Absence of Embedded Eigenvalues for Non-Local Schr\"odinger Operators

TL;DR

This paper develops advanced mathematical techniques to prove the absence of embedded eigenvalues in non-local Schr"odinger operators with decaying potentials, extending classical methods to non-local contexts.

Contribution

It introduces new applications of virial theorems, Mourre estimates, and an extended Birman-Schwinger principle to non-local Schr"odinger operators, previously used only for classical cases.

Findings

Derived conditions ruling out embedded eigenvalues.

Extended classical techniques to non-local operators.

Analyzed existence of edge eigenvalues in specific models.

Abstract

We consider non-local Schr\"odinger operators with kinetic terms given by several different types of functions of the Laplacian and potentials decaying to zero at infinity, and derive conditions ruling embedded eigenvalues out. Our goal in this paper is to advance techniques based on virial theorems, Mourre estimates, and an extended version of the Birman-Schwinger principle, previously developed for classical Schr\"odinger operators but thus far not used for non-local operators. We also present a number of specific cases by choosing particular classes of kinetic and potential terms, and discuss existence/non-existence of at-edge eigenvalues in a basic model case in function of the coupling parameter.