Multiplicity of virtual levels at the lower edge of the continuous spectrum of a two-particle Hamiltonian on a lattice

Abstract

We consider a system of two arbitrary quantum particles on a three-dimensional lattice with special dispersion functions (describing site-to-site particle transport), where the particles interact by a chosen attraction potential. We study how the number of eigenvalues of the family of the operators h(k) depends on the particle interaction energy and the total quasimomentum $$k \in \mathbb{T}^3$$ (where $$\mathbb{T}^3$$ is a three-dimensional torus). Depending on the particle interaction energy, we obtain conditions under which the left edge of the continuous spectrum is simultaneously a multiple virtual level and an eigenvalue of the operator h(0).