On the finiteness of the discrete spectrum of a 3x3 operator matrix

TL;DR

This paper investigates the spectral properties of a 3x3 operator matrix modeling a three-particle lattice system, establishing conditions for finitely many discrete eigenvalues below the essential spectrum.

Contribution

It characterizes the essential spectrum structure and derives conditions ensuring a finite number of discrete eigenvalues for the operator.

Findings

Describes the essential spectrum via generalized Friedrichs models

Derives a symmetric Weinberg equation for eigenvectors

Provides conditions for finiteness of discrete eigenvalues

Abstract

An operator matrix $H$ associated with a lattice system describing three particles in interactions, without conservation of the number of particles, is considered. The structure of the essential spectrum of $H$ is described by the spectra of two families of the generalized Friedrichs models. A symmetric version of the Weinberg equation for eigenvectors of $H$ is obtained. The conditions which guarantee the finiteness of the number of discrete eigenvalues located below the bottom of the three-particle branch of the essential spectrum of $H$ is found.