Nonexistence of usual wave operators for fractional Laplacian and slowly decaying potentials

TL;DR

This paper proves that for quantum systems with fractional Laplacians and slowly decaying potentials, the usual wave operators do not exist, highlighting the boundary between short-range and long-range interactions.

Contribution

It establishes the nonexistence of wave operators for fractional Laplacian systems with slowly decaying potentials under certain assumptions, clarifying the borderline between interaction types.

Findings

Wave operators do not exist for these systems.

The result delineates the boundary between short-range and long-range potentials.

Assumes the existence and asymptotic completeness of Dollard-type modified wave operators.

Abstract

We consider quantum systems described by the fractional powers of the negative Laplacian and the interaction potentials. When a slowly decaying potential function is given, we prove the nonexistence of the wave operators, under the assumption that the Dollard-type modified wave operators exist and that they are asymptotically complete. This nonexistence indicates the borderline between short-range and long-range behavior.