Wave front set for solutions to perturbed harmonic oscillators

TL;DR

This paper characterizes the wave front set of solutions to perturbed harmonic oscillator Schrödinger equations using classical scattering data and propagator analysis, revealing how singularities recur at specific periods.

Contribution

It introduces a novel characterization of wave front sets for solutions to perturbed harmonic oscillators, linking singularity propagation to classical scattering and unperturbed propagator behavior.

Findings

Wave front set described via classical scattering data

Recurrence of singularities at period t=π

Propagation of singularities analyzed for short-range perturbations

Abstract

We consider Schr\"odinger equations with variable coefficients and the harmonic potential. We suppose the perturbation is short-range type in the sense of [Nakamura 2004]. We characterize the wave front set of the solutions to the equation in terms of the classical scattering data and the propagator of the unperturbed harmonic oscillator. In particular, we give a "recurrence of singularities" type theorem for the propagation of the period $t=\pi$.