Investigation of the Number of Negative Eigenvalues of a Third-Order Operator Matrix on a Noninteger Lattice

In this paper, a family of block operator matrices {A₇} (K), K { (- {/h; /h]}^3}, associated with the Hamiltonian of a system with a non-conserved number of particles not exceeding three on a non-integer lattice { ({hZ) }^3} with step h > 0, is considered. It is established that the operator {A₇} (0), 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 {A₇} (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 {N₇} (z) of eigenvalues of the operator {A₇} (0) lying below z, z 0 as the spectral parameter z - 0.