Estimates on the number of eigenvalues of two-particle discrete Schrödinger operators

Abstract

Two-particle discrete Schrödinger operators $H(k)=H_0(k)-V$ on the three-dimensional lattice ${\mathbb{Z}}^3,$ $k$ being the two-particle quasi-momentum, are considered. An estimate for the number of the eigenvalues lying outside of the band of $H_0(k)$ via the number of eigenvalues of the potential operator $V$ bigger than the width of the band of $H_0(k)$ is obtained. The existence of non negative eigenvalues below the band of $H_0(k)$ is proven for nontrivial values of the quasi-momentum $k\in \T^3\equiv (-π,π]^3$, provided that the operator H(0) has either a zero energy resonance or a zero eigenvalue. It is shown that the operator $H(k), k\in \T^3,$ 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 $k\in \T^3$ is $\pi .$

12 pages