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

We consider a family of discrete Schrö dinger 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 Ek-1(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 Ek-1(𝔢m(k)) is of dimension d−1 or d−2, we obtain the necessary and sufficient conditions for the existence of infinite discrete spectrum of hd(k) , while in the case dimEk-1(𝔢m(k)) ≤ d-3, the discrete spectrum of hd(k) is finite.