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

Abstract

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 ℱa(L2(𝕋3)) over L2(𝕋3). 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. (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 γγ*. (ii) In the case γ>γ* , we also establish the following asymptotics for the number N(z) of eigenvalues of H below z0, for all γ>γ*.