Propagation of anisotropic Gabor singularities for Schrödinger type equations

Abstract

We show results on propagation of anisotropic Gabor wave front sets for solutions to a class of evolution equations of Schrödinger 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.

40 pages