ASYMPTOTIC BEHAVIOR OF EIGENVALUES IN DISCRETE SCHRÖDINGER OPERATORS WITH POINT INTERACTION POTENTIAL
Keywords:
Schrödinger operator, spectrum, eigenvalue, Fredholm determinant, eigenvalue asymptoticsAbstract
We investigate the one-particle discrete Schrödinger operator with Dirac delta potential on the -dimensional cubic lattice. We show that the operator the operator has a unique eigenvalue and obtain an asymptotics expansion for this eigenvalue as wight of potential approaches the infinity.
Downloads
References
Albeverio S., Lakaev S. N., Makarov K. A., Muminov Z. I.: The Threshold Effects for the Two-particle Hamiltonians on Lattices, Comm.Math.Phys. 262, 2006. P. 91–115.
Albeverio S., S.N. Lakaev, Muminov Z.I. Schrödinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics. Ann. Henri Poincare, 5 (4), 2004. P.743-772.
Berkolaiko G., Carlson R., Fulling S.A. and Kuchment P.A. (eds.): Quantum Graphs and Their Applications, Contemp. Math. 2006.
Berkolaiko G. and Kuchment P.A.: Introduction to Quantum Graphs, AMS Mathematical Surveys and Monographs, 186, (2012).
Bellissard J. and Schulz-Baldes H. Scattering theory for lattice operators in dimension , Reviews in Mathematical Physics, 24(8), 2012.
Chung F.: Spectral Graph Theory, CBMS Regional Conf. Series Math., Washington DC 1997.
Damanik D., Hundertmark D., Killip R., Simon B. Variational estimates for discrete Schrodinger operators with potentials of indefinite sign. Comm.Math.Phys. Vol. 238, 2003. P. 545-562.
Exner P., Kuchment P.A. and Winn B. : On the location of spectral edges in Z-peridoc media, J. Phys. A 43, 474022 (2010).
Exner P., Keating J.P., Kuchment P.A., Sunada T. and Teplyaev A. (eds.): Analysis on Graphs and Its Applications, Proc. Symp. Pure Math., vol. 77, AMS Providence, 2008.
Faria da Veiga P. A., Ioriatti L., and O’Carroll M.: Energy-momentum spectrum of some two-particle lattice Schrödinger Hamiltonians. Phys. Rev. E (3) 66, 2002.
Grigor’yan A. : Heat kernels on manifolds, graphs and fractals, in: European Congress of Mathematics, Barcelona, July 10-14, 2000, Progress in Mathematics 201, BirkhAauser, 2001. P. 393-406.
Hiroshima F., Muminov Z., Kuljanov U., “Threshold of discrete Schrödinger operators with delta potentials on N-dimensional lattice, ”Linear and Multilinear Algebra. 2020. P. 1563-5139.
Hiroshima F., Sasaki I., Shirai T. and Suzuki A.: Note on the spectrum of discrete Schrödinger operators, J. Math-for-Industry 4. 2012. P.105-108.
Kholmatov Sh.Yu., Lakaev S.N., Almuratov F.M.: Bound states of Schrödinger-type operators on one and two dimensional lattices. J. Math. Anal. Appl. 503 2021.
Korotyaev E. and Saburova N.: Schrödinger operators on periodic discrete graphs, arXiv:1307.1841 (2013).
Lakaev S.N., Bozorov I.N.: The number of bound states of one particle Hamilonian on a three-dimensional lattice. Theoretical and Mathematical physics, 158(3): 2009.P.360-376.
Lakaev S.N., Bozorov I.N. “On the number and location of eigenvalues oneparticle Hamiltonian on a one-dimensional lattice,”Uzbek Mathematical Journal, 2, [in Russian].2007.P.70-80.
Lakaev S.N., Tilovova Sh.M. Merging of eigenvalues and resonances of a two-particle Schrödinger operator. Theoretical and Mathematical Physics. 101:2:13201331. 1994.
Muminov Z.E., Alladustov Sh.U., and Lakaev Sh.S., “Threshold Analysis of the Three Dimensional Lattice Schrödinger Operator with Non-local Potential, ”Lobachevskii J. Math., 41 (6), 2020. P. 1094-1102.
Muminov Z.E., Alladustov Sh.U., and Lakaev Sh.S. “Spectral and threshold analysis of a small rank perturbation of the discrete Laplacian,”J. Math. Anal. Appl., 496 (2), 124827 (2021).
Post O.: Spectral Analysis on Graph-Like Spaces, Lecture Notes in Mathematics 2039, Springer, 2012.
Yafaev D. R.: Scattering theory: Some old and new problems, Lecture Notes in Mathematics, 1735. Springer-Verlag, Berlin, 169 pp, 2000.
Z. Muminov, Sh. Lakaev.: On negative eigenvalues of the discrete Schrödinger operator with non-local potential. Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences. 3 (1) Article 4, 2020.
F.A. Madatova, Sh.S. Lakaev.: Asymptotics for the eigenvalue of the discrete Schrödinger operator on the d-dimensional lattice. Uz.Math. Journal. 2022. P.61-69.
Downloads
Published
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
License Terms of our Journal
