On the essential and discrete spectrum of a model operator related to three-particle discrete Schr\"odinger operators

TL;DR

This paper analyzes a model operator related to three-particle discrete Schrödinger operators on a lattice, describing its essential spectrum and eigenvalue distribution, especially under conditions involving zero eigenvalues and resonances.

Contribution

It characterizes the essential spectrum and eigenvalue count of the model operator, revealing finite or infinite eigenvalues below the spectrum depending on zero eigenvalues or resonances.

Findings

Finite eigenvalues below spectrum when zero eigenvalues are present

Infinite eigenvalues accumulate at spectrum bottom with zero energy resonance

Essential spectrum described via two Friedrichs models

Abstract

A model operator $H$ corresponding to a three-particle discrete Schr\"odinger operator on a lattice $\Z^3$ is studied. The essential spectrum is described via the spectrum of two Friedrichs models with parameters $h_\alpha(p),$ $\alpha=1,2,$ $p \in \T^3=(-\pi,\pi]^3.$ 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.