Semiclassical propagation through cone points

TL;DR

This paper develops a framework for understanding how high-frequency waves diffract at cone points, showing that regularity mostly propagates with minimal loss even in non-symmetric cases, with applications to quantum and wave equations.

Contribution

It introduces a semiclassical cone calculus that unifies cone and b-regularity, providing new propagation estimates and applications to inverse square potentials and Dirac-Coulomb equations.

Findings

Semiclassical regularity propagates through cone points with minimal loss.

Improved regularity along strictly diffractive geodesics.

Sharp propagation estimates for the semiclassical conic Laplacian.

Abstract

We introduce a general framework for the study of the diffraction of waves by cone points at high frequencies. We prove that semiclassical regularity propagates through cone points with an almost sharp loss even when the underlying operator has leading order terms at the conic singularity which fail to be symmetric. We moreover show improved regularity along strictly diffractive geodesics. Applications include high energy resolvent estimates for complex- or matrix-valued inverse square potentials and for the Dirac-Coulomb equation. We also prove a sharp propagation estimate for the semiclassical conic Laplacian. The proofs use the semiclassical cone calculus, introduced recently by the author, and combine radial point estimates with estimates for a scattering problem on an exact cone. A second microlocal refinement of the calculus captures semiclassical conormal regularity at the cone point and thus facilitates a unified treatment of semiclassical cone and b-regularity.