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

Speaker: Elena Cordero, Università degli Studi di Torino

Abstract: In this talk, we develop a phase-space framework for studying linear PDE evolutions via their Wigner kernels and Gabor matrices. Focusing on propagators from the complex heat, wave, and Hermite equations, we derive explicit formulas describing how solution operators act on localized wave packets, revealing the mechanisms behind diffusion, dispersion, and oscillations. A key result is a precise characterization of off-diagonal decay (sparsity) in Gabor matrices, directly tied to numerical efficiency in frame-based discretizations. For parabolic flows we obtain Gaussian decay with explicit constants, while for hyperbolic flows the analysis captures sharp propagation along characteristic cones, lacuna behavior, and the emergence of “ghost frequencies’’ in Wigner representations. The formulas enable numerical evaluation and visualization of phase-space geometry, offering a unified, computation-oriented view of diffusion, wave propagation, and metaplectic dynamics with implications for high-resolution schemes, sparse representations, and model reduction. This is joint work with Gianluca Giacchi (USI) and Luigi Rodino (UniTo).

Biography: Elena Cordero is Full Professor of Mathematical Analysis at the Università degli Studi di Torino. Her research focuses on harmonic and time-frequency analysis, with applications to Schrödinger equations, pseudodifferential and Fourier integral operators, and the metaplectic representation of operators. She is the author of more than 90 scientific publications and a monograph published by De Gruyter. She has delivered numerous invited and plenary talks at international conferences. She is currently on the Editorial Board of the Journal of Fourier Analysis and Applications and of AIMS Mathematics and she was recently honored with membership in the Accademia delle Scienze of Turin.

Host: Dr. Gianluca Giacchi