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

Abstract

We consider the family H(k) of two-particle discrete Schrodinger 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 ℤd, d ≥ 3, interacting via a short-range attractive pair potential. We prove that the eigenvalues of the Schrodinger 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 Schrodinger operator H+(k), k ∈ \(\mathbb{T}^d \), corresponding to a two-particle system with repulsive interaction.