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

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)