ON THE EIGENVALUES OF A 2 x 2 BLOCK OPERATOR MATRIX

A 2 x 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 H-22 (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 H-22 lying below z < 0; the following asymptotics is found lim(z ->-0) N (z) vertical bar log vertical bar z vertical bar vertical bar (1) = U-0 (0 < U-0 < infinity). 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.