Wigner and Gabor phase-space analysis of propagators for evolution equations

TL;DR

This paper analyzes the phase-space behavior of propagators for various evolution equations using Wigner and Gabor analysis, deriving explicit formulas and decay estimates that reveal their geometric and dispersive properties.

Contribution

It provides explicit formulas and decay estimates for Wigner kernels and Gabor matrices of evolution equation propagators, extending phase-space analysis to a broad class of PDEs.

Findings

Gabor matrices exhibit exponential off-diagonal decay or quasi-diagonality.

Explicit formulas show Gaussian decay and spreading for the heat equation.

Gabor matrices for Hermite equations undergo pure rotation, reflecting symplectic structure.

Abstract

We study the Wigner kernel and the Gabor matrix associated with the propagators of a broad class of linear evolution equations, including the complex heat, wave, and Hermite equations. Within the framework of time-frequency analysis, we derive explicit expressions for the Wigner kernels of Fourier multipliers and establish quantitative decay estimates for the corresponding Gabor matrices. These results are obtained under symbol regularity conditions formulated in the Gelfand-Shilov scale and ensure exponential off-diagonal decay or quasi-diagonality of the matrix representation. We believe this approach can be extended to more general symbols in the pseudodifferential setting, improving the existing results in terms of their Gabor matrix decay. For the complex heat equation, we obtain closed-form formulas exhibiting both dissipative and oscillatory behavior governed respectively by the real and imaginary parts of the diffusion parameter. The modulus of the Gabor matrix is shown to display Gaussian decay and temporal spreading consistent with diffusion phenomena. In contrast, the complex Hermite equation is analyzed via H\"ormander's metaplectic semigroup, where the propagator decomposes as the product of a real Hermite semigroup and a fractional Fourier transform. In this setting, the Gabor matrix retains its Gaussian shape while undergoing a pure rotation on the time-frequency plane, reflecting the symplectic structure of the underlying flow. The analysis provides a unified operator-theoretic and phase-space perspective on parabolic and hyperbolic evolution equations, linking the geometry of their symbols with the sparsity and localization properties of their Gabor representations. Explicit formulas are given in a form suitable for numerical computation and visualization of phase-space dynamics.