Diffractive Theorems for the Wave Equation with Inverse Square Potential

TL;DR

This paper proves the existence of diffractive fronts in the wave equation with inverse square potential, showing how singularities propagate in higher dimensions using microlocal energy estimates.

Contribution

It establishes the presence of diffractive fronts in the wave equation with inverse square potential and extends singularity propagation results to higher dimensions.

Findings

Diffractive fronts are present in the fundamental solution with inverse square potential.

Singularity propagation is restricted to these diffractive fronts in higher dimensions.

Microlocal energy estimates are used to prove the results.

Abstract

We first establish the presence of a diffractive front in the fundamental solution of the wave operator with a diract delta intial condition in two dimensional euclidean space caused by the potentials perturbation on the spherical laplacian. This motivates a result which restricts the propagation of singularities for the wave operator with a more general potential to precisely these diffractive fronts higher dimensional euclidean spaces. This is proven using microlocal energy estimates.