Propagation of exponential phase space singularities for Schr\"odinger equations with quadratic Hamiltonians

TL;DR

This paper investigates how exponential phase space singularities propagate in Schrödinger equations with quadratic Hamiltonians, extending the understanding of singularity behavior in the context of Gelfand--Shilov spaces.

Contribution

It introduces the propagation of Gelfand--Shilov wave front sets for Schrödinger equations with quadratic Hamiltonians, linking singularity propagation to the singular space of the quadratic form.

Findings

Propagation determined by the singular space of the quadratic form

Extension of singularity propagation results to Gelfand--Shilov spaces

Characterization of singularities via the Gelfand--Shilov wave front set

Abstract

We study propagation of phase space singularities for the initial value Cauchy problem for a class of Schr\"odinger equations. The Hamiltonian is the Weyl quantization of a quadratic form whose real part is non-negative. The equations are studied in the framework of projective Gelfand--Shilov spaces and their distribution duals. The corresponding notion of singularities is called the Gelfand--Shilov wave front set and means the lack of exponential decay in open cones in phase space. Our main result shows that the propagation is determined by the singular space of the quadratic form, just as in the framework of the Schwartz space, where the notion of singularity is the Gabor wave front set.