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 A_h(K), K ∈ ( - π /h;π /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 A_h(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 A_h(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_h(z) of eigenvalues of the operator A_h(0) lying below $$z,$$ z 0 as the spectral parameter z → - 0.