Abstract:
In this paper, a family of block operator matrices ${\mathcal A}_{\mathrm h}(K),$$K \in (-\pi/{\mathrm h}; \pi/{\mathrm h}]^3$, associated with the Hamiltonian of a system with a non-conserved number of particles not exceeding three on a non-integer lattice $({\mathrm h} {\mathbb Z})^3$ with step ${\mathrm h}>0$, is considered. It is established that the operator ${\mathcal A}_{\mathrm h}({\mathbf 0}),$${\mathbf 0}:=(0,0,0),$ has a finite number of negative eigenvalues if the corresponding generalized Friedrichs model has a zero eigenvalue. It is shown that the operator ${\mathcal A}_{\mathrm h}({\mathbf 0})$ possesses an infinite number of negative eigenvalues accumulating at zero (the Efimov effect) if the generalized Friedrichs model has a zero-energy resonance. An asymptotic formula is obtained for the number $N_{\mathrm h}(z)$ of eigenvalues of the operator ${\mathcal A}_{\mathrm h}({\mathbf 0})$ lying below $z,$$z \leq 0$ as the spectral parameter $z\to -0.$
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