Estimates on the number of eigenvalues of two-particle discrete Schr\"odinger operators

TL;DR

This paper provides estimates on the number of eigenvalues of two-particle discrete Schrödinger operators on a 3D lattice, analyzing their spectral properties depending on quasi-momentum and potential characteristics.

Contribution

It introduces bounds on eigenvalues outside the essential spectrum and investigates conditions for the existence of eigenvalues below the band, including accumulation phenomena.

Findings

Eigenvalue estimates outside the band based on potential eigenvalues

Existence of non-negative eigenvalues below the band for certain quasi-momenta

Infinite eigenvalues accumulating at the band bottom when a quasi-momentum coordinate equals pi

Abstract

Two-particle discrete Schr\"{o}dinger operators $H(k)=H_{0}(k)-V$ 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 $H_{0}(k)$ 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 $k\in \T^3\equiv (-\pi,\pi]^3$, 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 $\pi.$