The Spectrum of Non-Local Discrete Schroedinger Operators with a delta-Potential

TL;DR

This paper investigates the spectral edge behavior of non-local discrete Schrödinger operators with delta-potentials, revealing asymmetries and classifying spectral phenomena depending on the operator's non-local characteristics and lattice dimension.

Contribution

It introduces a framework for analyzing spectral edges of non-local discrete Schrödinger operators, highlighting asymmetries and dependence on the defining function and lattice dimension.

Findings

Spectral edge behaviors differ at the two ends of the spectrum for non-local operators.

An asymmetry in spectral behavior is observed compared to local Schrödinger operators.

A classification scheme for spectral edge phenomena is proposed.

Abstract

The behaviour of the spectral edges (embedded eigenvalues and resonances) is discussed at the two ends of the continuous spectrum of non-local discrete Schr\"odinger operators with a $\delta$-potential. These operators arise by replacing the discrete Laplacian by a strictly increasing $C^1$-function of the discrete Laplacian. The dependence of the results on this function and the lattice dimension are explicitly derived. It is found that while in the case of the discrete Schr\"odinger operator these behaviours are the same no matter which end of the continuous spectrum is considered, an asymmetry occurs for the non-local cases. A classification with respect to the spectral edge behaviour is also offered.