On the essential and discrete spectrum of a model operator related to three-particle discrete Schrödinger operators

Abstract

A model operator $H$ corresponding to a three-particle discrete Schrödinger operator on a lattice ${\mathbb{Z}}^3$ is studied. The essential spectrum is described via the spectrum of two Friedrichs models with parameters $h_{\alpha }(p),$ $\alpha =1,2,$ $p \in \T^3=(-π,π]^3.$ The following results are proven: 1) The operator $H$ has a finite number of eigenvalues lying below the bottom of the essential spectrum in any of the following cases: (i) both operators $h_{\alpha }(0),\alpha =1,2,$ have a zero eigenvalue; (ii) either $h_1(0)$ or $h_2(0)$ has a zero eigenvalue. 2) The operator $H$ has infinitely many eigenvalues lying below the bottom and accumulating at the bottom of the essential spectrum, if both operators $h_{\alpha }(0),\alpha =1,2,$ have a zero energy resonance.