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Estimates on the number of eigenvalues of two-particle discrete Schrodinger operators
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ESTIMATES ON THE NUMBER OF EIGENVALUES OF TWO-PARTICLE
DISCRETE SCHRO?DINGER OPERATORS
SERGIO ALBEVERIO
1,2,3
, SAIDAKHMAT N. LAKAEV
4,5
, AND JANIKUL I. ABDULLAEV
5
ABSTRACT. Two-particle discrete Schro?dinger operators H(k) = H0(k) ? V on the
three-dimensional lattice Z
3, k being the two-particle quasi-momentum, are considered.
An estimate for the number of the eigenvalues lying outside of the band of H0(k) via the
number of eigenvalues of the potential operator V bigger than the width of the band of
H0(k) is obtained. The existence of non negative eigenvalues below the band of H0(k) is
proven for nontrivial values of the quasi-momentum k ∈ T3 ≡ (?π, π]3, provided that
the operator H(0) has either a zero energy resonance or a zero eigenvalue. It is shown that
the operator H(k), k ∈ T3, has infinitely many eigenvalues accumulating at the bottom
of the band from below if one of the coordinates k(j), j = 1, 2, 3, of k ∈ T3 is π.
Subject Classification: Primary: 81Q10, Secondary: 35P20, 47N50
Keywords and phrases: Spectral properties, two-particle, discrete Schro?dinger oper-
ators, number of eigenvalues, band spectrum, Birman-Schwinger principle, zero energy
resonance, zero eigenvalue, low-lying excitation spectrum.
1. INTRODUCTION
The study of the low-lying excitation spectrum for lattice Hamiltonians of systems with
an infinite number of degrees of freedom has recently attracted considerable attention (see,
e.g., [7, 10, 20]).
See also [3, 4, 5, 6, 9, 11, 13, 19] for general expositions and the discussion of particular
problems of the theory of discrete Schro?dinger operators on lattices, including applications
to solid state physics.
The main aim of the present paper is to provide a thorough analysis of the dependence
of the number of eigenvalues of a two-particle lattice Schro?dinger operator with emphasis
on its non-trivial dependence on the total quasi-momentum and threshold phenomena.
In the
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