Propagation of anisotropic Gabor wave front sets

TL;DR

This paper investigates how the anisotropic Gabor wave front set propagates under linear operators with specific kernels, extending classical results to more general evolution equations including those with constant coefficient differential operators.

Contribution

It introduces a propagation result for the anisotropic Gabor wave front set under broad classes of linear operators, generalizing previous Schrödinger equation analyses.

Findings

Propagation of anisotropic Gabor wave front set established

Applicable to evolution equations with constant coefficient operators

Generalizes classical Schrödinger equation results

Abstract

We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schr\"odinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.