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Sparsity properties for phase-space representations of several types ofoperators have been extensively studied in recent papers, includingpseudodifferential, Fourier integral and metaplectic operators, withapplications to time-frequency analysis of Schr\"odinger-type evolutionequations. It has been proved that such operators are approximatelydiagonalized by Gabor wave packets. While the latter are expected to undergosome spreading phenomenon, there is no record of this issue in theaforementioned results. In this paper we prove refined estimates for the Gabormatrix of metaplectic operators, also of generalized type, where sparsity,spreading and dispersive properties are all noticeable. We provide applicationsto the propagation of singularities for the Schr\"odinger equation.

Dispersion, spreading and sparsity of Gabor wave packets for metaplectic and Schr\"odinger operators