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Propagation for Schrödinger operators with potentials singular along a hypersurface

Jeffrey Galkowski, Jared Wunsch

Abstract

In this article, we study propagation of defect measures for Schrödinger operators, $-h^2Δ_g+V$, 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.

Propagation for Schrödinger operators with potentials singular along a hypersurface

Abstract

In this article, we study propagation of defect measures for Schrödinger operators, , on a Riemannian manifold of dimension with having conormal singularities along a hypersurface in the sense that derivatives along vector fields tangent to preserve the regularity of . We show that the standard propagation theorem holds for bicharacteristics travelling transversally to the surface whenever the potential is absolutely continuous. Furthermore, even when bicharacteristics are tangent to at exactly first order, as long as the potential has an absolutely continuous first derivative, standard propagation continues to hold.
Paper Structure (14 sections, 21 theorems, 156 equations)

This paper contains 14 sections, 21 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Conormal distributions
  3. On the bicharacteristic flow
  4. The bicharacteristic flow in the hyperbolic set
  5. The bicharacteristic flow near glancing
  6. Semiclassical preliminaries
  7. Defect measures
  8. Tangential operators
  9. Technical estimates
  10. Elliptic Estimates
  11. $L^\infty L^2$ estimates
  12. Estimates for the Schrödinger equation
  13. Defect measures for the Schrödinger equation
  14. Propagation for $\mathcal{C}^1$ potentials

Key Result

Theorem 1.1

If $V \in IW^{1,1}$, then through every point in $\mathcal{H}$ there exists a unique maximal integral curve of $H_p$ in $\mathcal{H}$. If $V \in IW^{2,1}$ then through every point in $\mathcal{H} \cup \mathcal{G}_2$ there exists a unique maximal integral curve of $H_p$ in $\mathcal{H} \cup \mathcal{

Theorems & Definitions (40)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • ...and 30 more