Propagation of anisotropic Gelfand-Shilov wave front sets

TL;DR

This paper investigates how the anisotropic Gelfand-Shilov wave front set propagates under linear operators with specific ultradistribution kernels, extending results to generalized evolution equations beyond the Schrödinger case.

Contribution

It establishes propagation results for anisotropic Gelfand-Shilov wave front sets under operators with ultradistribution kernels, generalizing classical evolution equations.

Findings

Propagation of wave front set under specified operators

Application to generalized evolution equations

Extension beyond classical Schrödinger equation

Abstract

We show a result on propagation of the anisotropic Gelfand--Shilov wave front set for linear operators with Schwartz kernel which is a Gelfand--Shilov ultradistribution of Beurling type. This anisotropic wave front set is parametrized by two positive parameters relating the space and frequency variables. The anisotropic Gelfand--Shilov wave front set of the Schwartz kernel of the operator 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 a partial differential operator defined by a symbol which is a polynomial with real coefficients and order at least two.