On the eigenvalues of a 2 × 2 block operator matrix

Abstract

A 2 × 2 block operator matrix H acting in the direct sum of one- and two-particle subspaces of a Fock space is considered. The existence of infinitely many negative eigenvalues of H22 (the second diagonal entry of H) is proved for the case where both of the associated Friedrichs models have a zero energy resonance. For the number N (z) of eigenvalues of H22 lying below z < 0; the following asymptotics is found (Formula presented.) Under some natural conditions the infiniteness of the number of eigenvalues located respectively inside, in the gap, and below the bottom of the essential spectrum of H is proved.