered operators,, J. Funct. Anal., 152 (1989), 202.  doi: 10.1006/jfan.1997.3154.  Google Scholar [8] F. Hiroshima, I. Sasaki, T. Shirai and A. Suzuki, Note on the spectrum of discrete Schrödinger operators,, J. Math-for-Industry, 4 (2012), 105.   Google Scholar [9] L. Hörmander, Lower bounds at infinity for solutions of differential equations with constant coefficients,, Israel J. Math., 16 (1973), 103.  doi: 10.1007/BF02761975.  Google Scholar [10] L. Hörmander, The Analysis of Linear Partial Differential Operators III, Pseudo-Differential Operators,, Classics in Mathematics. Springer, (2007).   Google Scholar [11] H. Isozaki and E. Korotyaev, Inverse problems, trace formulae for discrete Schrödinger operators,, Ann. Henri Poincaré, 13 (2012), 751.  doi: 10.1007/s00023-011-0141-0.  Google Scholar [12] H. Isozaki and H. Morioka, Inverse scattering at a fixed energy for discrete Schrödinger operators on the square lattice,, preprint, ().   Google Scholar [13] T. Kato, Growth properties of solutions of the reduced wave equation with a variable coefficient,, CPAM, 12 (1959), 403.  doi: 10.1002/cpa.3160120302.  Google Scholar [14] S. G. Krantz, Function Theory of Several Complex Variables,, John Wiley and Sons Inc., (1982).   Google Scholar [15] P. Kuchment and B. Vainberg, On absence of embedded eigenvalues for Schrödinger operators with perturbed periodic potentials,, Comm. PDE, 25 (2000), 1809.  doi: 10.1080/03605300008821568.  Google Scholar [16] W. Littman, Decay at infinity of solutions to partial differential equations with constant coefficients,, Trans. Amer. Math. Soc., 123 (1966), 449.  doi: 10.1090/S0002-9947-1966-0197951-7.  Google Scholar [17] W. Littman, Decay at infinity of solutions to higher order partial differential equations: Removal of the curvature assumption,, Israel J. Math., 8 (1970), 403.  doi: 10.1007/BF02798687.  Google Scholar [18] M. Murata, Asymptotic behaviors at infinity of solutions to certain linear partial differential equations,, J. Fac. Sci. Univ. Tokyo Sec. IA, 23 (1976), 107.   Google Scholar [19] F. Rellich, Über das asymptotische Verhalten der Lösungen von $\Delta u + \lambda u = 0$ in unendlichen Gebieten,, Jahresber. Deitch. Math. Verein., 53 (1943), 57.   Google Scholar [20] S. N. Roze, The spectrum of a second-order elliptic operator,, Math. Sb., 80 (1969), 195.   Google Scholar [21] W. Shaban and B. Vainberg, Radiation conditions for the difference Schrödinger operators,, Applicable Analysis, 80 (2001), 525.  doi: 10.1080/00036810108841007.  Google Scholar [22] I. R. Shafarevich, Basic Algebraic Geometry 1,, $2^{nd}$ edition, (1994).   Google Scholar [23] F. Treves, Differential polynomials and decay at infinity,, Bull. Amer. Math. Soc., 66 (1960), 184.  doi: 10.1090/S0002-9904-1960-10423-5.  Google Scholar [24] E. Vekua, On metaharmonic functions,, Trudy Tbiliss. Mat. Inst., 12 (1943), 105.   Google Scholar [25] M. Zworski, Semiclassical Analysis,, Graduate studies in Mathematics, (2012).   Google Scholar
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