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

show all references

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

[1]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[2]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[3]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[4]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[5]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316

[6]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002

[7]

Lingyu Li, Jianfu Yang, Jinge Yang. Solutions to Chern-Simons-Schrödinger systems with external potential. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021008

[8]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298

[9]

Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, 2021, 20 (2) : 867-884. doi: 10.3934/cpaa.2020294

[10]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374

[11]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[12]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[13]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[14]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[15]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[16]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[17]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[18]

Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger Equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020392

[19]

Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125

[20]

Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328

2019 Impact Factor: 1.373

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]