The Threshold Effects for the Two Particle Discrete Schrodinger Operators on Lattices

We study the Schrodinger operators H-gamma lambda(K), with K is an element of T-d, the fixed quasimomentum of the particles pair, associated with a system of two identical bosons on the d-dimensional lattice Z(d), d >= 3 with on one site and on nearest-neighboring-site interactions of magnitudes gamma is an element of R and lambda is an element of R, respectively. We partition the (gamma,lambda)- plane into connected components such that, in each connected components the number of eigenvalues of the Schrodinger operator H-gamma lambda(0) remains constant. Moreover, we establish that the operator H-lambda gamma(0) has in each boundary of the connected components either a threshold eigenvalue or a threshold resonance. We also find a sharp lower bound for the number of isolated eigenvalues of H-gamma lambda(K) overall K is an element of T-d, on each boundary of the connected components.