Threshold analysis for Schrödinger operator on the one-and-two dimensional lattices

Abstract

We consider the discrete Schrödinger operator  on the lattice  with a non-local potential constructed via the composition of Dirac delta function and the shift operator. The existence of lower eigenvalue behaviors on the parameters  of the operator is explicitly derived. We investigate the threshold resonance and embedded eigenvalue problem on the manifold in , being a downward parabola, on which the lowest eigenvalue of the operator gets absorbed into the essential spectrum.  We show that if the lowest eigenvalue is absorbed into the essential spectrum, it can turns to a threshold resonance ( resp. super-threshold resonance) at the left -intercept of the parabola, while to the regular at the other points of the parabola when  (resp ).