Dispersion, spreading and sparsity of Gabor wave packets

Abstract: Sparsity properties for phase-space representations of several types of operators (including pseudodifferential, metaplectic and Fourier integral operators) have been extensively studied in recent articles, with applications to the analysis of dispersive evolution equation. It has been proved that such operators are approximately diagonalized by Gabor wave packets - equivalently, the corresponding phase-space representations (Gabor matrix/kernel) can be thought of as sparse infinite-dimensional matrices. While wave packets are expected to undergo some spreading and dispersion phenomena, there is no record of these issues in the aforementioned estimates. We recently proved refined estimates for the Gabor matrix of metaplectic operators, also of generalized type, where sparsity, spreading and dispersive properties are all simultaneously noticeable. We also provide applications to the propagation of singularities for the Schr\"odinger equation; in this connection, our results can be regarded as a microlocal refinement of known estimates. The talk is based on joint work with Elena Cordero and Fabio Nicola.

classical analysis and ODEsfunctional analysisrepresentation theoryspectral theory

Audience: researchers in the topic

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