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Scientific Journal of Samarkand University

Abstract

On the d- dimensional lattice 2 , 1 ,  d d Z the discrete Schrödinger operator  H with non- local potential constructed via the Dirac delta function and shift operator is considered. The existence of negative eigenvalues on the parameters of the operator is explicity derived.

First Page

8

Last Page

12

References

1. S. Albeverio, S. N. Lakaev, K. A. Makarov, Z. I. Muminov: The Threshold Effects for the Two-particle Hamiltonians on Lattices, Comm.Math.Phys. 262 (2006), 91--115. 2. J. Bellissard and H. Schulz-Baldes: Scattering theory for lattice operators in dimension 3  d , arXiv:1109.5459v2, 2012 3. P. Exner, P.A. Kuchment and B. Winn: On the location of spectral edges in Z-peridoc media, J. Phys. A 43 (2010), 474022 4. 4. Faria da Veiga P. A., Ioriatti L., and OʻCarroll M.: Energy-momentum spectrum of some two-particle lattice Schrödinger Hamiltonians. Phys. Rev. E (3) 66, 016130, 9 pp. (2002). 5. 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 6. S.N. Lakaev, I.N. Bozorov.: The number of bound states of one particle Hamilonian on a three-dimensional lattice. Theoretical and Mathematical physics, 158(3): 360- 376 (2009)

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