Time-frequency analysis of the Dirac equation

TL;DR

This paper applies time-frequency analysis to study the Dirac equation, providing estimates in modulation spaces, addressing rough potentials, and establishing local well-posedness for nonlinear variants, including the Thirring model.

Contribution

It introduces a novel framework for analyzing the Dirac equation using time-frequency methods, extending results to rough potentials and nonlinear cases.

Findings

Established estimates in weighted modulation and Wiener amalgam spaces.

Proved local well-posedness for nonlinear Dirac equations with general nonlinearities.

Generalized previous results to include rough potentials and non-smooth functions.

Abstract

The purpose of this paper is to investigate several issues concerning the Dirac equation from a time-frequency analysis perspective. More precisely, we provide estimates in weighted modulation and Wiener amalgam spaces for the solutions of the Dirac equation with rough potentials. We focus in particular on bounded perturbations, arising as the Weyl quantization of suitable time-dependent symbols, as well as on quadratic and sub-quadratic non-smooth functions, hence generalizing the results in a recent paper by Kato and Naumkin. We then prove local well-posedness on the same function spaces for the nonlinear Dirac equation with a general nonlinearity, including power-type terms and the Thirring model. For this study we adopt the unifying framework of vector-valued time-frequency analysis as developed by Wahlberg; most of the preliminary results are stated under general assumptions and hence they may be of independent interest.