On the number of the discrete spectrum of two-particle discrete Schröodinger operators

Abstract

We consider a family of discrete Schrödinger operators hd(k), where k is the two-particle quasi-momentum varying in Td = (−π, π]d, associated to a system of two-particles on the d dimensional lattice Zd, d ≥ 1. The CwikelLieb-Rozenblum (CLR)-type estimates are written for hd(k) when the Fermi surface E−1 k (em(k)) of the associated dispersion relation is a one point set at em(k), the bottom of the essential spectrum. Moreover, when the Fermi surface E−1 k (em(k)) is of dimension d−1 or d−2, we obtain the necessary and su cient conditions for the existence of in nite discrete spectrum of hd(k), while in the case dimE−1 k (em(k)) ≤ d− 3, the discrete spectrum of h d(k) is nite.