Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice

Abstract

We consider the family H(k) of two-particle discrete Schrödinger operators depending on the quasimomentum of a two-particle system k ∈ $\mathbb{T}^d$, where $\mathbb{T}^d$ is a d-dimensional torus. This family of operators is associated with the Hamiltonian of a system of two arbitrary particles on the d-dimensional lattice ℤ^d, d ≥ 3, interacting via a short-range attractive pair potential. We prove that the eigenvalues of the Schrödinger operator H(k) below the essential spectrum are positive for all nonzero values of the quasimomentum k ∈ $\mathbb{T}^d$ if the operator H(0) is nonnegative. We establish a similar result for the eigenvalues of the Schrödinger operator H₊(k), k ∈ $\mathbb{T}^d$, corresponding to a two-particle system with repulsive interaction.