On the number of eigenvalues of a model operator in fermionic Fock space

Abstract

We consider a model describing a truncated operator H (truncated with respect to the number of particles) acting in the direct sum of zero-, one-, and two-particle subspaces of a fermionic Fock space ℱa(L2(\U0001d54b3)) over L2(\U0001d54b3). We admit a general form for the "kinetic" part of the hamiltonian H, which contains a parameter γ to distinguish the two identical particles from the third one.In this note:(i) We find a critical value γ* for the parameter γ that allows or forbids the Efimov effect (infinite number of bound states if the associated generalized Friedrichs model has a threshold resonance) and we prove that only for γ < γ* the Efimov effect is absent, while this effect exists for any γ > γ*.(ii) In the case γ > γ* we also establish the following asymptotics for the number N(z) of eigenvalues z below Emin, the lower limit of the essential spectrum of H: