On the microlocal phase-space concentration of Wigner distributions associated with Schr\"odinger evolutions

TL;DR

This paper extends the microlocal analysis of Schr"odinger evolutions using metaplectic Wigner distributions, providing new insights into phase space concentration and ghost frequencies.

Contribution

It generalizes microlocal properties to all metaplectic Wigner distributions, including Kohn-Nirenberg, and examines their behavior in Fourier integral operators with quadratic phase.

Findings

Extended microlocal results to all metaplectic Wigner distributions.

Analyzed phase space concentration of Wigner distributions in Schr"odinger evolutions.

Provided refined understanding of ghost frequencies in phase space.

Abstract

In this work, we investigate the microlocal properties of the evolutions of Schr\"odinger equations using metaplectic Wigner distributions. So far, only restricted classes of metaplectic Wigner distributions, satisfying particular structural properties, have allowed the analysis of microlocal properties. We first extend the microlocal results to all metaplectic Wigner distributions, including the well-known Kohn-Nirenberg quantization, and examine these findings in the framework of Fourier integral operators with quadratic phase. Finally, we analyze the phase space concentration of the (cross) Wigner distribution arising from the interaction of two states, with particular attention to interactions generated by certain Schr\"odinger evolutions. These contributions enable a more refined study of the so-called ghost frequencies.