The Threshold effects for the two-particle Hamiltonians on lattices

TL;DR

This paper investigates the spectral properties of two-particle Hamiltonians on three-dimensional lattices, establishing conditions under which nontrivial quasi-momentum values guarantee the existence of discrete spectrum below the essential spectrum.

Contribution

It proves that for a broad class of two-particle lattice Hamiltonians, certain spectral conditions at zero quasi-momentum imply non-empty discrete spectra for all non-zero quasi-momenta.

Findings

Discrete spectrum exists below the threshold for all non-zero quasi-momenta.

Conditions involve eigenvalues or virtual levels at zero quasi-momentum.

Positivity-preserving semigroups are crucial for the results.

Abstract

For a wide class of two-body energy operators $h(k)$ on the three-dimensional lattice $\bbZ^3$, $k$ being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for all nontrivial values $k$, $k\ne 0$, the discrete spectrum of $h(k)$ below its threshold is non-empty. The assumptions are: (i) the two-particle Hamiltonian $h(0)$ corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and (ii) the one-particle free Hamiltonians in the coordinate representation generate positivity preserving semi-groups.