The Infiniteness of the Number of Eigenvalues of the Schrodinger Operator of a System of Two Particles on a Lattice

We consider the Hamiltonian associated with a system of two particles (bosons) on a two-dimensional lattice Z(2) with a potential of a certain type. The Schrodinger operator H(k) of the system for k = pi = (pi, pi) (where k = (k(1), k(2))) is the total quasimomentum) has an infinite number of eigenvalues. It is shown that z(0)(p) = 4 - (v) over bar (0) is simple, z(1)(p) = 4 - (v) over bar (1) is a double, z(2)(pi) = 4 - (v) over bar (2) is a fourfold eigenvalue, while the remaining eigenvalues z(n)(p) = 4- (v) over bar (n), n >= 3, are fivefold. We prove that all multiple eigenvalues of the H(p) are split into non-degenerate eigenvalues. We obtain asymptotic formulas with the accuracy of beta(2) for eigenvalues of the Schrodinger operator H((pi - 2 beta, pi)).