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

TL;DR

This work develops a unified phase-space framework for evolution semigroups by computing explicit Wigner kernels and Gabor matrices for propagators of parabolic and hyperbolic equations, notably the complex heat and wave equations and the complex Hermite equation. Using Fourier multipliers and symbols in Gelfand–Shilov classes, it proves exponential off-diagonal decay and quasi-diagonality in the Gabor representation, linking symbol regularity to sparsity in the phase-space representation. The study combines operator-theoretic and metaplectic methods to derive closed-form formulas, quantify decay, and describe geometric features (anisotropic diffusion, rotations in time–frequency) that are amenable to numerical computation and visualization. The results illuminate how the underlying symbol geometry shapes sparsity and localization of propagators, with potential extensions to broader pseudodifferential settings and general constant-coefficient Cauchy problems.

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 Hörmander'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.

Figures (5)

• Figure 1: Upper bound on the Gabor matrix elements $|G_t((0,0),(y,\eta))|$ from \ref{['eq:GaborM']} for the real heat equation ($d=1$, $\alpha=1$, $\beta=0$) at times $t = 0, 1, 2, 5$. A logarithmic color scalelevels=[1e-8,1e-4,1e-2,1e-1,1] and white contour lines ensure visibility across the full dynamic range. At $t=0$, the bound matches the exact Gabor overlap (peak $=1$). As $t$ increases, the support spreads anisotropically: Fast decay in $y$ (rate $\varepsilon(t,1,0)$): narrow in position shift. Slow decay in $\eta$ (rate $\varepsilon(t,1,0)/2$): broad in frequency shift. This anisotropic decay confirms the sparsity of the Gabor representation for parabolic propagators.
• Figure 2: The magnitude of the Gabor matrix elements $G_t((0,0),(y,\eta))$ from \ref{['eq:Gt-forma1']} with parameters $d=1$, $\alpha+i\beta=1+i$ (top) and $\alpha+i\beta=0.1+i$ (bottom) at times $t = 0, 1, 2, 5$. A logarithmic color scalelevels=[1e-8,1e-4,1e-2,1e-1,1] and white contour lines ensure visibility across the full dynamic range. The predominance of the heat parameter$\alpha$ with respect to the Schrödinger parameter $\beta$ reflects on the angle between the horizontal axis and the axle shafts, which increases as $\alpha$ reduces. Moreover as $t$ increases, the support spreads anisotropically along the axle shafts.
• Figure 3: Kernels of Examples 4.5 and 4.6. Left: $\kappa_t^{(1)}(s,\xi)$ from Example 4.5. Center: $\kappa_t^{(2)}(\xi;s)$ in dimension $d=2$. Right: $\kappa_t^{(3)}(\xi;s)$ in dimension $d=3$. All plots are shown for $t=1$.
• Figure 4: The magnitude of the Gabor matrix $G_t(0,w)=\langle T_t\pi(0,0)g,\pi(y,\eta)g\rangle$ for the one-dimensional wave equation outlined for $y\in[-6,6]$, $\eta\in[-6,6]$ at times $t = 0,\,1,\,2,\, 5$, using a logarithmic color scale with levels=[1e-8,1e-4,1e-2,1e-1,1]. The early and intermediate stages of the one-dimensional wave evolution are captured: as $t$ increases, the spreading propagation typical of the solutions of wave equations manifests itself thorugh a stretching effect of the Gabor matrix.
• Figure 5: Plots in logarithmic color scale of $|G_t(z,w)| = |\langle R_{(\vartheta+i\mu)t}\pi(z)g,\,\pi(w)g\rangle|$ for the complex Hermite equation with random parameters $\vartheta=0.7$ and $\mu=1.3$, outlined for $z=(0,1)$. The panels correspond to times $t = 0\,1,\,2,\,5$. The peak of the Gabor matrix undergoes a clear rotation in the $(y,\eta)$-plane induced by the fractional Fourier transform component $\mathscr{F}_{\mu t}$, its Gaussian shape and width remain constant, and it undergoes an exponential dumping due to the coefficient $(2\sinh(\vartheta t))^{-1/2}$. Hence, unlike the complex heat equation displayed in Figure \ref{['fig:gabor-complex-heat']}, there is no diffusion or spreading, but a rigid rotation of the localization region in phase space.

Theorems & Definitions (68)

• Definition 2.1
• Lemma 2.2: Covariance property; see, e.g., Corollary 217 in Gos11
• Definition 2.3
• Lemma 2.4: Gaussian integral
• Definition 2.5
• Theorem 2.6
• Lemma 2.7
• proof
• Definition 3.1
• Theorem 3.2