Gabor Frame Decomposition of Evolution Operators and Applications

TL;DR

This paper analyzes the Gabor matrix representation of Schrödinger-type evolution operators, providing exact expressions, bounds, and decay properties, with applications to heat equations and harmonic oscillators, enhancing numerical evaluation methods.

Contribution

It introduces new bounds and decay estimates for Gabor matrices of evolution operators, extending previous work and improving numerical computation techniques.

Findings

Exact Gabor matrix for Heat Equation derived

Upper bounds for Gabor matrix decay established

Super-exponential decay shown for Harmonic Repulsor

Abstract

We compute the Gabor matrix for Schr\"odinger-type evolution operators. Precisely, we analyze the Heat Equation, already presented in \cite{2012arXiv1209.0945C}, giving the exact expression of the Gabor matrix which leads to better numerical evaluations. Then, using asymptotic integration techniques, we obtain an upper bound for the Gabor matrix in one-dimension for the generalized Heat Equation, new in the literature. Using Maple software, we show numeric representations of the coefficients' decay. Finally, we show the super-exponential decay of the coefficients of the Gabor matrix for the Harmonic Repulsor, together with some numerical evaluations. This work is the natural prosecution of the ideas presented in \cite{2012arXiv1209.0945C} and \cite{MR2502369}.