Propagation of anisotropic Gabor singularities for Schr\"odinger type equations

TL;DR

This paper investigates how anisotropic Gabor wave front sets propagate for solutions to Schr"odinger-type equations with anisotropic homogeneous Hamiltonians, establishing continuity on anisotropic Sobolev spaces and flow-based propagation results.

Contribution

It introduces a framework for analyzing anisotropic Gabor wave front set propagation in Schr"odinger equations with anisotropic homogeneous symbols, extending prior isotropic results.

Findings

Propagation follows the Hamilton flow of the principal symbol.

The propagator is continuous on anisotropic Shubin--Sobolev spaces.

Results apply to a class of evolution equations with anisotropic homogeneity.

Abstract

We show results on propagation of anisotropic Gabor wave front sets for solutions to a class of evolution equations of Schr\"odinger type. The Hamiltonian is assumed to have a real-valued principal symbol with the anisotropic homogeneity $a(\lambda x, \lambda^\sigma \xi) = \lambda^{1+\sigma} a(x,\xi)$ for $\lambda > 0$ where $\sigma > 0$ is a rational anisotropy parameter. We prove that the propagator is continuous on anisotropic Shubin--Sobolev spaces. The main result says that the propagation of the anisotropic Gabor wave front set follows the Hamilton flow of the principal symbol.