THE NUMBER OF EIGENVALUES OF THE DISCRETE SCHRÖDINGER OPERATOR
Keywords:
We study the discrete Schrodinger operators H_γλμ of a quantum particle moving in the one-dimensional lattice Z interacting with an indefinite sign external field, where the potential has both positive and negative values. We also find particular connected components of the γ-λ-µ space, where the number of eigenvalues are preserved. Interestingly, this division is only dependent on the first two coefficients of the asymptotic expansion of the Fredholm determinant corresponding to the operator H_γλμ near the threshold (the left edge of the essential spectrum).Abstract
We study the discrete Schrodinger operators of a quantum particle moving in the one-dimensional lattice interacting with an indefinite sign external field, where the potential has both positive and negative values. In this paper we consider the family of Schrodinger operators, associated to the Bose-Hubbard Hamiltonian of a single boson on the one dimensional lattice with on-site interaction , nearest-neighbor interaction and next nearest-neighbor interaction . We find suffiently conditions for the parameters , such that the operator has no one eigenvalues below the essential spectrum and three eigenvalues above the essential spectrum.
References
S. N. Lakaev, A. M. Khalkhuzhaev, Sh. S. Lakaev: Asymptotic behavior of an eigenvalue of the two-particle discrete Schrödinger operator, Teoret. Mat. Fiz., 171:3 (2012), 438-451
Saidakhmat N. Lakaev, Ender Ozdemir: The existence and location of eigenvalues of the one particle discrete Schrödinger operator, Hacettepe J. Math. and Statistics, 45(2016),1963-1703.
M. Reed and B. Simon: Methods of modern mathematical physics. IV: Analysis of operators. Academic Press, N.Y., 1978.
