Existence and nonexistence of wave operators for time-decaying harmonic oscillators

TL;DR

This paper investigates the conditions under which wave operators exist or not for particles influenced by time-decaying harmonic potentials, revealing a specific threshold related to potential decay and particle mass.

Contribution

It identifies a precise threshold for the existence of wave operators in time-decaying harmonic oscillators, linking it to potential decay rate and particle mass.

Findings

Threshold for wave operator existence is 1/(1-λ) for 0 ≤ λ < 1/2

Decaying harmonic potentials alter the boundary between short-range and long-range potentials

Particle mass and potential coefficient determine wave operator behavior

Abstract

Controlled time-decaying harmonic potentials decelerate the velocity of the charged particle but the particle never be trapped by this harmonic potentials. This physical phenomena changes threshold between the short range class of potential and long-range class of potential in the sense of the existence of physical wave operators. In this paper, we reveal such a threshold is $1/(1-\lambda)$ for some $0\leq \lambda<1/2$ , which is determined by the mass of the particle and a coefficient of harmonic potential.