Propagation for Schrödinger Operators with Potentials Singular Along a Hypersurface

Jeffrey Galkowski; Jared Wunsch

doi:10.1007/s00205-024-01965-1

Propagation for Schrödinger Operators with Potentials Singular Along a Hypersurface

Jeffrey Galkowski^*, Jared Wunsch

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In this article, we study the propagation of defect measures for Schrödinger operators -h²Δ_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.

Original language	English (US)
Article number	37
Journal	Archive for Rational Mechanics and Analysis
Volume	248
Issue number	3
DOIs	https://doi.org/10.1007/s00205-024-01965-1
State	Published - Jun 2024

Funding

The authors are grateful to two anonymous referees for helpful comments on the manuscript. JG acknowledges support from EPSRC grants EP/V001760/1 and EP/V051636/1. JW was partially supported by Simons Foundation grant 631302, NSF grant DMS\u20132054424, and a Simons Fellowship.

ASJC Scopus subject areas

Analysis
Mathematics (miscellaneous)
Mechanical Engineering

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abstract = "In this article, we study the propagation of defect measures for Schr{\"o}dinger operators -h2Δg+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.",

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Propagation for Schrödinger Operators with Potentials Singular Along a Hypersurface

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