On the discrete spectrum of two-particle discrete Schr\"odinger operators

TL;DR

This paper investigates the spectral properties of two-particle discrete Schrödinger operators on multi-dimensional lattices, establishing conditions for the existence of an infinite discrete spectrum depending on the quasi-momentum.

Contribution

It provides necessary and sufficient conditions for the existence of an infinite discrete spectrum for these operators, depending on the quasi-momentum.

Findings

Conditions for infinite discrete spectrum when quasi-momentum is outside a specific domain.

Spectral properties depend on the quasi-momentum and potential.

Extension of spectral theory to multi-dimensional lattice operators.

Abstract

In the present paper our aim is to explore some spectral properties of the family two-particle discrete Schr\"odinger operators $h^{\mathrm{d}}(k)=h^{\mathrm{d}}_0(k)+ \mathbf{v},$ $k\in \T^\mathrm{d},$ on the $\mathrm{d}$ dimensional lattice $\Z^{\mathrm{d}},$ $\mathrm{d}\geq 1,$ $k$ being the two-particle quasi-momentum. Under some condition in the case $k\in \T^{\mathrm{d}}\setminus (-\pi,\pi)^{\mathrm{d}},$ we establish necessary and sufficient conditions for existence of infinite discrete spectrum of the operator $h^{\mathrm{d}}(k)$.