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