Abstract
A system of three quantum particles on the three‐dimensional lattice ℤ^3^ 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 ℤ^3^ 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 h~α~ (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 h~α~ (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)