Propagation for Schrödinger Operators with Potentials Singular Along a Hypersurface

Abstract

Abstract In this article, we study the propagation of defect measures for Schrödinger operators $$-h^2\Delta _g+V$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> 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.

Locations

• Archive for Rational Mechanics and Analysis - View - PDF
• PubMed - View