Embedded Eigenvalues of a Hamiltonian in Bosonic Fock Space

Abstract

We consider a model operator ${\mathbf H}_{\mu\lambda},$ $\mu,\lambda \geq 0$ associated with the energy operator of a lattice system describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. The existence of infinitely many negative eigenvalues of ${\mathbf H}_{0\lambda}$ is proved for the case where the associated Friedrichs model have a zero energy resonance and an asymptotics of the form ${\mathcal U}_0 |\log|z||$ for the number of eigenvalues of ${\mathbf H}_{0\lambda}$ lying below $z<0,$ is obtained. We find the conditions for the infiniteness of the number of eigenvalues located inside (in the gap, in the below of the bottom) of the essential spectrum of ${\mathbf H}_{\mu\lambda}$.