On the structure of the essential spectrum for the three‐particle Schrödinger operators on lattices

Abstract

AbstractA system of three quantum particles on the three‐dimensional lattice ℤ3 with arbitrary dispersion functions having not necessarily 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 ℤ3 in the coordinate and momentum representations are described as bounded self‐adjoint operators on the corresponding Hilbert spaces. For all sufficiently small values of the two‐particle quasi‐momentum k ∈ (–π, π ]3 the finiteness of the number of eigenvalues of the two‐particle discrete Schrödinger operator hα (k) below the continuous spectrum is established. The location of the essential spectrum of the three‐particle discrete Schrödinger operator H (K), K ∈ (–π,π ]3 being the three‐particle quasi‐momentum, is described by means of the spectrum of the two‐particle discrete Schrödinger operator hα (k), k ∈ (–π, π ]3. It is established that the essential spectrum of the three‐particle discrete Schrödinger operator H (K), K ∈ (–π, π ]3, consists of finitely many bounded closed intervals. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)