Abstract
In this paper, a family of block operator matrices \({{\mathcal{A}}_{{\text{h}}}}(K),\) \(K \in {{( - \pi {\text{/h}};\pi {\text{/h}}]}^{3}}\) , associated with the Hamiltonian of a system with a non-conserved number of particles not exceeding three on a non-integer lattice \({{({\text{h}}\mathbb{Z})}^{3}}\) with step \({\text{h}} > 0\) , is considered. It is established that the operator \({{\mathcal{A}}_{{\text{h}}}}({\mathbf{0}}),\) \({\mathbf{0}}: = (0,0,0),\) has a finite number of negative eigenvalues if the corresponding generalized Friedrichs model has a zero eigenvalue. It is shown that the operator \({{\mathcal{A}}_{{\text{h}}}}({\mathbf{0}})\) 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 \({{N}_{{\text{h}}}}(z)\) of eigenvalues of the operator \({{\mathcal{A}}_{{\text{h}}}}({\mathbf{0}})\) lying below \(z,\) \(z \leqslant 0\) as the spectral parameter \(z \to - 0.\)
