Propagation for Schr\"odinger operators with potentials singular along a hypersurface

TL;DR

This paper investigates how defect measures propagate for Schr"odinger operators with potentials that have singularities along a hypersurface, extending known propagation results to cases with conormal singularities.

Contribution

It establishes propagation of defect measures for Schr"odinger operators with conormal singularities along a hypersurface, including cases with tangent bicharacteristics and less regular potentials.

Findings

Propagation theorem holds for bicharacteristics crossing the hypersurface.

Propagation continues for tangent bicharacteristics if the potential's first derivative is absolutely continuous.

Standard propagation results extend to potentials with conormal singularities.

Abstract

In this article, we study propagation of defect measures for Schr\"odinger operators, $-h^2\Delta_g+V$, on a Riemannian manifold $(M,g)$ of dimension $n$ with $V$ having conormal singularities along a hypersurface $Y$ in the sense that derivatives along vector fields tangent to $Y$ preserve the regularity of $V$. We show that the standard propagation theorem holds for bicharacteristics travelling transversally to the surface $Y$ whenever the potential is absolutely continuous. Furthermore, even when bicharacteristics are tangent to $Y$ at exactly first order, as long as the potential has an absolutely continuous first derivative, standard propagation continues to hold.