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

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 F-a (L-2(T-3)) over L-2(T-3). We admit a general form for the "kinetic" part of the hamiltonian H, which contains a parameter gamma to distinguish the two identical particles from the third one. In this note: (i) We find a critical value gamma* for the parameter gamma 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 gamma < gamma* the Efimov effect is absent, while this effect exists for any gamma > gamma*. (ii) In the case gamma > gamma* we also establish the following asymptotics for the number N(z) of eigenvalues z below E-min, the lower limit of the essential spectrum of H : lim(z -> Emin-) N(z)/vertical bar log vertical bar Emin - z vertical bar vertical bar = u(0)(gamma) (u(0)(gamma) > 0), for all(gamma) > gamma*.