Discrete Eigenvalues of a $2 \times 2$ Operator Matrix

TL;DR

This paper analyzes the discrete eigenvalues of a $2 imes 2$ operator matrix in a bosonic Fock space, identifying critical coupling constants where the number of eigenvalues changes from finite to infinite.

Contribution

It characterizes the spectrum of a $2 imes 2$ operator matrix, identifying critical coupling values that determine the finiteness or infiniteness of eigenvalues.

Findings

Location and bounds of the essential spectrum are described.

Existence of critical coupling constants with infinitely many eigenvalues.

Finiteness of eigenvalues outside critical coupling values.

Abstract

We consider a $2\times2$ block operator matrix ${\mathcal A}_\mu$ $($$\mu>0$ is a coupling constant$)$ acting in the direct sum of one- and two-particle subspaces of a bosonic Fock space. The location of the essential spectrum of ${\mathcal A}_\mu$ is described and its bounds are estimated. It is shown that there exist the critical values $\mu_l^0(\gamma)$ with $\gamma>0$ and $\mu_r^0(\gamma)$ with $\gamma<12$ of the coupling constant $\mu>0$ such that for all $\gamma>0$ $(\gamma<12)$ the operator ${\mathcal A}_\mu$ with $\mu=\mu_l^0(\gamma)$ $(\mu=\mu_r^0(\gamma)$ has infinitely many eigenvalues on the l.h.s. $($r.h.s.$)$ of the its essential spectrum. We prove that for all $\mu \not\in \{\mu_l^0(\gamma),\mu_r^0(\gamma)\}$ the operator ${\mathcal A}_\mu$ has finitely many discrete eigenvalues on the l.h.s. and r.h.s. of its essential spectrum.