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 a (L 2 (3)) over L 2 (3). 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 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 of H below z < E m i n = inf σ e s s H: l im z → E min N z / log E m i n - z = U 0 γ U 0 γ > 0, for all γ > γ .

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