On the Discrete Spectrum of a Model Operator in Fermionic Fock Space

We consider a model operator H associated with a system describing three particles in interaction, without conservation of the number of particles. The operator H acts in the direct sum of zero-, one-, and two-particle subspaces of the 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.., which contains a parameter.. to distinguish the two identical particles from the third one. (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 of H below z < E-min = inf sigma(ess) (H) : lim(z -> Emin) (N(z)/vertical bar log vertical bar E-min - z parallel to) = U-0(gamma) (U-0(gamma) > 0), for all gamma > gamma*.