On the eigenvalues of a 2×2 block operator matrix

Abstract

A $2×2$ block operator matrix ${\mathbf H}$ 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 $H_{22}$ (the second diagonal entry of ${\mathbf H}$) 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 $H_{22}$ lying below z<0, the following asymptotics is found $\lim\limits_{z\to -0} N(z) |\log|z||^{-1}=\,{\mathcal U}_0 \quad (0\lt {\mathcal U}_0\lt \infty).$ 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 ${\mathbf H}$ is proved.