Threshold between short and long-range potentials for non-local Schr\"odinger operators

TL;DR

This paper develops scattering theory for a class of non-local Schrödinger operators, identifying the threshold between short and long-range potentials by analyzing wave operators for fractional Laplacians.

Contribution

It extends scattering theory to non-local Schrödinger operators with functions of the Laplacian beyond Bernstein functions, clarifying the decay conditions for potentials.

Findings

Established existence and non-existence of wave operators.

Identified the decay threshold distinguishing short and long-range potentials.

Extended the class of functions of the Laplacian considered in scattering theory.

Abstract

We develop scattering theory for non-local Schr\"odinger operators defined by functions of the Laplacian that include its fractional power $(-\Delta)^\rho$ with $0<\rho\leqslant1$. In particular, our function belongs to a wider class than the set of Bernstein functions. By showing the existence and non-existence of the wave operators, we clarify the threshold between the short and long-range decay conditions for perturbational potentials.