About the negative eigenvalues of the discrete Schr¨odinger operator with non-local potential in d-dimensional case

Abstract

Eigenvalue behaviour of a family of discrete Schr¨odinger operators Hλµ depending on parameters λ; µ 2 R is studied on the d-dimensional lattice Zd; (d ≥ 3). The non-local potential is described by the Kronecker delta function and the shift operator. The existence of eigenvalues below the essential spectrum and their dependence on the parameters are explicitly proven. We also show that the essential spectrum absorbs the threshold eigenvalue and there exists a particular parabola, on whose left intercept the threshold becomes an embedded eigenvalue and the threshold resonance at its other points.