Asymptotics of the eigenvalues of a discrete Schrödinger operator with zero-range potential

Abstract

We consider a family of discrete Schrödinger operators , . These operators are associated with the Hamiltonian of a system of two identical quantum particles (bosons) moving on the -dimensional lattice , , and interacting by means of a pairwise zero-range (contact) attractive potential . It is proved that for any there is a number which is a threshold value of the coupling constant; for the operator , , has a unique eigenvalue placed to the left of the essential spectrum. The asymptotic behaviour of is found as and as and also as for every value of the quasi-momentum belonging to the manifold , where .