On the structure of the essential spectrum of the three-particle Schrödinger operators on a lattice

Abstract

A system of three quantum particles on the three-dimensional lattice ${\mathbb{Z}}^3$ 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 ${\mathbb{Z}}^3$ 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 Schrödinger operator $h_{\alpha }(k)$ below the continuous spectrum is established. A location of the essential spectrum of the three-particle discrete Schrödinger 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.