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On the discrete spectrum of a model operator in fermionic Fock space. (English) Zbl 1433.81086

Summary: 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 \(\mathcal{F}_a(L^2(\mathbb{T}^3))\) over \(L^2(\mathbb{T}^3)\). 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. (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 \(z < E_{\min} = \inf \sigma_{\mathrm{ess}} \left(H\right) : \lim_{z \rightarrow E_{\min}} \left(N \left(z\right) / \left|\log \left|E_{\min} - z\right|\right|\right) =\mathcal{U}_0 \left(\gamma\right) \left(\mathcal{U}_0 \left(\gamma\right) > 0\right)\), for all \(\gamma > \gamma^*\).

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47A10 Spectrum, resolvent
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