On the structure of the essential spectrum of the three-particle Schr\"{o}dinger operators on a lattice

TL;DR

This paper analyzes the essential spectrum of three-particle Schrödinger operators on a lattice, describing its structure and establishing finiteness of eigenvalues for small nonzero two-particle quasi-momenta.

Contribution

It provides a detailed description of the essential spectrum of three-particle lattice Schrödinger operators and shows the spectrum consists of finitely many bounded intervals.

Findings

Finiteness of eigenvalues below the continuous spectrum for small nonzero two-particle quasi-momenta.

The essential spectrum of the three-particle operator comprises finitely many bounded intervals.

Location of the essential spectrum is characterized via the two-particle spectrum.

Abstract

A system of three quantum particles on the three-dimensional lattice $\Z^3$ with arbitrary "dispersion functions" having non-compact support and interacting via short-range pair potentials is considered. The energy operators of the systems of the two-and three-particles on the lattice $\Z^3$ in the coordinate and momentum representations are described as bounded self-adjoint operators on the corresponding Hilbert spaces. For all sufficiently small nonzero values of the two-particle quasi-momentum $k\in (-\pi,\pi]^3$ the finiteness of the number of eigenvalues of the two-particle discrete Schr\"odinger operator $h_\alpha(k)$ below the continuous spectrum is established. A location of the essential spectrum of the three-particle discrete Schr\"odinger operator $H(K),K\in (-\pi,\pi]^3$ the three-particle quasi-momentum, by means of the spectrum of $h_\alpha(k)$ is described. It is established that the essential spectrum of $H(K), K\in (-\pi,\pi]^3$ consists of a finitely many bounded closed intervals.