Gabor analysis for Schrodinger equations and propagation of singularities

TL;DR

This paper develops a Gabor analysis-based global expression for the Schrödinger propagator, establishing boundedness in modulation spaces and analyzing the propagation of micro-singularities using time-frequency methods.

Contribution

It introduces a novel Gabor analysis approach to express the Schrödinger propagator globally and studies micro-singularity propagation with time-frequency techniques.

Findings

Provides a global Gabor-based expression for the propagator

Establishes boundedness of the propagator in modulation spaces

Demonstrates propagation of micro-singularities using time-frequency analysis

Abstract

We consider the Schr\"odinger equation \begin{equation*} i \displaystyle\frac{\partial u}{\partial t} +Hu=0,\quad H=a(x,D), \end{equation*} where the Hamiltonian $a(z)$, $z=(x,\xi)$, is assumed real-valued and smooth, with bounded derivatives $|\partial^\alpha a(z)|\leq C_\alpha$, for every $|\alpha|\geq 2$, $z\in\mathbb{R}^{2d}$. For such equation results are known concerning well-posedness of the Cauchy problem for initial data in $L^2(\mathbb{R}^d)$ and local representation of the propagator $e^{it H}$ by means of Fourier integral operators. In the present paper we give a global expression for $e^{itH}$ in terms of Gabor analysis and we deduce boundedness in modulation spaces. Moreover, by using time-frequency techniques, we obtain a result of propagation of micro-singularities for $e^{itH}$.