Gabor frames of Gaussian beams for the Schrödinger equation

The present paper is devoted to the semiclassical analysis of linear Schrödinger equations from a Gabor frame perspective. We consider (time-dependent) smooth Hamiltonians with at most quadratic growth. Then we construct higher order parametrices for the corresponding Schrödinger equations by means of ħ-Gabor frames, as recently defined by M. de Gosson, and we provide precise L2-estimates of their accuracy, in terms of the Planck constant ħ. Nonlinear parametrices, in the spirit of the nonlinear approximation, are also presented. Numerical experiments are exhibited to compare our results with the early literature.