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 ${ℱ}_a(L^{\mathrm{2}}({𝕋}^{\mathrm{3}}))$ over $L^{\mathrm{2}}({𝕋}^{\mathrm{3}})$. 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. (i) We find a critical value ${\gamma }^{\mathrm{*}}$ 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 the Efimov effect is absent, while this effect exists for any $\gamma >{\gamma }^{\mathrm{*}}$. (ii) In the case $\gamma >{\gamma }^{\mathrm{*}}$ , we also establish the following asymptotics for the number $N(z)$ of eigenvalues of $H$ below ${𝒰}_0\left(\gamma \right)\left({𝒰}_0\left(\gamma \right)>0\right)$, for all $\gamma >{\gamma }^{*}$.