Propagation of the Gabor Wave Front Set for Schr\"odinger Equations with non-smooth potentials

TL;DR

This paper studies how singularities in solutions to Schrödinger equations with non-smooth potentials propagate, using Gabor wave front sets within modulation spaces to establish well-posedness and microlocal analysis.

Contribution

It introduces a framework for analyzing Schrödinger equations with non-smooth potentials using Gabor wave front sets, extending microlocal propagation results to less regular symbols.

Findings

Well-posedness established in modulation spaces

Propagation of singularities characterized via Gabor wave front sets

Applicable to Schrödinger equations with non-smooth potentials

Abstract

We consider Scr\"odinger equations with real-valued smooth Hamiltonians, and non-smooth bounded pseudo-differential potentials, whose symbols may be not even differentiable. The well-posedness of the Cauchy problem is proved in the frame of the modulation spaces, and results of micro-local propagation of singularities are given in terms of Gabor wave front sets.