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

TL;DR

This paper investigates how phase space singularities evolve in Schr"odinger equations with quadratic Hamiltonians, revealing that singularities propagate along specific flows within the singular space, refining understanding of wave front sets.

Contribution

It establishes the role of the singular space in the propagation of phase space singularities for quadratic Hamiltonian Schr"odinger equations, extending previous wave front set analyses.

Findings

Singularities are confined within the singular space.

Propagation occurs along the Hamilton flow of the imaginary part.

Refines the understanding of wave front set behavior in quadratic cases.

Abstract

We study propagation of phase space singularities for a Schr\"odinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with non-negative real part. Phase space singularities are measured by the lack of polynomial decay of given order in open cones in the phase space, which gives a parametrized refinement of the Gabor wave front set. The main result confirms the fundamental role of the singular space associated to the quadratic form for the propagation of phase space singularities. The singularities are contained in the singular space, and propagate in the intersection of the singular space and the initial datum singularities along the flow of the Hamilton vector field associated to the imaginary part of the quadratic form.