Estimates on the number of eigenvalues of two-particle discrete Schrödinger operators
Sergio Albeverio, Saidakhmat N. Lakaev, Janikul I. Abdullaev
Abstract
Two-particle discrete Schrödinger operators H(k)=H0(k)-V on the three-dimensional lattice 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∈ 3 (-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∈ 3, 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∈ 3 is π.
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