Threshold Analysis of the Three Dimensional Lattice Schrödinger Operator with Non-Local Potential

Abstract

We consider a family of the discrete Schrodinger operators $$H_{\lambda\mu}$$ , depending on parameters, in the $$3$$ -dimensional lattice, $$\mathbb{Z}^{3}$$ with a non-local potential constructed via the Dirac delta function and the shift operator. The existence of eigenvalues outside the essential spectrum and their dependence on the parameters of the operator are explicitly derived. The threshold eigenvalue is proven to be absorbed into the essential spectrum and it turns into an embedded eigenvalue at the left intercept of a particular parabola, and the threshold resonance at the other points of the parabola.