Propagation of Gabor singularities for Schr\"odinger equations with quadratic Hamiltonians

TL;DR

This paper investigates how Gabor wave front sets evolve under Schr"odinger equations with quadratic Hamiltonians, revealing the role of the singular space and providing conditions for regularization of solutions.

Contribution

It establishes the propagation of Gabor singularities within the singular space and links this to the flow of the Hamilton vector field, offering new insights into the regularity of solutions.

Findings

Gabor singularities are contained in the singular space for positive times

Propagation follows the flow of the Hamilton vector field of the imaginary part

Conditions for initial data imply regularization to Schwartz space

Abstract

We study propagation of the Gabor wave front set for a Schr\"odinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with non-negative real part. We point out that the singular space associated to the quadratic form plays a crucial role for the understanding of this propagation. We show that the Gabor singularities of the solution to the equation for positive times are always contained in the singular space, and that they propagate in this set along the flow of the Hamilton vector field associated to the imaginary part of the quadratic form. As an application we obtain for the heat equation a sufficient condition on the Gabor wave front set of the initial datum tempered distribution that implies regularization to Schwartz regularity for positive times.