Nonexistence of wave operators via strong propagation estimates for Schr\"{o}dinger operators with sub-quadratic repulsive potentials

TL;DR

This paper establishes the nonexistence of wave operators for Schrödinger operators with sub-quadratic repulsive potentials when the external potential's decay rate exceeds a certain threshold, using new propagation estimates.

Contribution

It introduces new propagation estimates for the time propagator and proves the nonexistence of wave operators beyond a specific decay threshold for sub-quadratic potentials.

Findings

Wave operators do not exist if the external potential decay rate is greater than or equal to a certain threshold.

New propagation estimates are developed for the time propagator in this context.

The threshold decay rate for the external potential is confirmed to be optimal.

Abstract

Sub-quadratic repulsive potentials accelerate quantum particles and can relax the decay rate in the $x$ of the external potentials $V$ that guarantee the existence of the quantum wave operators. In the case where the sub-quadratic potential is $- |x|^{\alpha} $ with $0< \alpha < 2$ and the external potential satisfies $|V(x) | \leq C (1+|x|) ^{-(1- \alpha /2) - \varepsilon} $ with $\varepsilon>0$, Bony et. al [3] determined the existence and completeness of the wave operators, and Itakura [12, 13, 14] then obtained their results using stationary scattering theory for more generalized external potentials. Based on their results, we naturally expect the following. If the decay power of the external potential $V$ is less than ${ -(1- \alpha /2) } $, V is included in the short-range class. If the decay power is greater than or equal to ${ -(1- \alpha /2) } $, $V$ is included in the long-range class. In this study, we first prove the new propagation estimates for the time propagator that can be applied to scattering theory. Second, we prove that the wave operators do not exist if the power is greater than or equal to $-(1- \alpha /2)$ and that the threshold expectation of ${ -(1- \alpha /2) } $ is true using the new propagation estimates.