Quantum scattering with time-decaying harmonic oscillators

TL;DR

This paper studies quantum scattering in systems with time-decaying harmonic potentials, demonstrating the existence and completeness of wave operators under certain decay conditions for short-range potentials.

Contribution

It introduces a framework for quantum scattering with controlled, time-decaying harmonic potentials and proves wave operator properties for specific decay rates of the potential.

Findings

Wave operators exist and are complete for certain short-range potentials.

Deceleration of quantum particles occurs without trapping.

Theoretical conditions for potential decay rates are established.

Abstract

By controlling coefficients and decaying order of time-decaying harmonic potentials, the velocity of a quantum particle is decelerated by the effect of harmonic potentials but the particle is non-trapping. In this paper, we consider the quantum system with controlled harmonic potentials. By defining the wave operators for this system and suitable range of these, we can prove the existence and completeness of wave operators with respect to the short-range potentials $V(t,x)$ satisfying $|V(t,x)| = o ((1+|x|)^{-1/(1- \lambda)})$, for some $\lambda$, $0< \lambda < 1/2$.