Propagation of space-time singularities for perturbed harmonic oscillators

TL;DR

This paper investigates how space-time singularities propagate in a quantum harmonic oscillator with time-dependent perturbations, using semiclassical analysis and reformulated wave front sets to compare with unperturbed systems.

Contribution

It introduces a semiclassical reformulation of the wave front set to analyze singularity propagation in perturbed quantum harmonic oscillators, extending previous spatial singularity analysis methods.

Findings

Characterization of space-time singularity propagation in perturbed systems

Extension of wave front set analysis to time-dependent scenarios

Comparison with unperturbed harmonic oscillator behavior

Abstract

We discuss propagation of space-time singularities for the quantum harmonic oscillator with time-dependent metric and potential perturbations. Reformulating the quasi-homogeneous wave front set according to Lascar (1977) in a semiclassical manner, we obtain a characterization of its appearance in comparison with the unperturbed system. The idea of our proof is based on the argument of Nakamura (2009), which was originally devised for the analysis of spatial singularities of the Schr\"odinger equation, however, the application is non-trivial since the time is no more a parameter, but takes a part in the base variables.