Propagation for Schrödinger Operators with Potentials Singular Along a Hypersurface

Abstract

In this article, we study the propagation of defect measures for Schrödinger 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 tangential 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 tangential to Y at exactly first order, as long as the potential has an absolutely continuous first derivative, standard propagation continues to hold.