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 and 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), 1-25. doi: 10.1007/BF02795485.

[2]

S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. d'Anal. Math., 30 (1976), 1-38. doi: 10.1007/BF02786703.

[3]

E. M. Chirka, Complex Analytic Sets, Mathematics and Its applications, Kluwer Academic Publishers, Dordrecht, 1989.

[4]

J. Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc., 284 (1984), 787-794. doi: 10.1090/S0002-9947-1984-0743744-X.

[5]

D. M. Eidus, The principle of limiting absorption, Amer. Math. Soc. Transl. Set. 2, 47 (1962), 157-191.

[6]

M. S. Eskina, The direct and the inverse scattering problem for a partial difference equation, Soviet Math. Doklady, 7 (1966), 193-197.

[7]

C. Gérard and F. Nier, The Mourre theory for analytically fibered operators, J. Funct. Anal., 152 (1989), 202-219. doi: 10.1006/jfan.1997.3154.

[8]

F. Hiroshima, I. Sasaki, T. Shirai and A. Suzuki, Note on the spectrum of discrete Schrödinger operators, J. Math-for-Industry, 4 (2012), 105-108.

[9]

L. Hörmander, Lower bounds at infinity for solutions of differential equations with constant coefficients, Israel J. Math., 16 (1973), 103-116. doi: 10.1007/BF02761975.

[10]

L. Hörmander, The Analysis of Linear Partial Differential Operators III, Pseudo-Differential Operators, Classics in Mathematics. Springer, Berlin, 2007.

[11]

H. Isozaki and E. Korotyaev, Inverse problems, trace formulae for discrete Schrödinger operators, Ann. Henri Poincaré, 13 (2012), 751-788. doi: 10.1007/s00023-011-0141-0.

[12]

H. Isozaki and H. Morioka, Inverse scattering at a fixed energy for discrete Schrödinger operators on the square lattice, preprint, arXiv:math/12084483.

[13]

T. Kato, Growth properties of solutions of the reduced wave equation with a variable coefficient, CPAM, 12 (1959), 403-425. doi: 10.1002/cpa.3160120302.

[14]

S. G. Krantz, Function Theory of Several Complex Variables, John Wiley and Sons Inc., 1982.

[15]

P. Kuchment and B. Vainberg, On absence of embedded eigenvalues for Schrödinger operators with perturbed periodic potentials, Comm. PDE, 25 (2000), 1809-1826. doi: 10.1080/03605300008821568.

[16]

W. Littman, Decay at infinity of solutions to partial differential equations with constant coefficients, Trans. Amer. Math. Soc., 123 (1966), 449-459. doi: 10.1090/S0002-9947-1966-0197951-7.

[17]

W. Littman, Decay at infinity of solutions to higher order partial differential equations: Removal of the curvature assumption, Israel J. Math., 8 (1970), 403-407. doi: 10.1007/BF02798687.

[18]

M. Murata, Asymptotic behaviors at infinity of solutions to certain linear partial differential equations, J. Fac. Sci. Univ. Tokyo Sec. IA, 23 (1976), 107-148.

[19]

F. Rellich, Über das asymptotische Verhalten der Lösungen von $\Delta u + \lambda u = 0$ in unendlichen Gebieten, Jahresber. Deitch. Math. Verein., 53 (1943), 57-65.

[20]

S. N. Roze, The spectrum of a second-order elliptic operator, Math. Sb., 80 (122) (1969), 195-209.

[21]

W. Shaban and B. Vainberg, Radiation conditions for the difference Schrödinger operators, Applicable Analysis, 80 (2001), 525-556. doi: 10.1080/00036810108841007.

[22]

I. R. Shafarevich, Basic Algebraic Geometry 1, $2^{nd}$ edition, Springer-Verlag, Heidelberg, 1994.

[23]

F. Treves, Differential polynomials and decay at infinity, Bull. Amer. Math. Soc., 66 (1960), 184-186. doi: 10.1090/S0002-9904-1960-10423-5.

[24]

E. Vekua, On metaharmonic functions, Trudy Tbiliss. Mat. Inst., 12 (1943), 105-174, (in Russian, Georgian, and English summary).

[25]

M. Zworski, Semiclassical Analysis, Graduate studies in Mathematics, 138, A. M. S., Providence, R. I., 2012.

show all references

References:
[1]

S. Agmon, Lower bounds for solutions of Schrödinger equations, J. d'Anal. Math., 23 (1970), 1-25. doi: 10.1007/BF02795485.

[2]

S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. d'Anal. Math., 30 (1976), 1-38. doi: 10.1007/BF02786703.

[3]

E. M. Chirka, Complex Analytic Sets, Mathematics and Its applications, Kluwer Academic Publishers, Dordrecht, 1989.

[4]

J. Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc., 284 (1984), 787-794. doi: 10.1090/S0002-9947-1984-0743744-X.

[5]

D. M. Eidus, The principle of limiting absorption, Amer. Math. Soc. Transl. Set. 2, 47 (1962), 157-191.

[6]

M. S. Eskina, The direct and the inverse scattering problem for a partial difference equation, Soviet Math. Doklady, 7 (1966), 193-197.

[7]

C. Gérard and F. Nier, The Mourre theory for analytically fibered operators, J. Funct. Anal., 152 (1989), 202-219. doi: 10.1006/jfan.1997.3154.

[8]

F. Hiroshima, I. Sasaki, T. Shirai and A. Suzuki, Note on the spectrum of discrete Schrödinger operators, J. Math-for-Industry, 4 (2012), 105-108.

[9]

L. Hörmander, Lower bounds at infinity for solutions of differential equations with constant coefficients, Israel J. Math., 16 (1973), 103-116. doi: 10.1007/BF02761975.

[10]

L. Hörmander, The Analysis of Linear Partial Differential Operators III, Pseudo-Differential Operators, Classics in Mathematics. Springer, Berlin, 2007.

[11]

H. Isozaki and E. Korotyaev, Inverse problems, trace formulae for discrete Schrödinger operators, Ann. Henri Poincaré, 13 (2012), 751-788. doi: 10.1007/s00023-011-0141-0.

[12]

H. Isozaki and H. Morioka, Inverse scattering at a fixed energy for discrete Schrödinger operators on the square lattice, preprint, arXiv:math/12084483.

[13]

T. Kato, Growth properties of solutions of the reduced wave equation with a variable coefficient, CPAM, 12 (1959), 403-425. doi: 10.1002/cpa.3160120302.

[14]

S. G. Krantz, Function Theory of Several Complex Variables, John Wiley and Sons Inc., 1982.

[15]

P. Kuchment and B. Vainberg, On absence of embedded eigenvalues for Schrödinger operators with perturbed periodic potentials, Comm. PDE, 25 (2000), 1809-1826. doi: 10.1080/03605300008821568.

[16]

W. Littman, Decay at infinity of solutions to partial differential equations with constant coefficients, Trans. Amer. Math. Soc., 123 (1966), 449-459. doi: 10.1090/S0002-9947-1966-0197951-7.

[17]

W. Littman, Decay at infinity of solutions to higher order partial differential equations: Removal of the curvature assumption, Israel J. Math., 8 (1970), 403-407. doi: 10.1007/BF02798687.

[18]

M. Murata, Asymptotic behaviors at infinity of solutions to certain linear partial differential equations, J. Fac. Sci. Univ. Tokyo Sec. IA, 23 (1976), 107-148.

[19]

F. Rellich, Über das asymptotische Verhalten der Lösungen von $\Delta u + \lambda u = 0$ in unendlichen Gebieten, Jahresber. Deitch. Math. Verein., 53 (1943), 57-65.

[20]

S. N. Roze, The spectrum of a second-order elliptic operator, Math. Sb., 80 (122) (1969), 195-209.

[21]

W. Shaban and B. Vainberg, Radiation conditions for the difference Schrödinger operators, Applicable Analysis, 80 (2001), 525-556. doi: 10.1080/00036810108841007.

[22]

I. R. Shafarevich, Basic Algebraic Geometry 1, $2^{nd}$ edition, Springer-Verlag, Heidelberg, 1994.

[23]

F. Treves, Differential polynomials and decay at infinity, Bull. Amer. Math. Soc., 66 (1960), 184-186. doi: 10.1090/S0002-9904-1960-10423-5.

[24]

E. Vekua, On metaharmonic functions, Trudy Tbiliss. Mat. Inst., 12 (1943), 105-174, (in Russian, Georgian, and English summary).

[25]

M. Zworski, Semiclassical Analysis, Graduate studies in Mathematics, 138, A. M. S., Providence, R. I., 2012.

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