Gabor Frames of Gaussian Beams for the Schr\"odinger equation

TL;DR

This paper develops a Gabor frame-based semiclassical analysis for linear Schrödinger equations with quadratic growth Hamiltonians, constructing higher order parametrices and providing precise $L^2$ estimates, including nonlinear approximations and numerical validation.

Contribution

It introduces a novel Gabor frame approach to semiclassical Schrödinger equations, constructing higher order parametrices with explicit error bounds and extending to nonlinear cases.

Findings

Higher order parametrices achieve accurate $L^2$ estimates.

Gabor frame methods effectively analyze Schrödinger equations.

Numerical experiments validate theoretical results.

Abstract

The present paper is devoted to the semiclassical analysis of linear Schr\"odinger 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 Schr\"odinger equations by means of $\hbar$-Gabor frames, as recently defined by M. de Gosson, and we provide precise $L^2$-estimates of their accuracy, in terms of the Planck constant $\hbar$. Nonlinear parametrices, in the spirit of the nonlinear approximation, are also presented. Numerical experiments are exhibited to compare our results with the early literature.