Semiclassical diffraction by conormal potential singularities

TL;DR

This paper studies how singularities in solutions to the semiclassical Schrödinger equation propagate when the potential has conormal singularities, revealing reflection and detachment phenomena along bicharacteristics.

Contribution

It establishes propagation of singularities in the presence of conormal potentials, including reflection and detachment behaviors, extending semiclassical analysis.

Findings

Wavefront set propagates along broken bicharacteristics

Reflected wavefronts are weaker by a power of h

Wavefront set can detach from the hypersurface for regular potentials

Abstract

We establish propagation of singularities for the semiclassical Schr\"odinger equation, where the potential is conormal to a hypersurface. We show that semiclassical wavefront set propagates along generalized broken bicharacteristics, hence reflection of singularities may occur along trajectories reaching the hypersurface transversely. The reflected wavefront set is weaker, however, by a power of $h$ that depends on the regularity of the potential. We also show that for sufficiently regular potentials, wavefront set may not stick to the hypersurface, but rather detaches from it at points of tangency to travel along ordinary bicharacteristics.