Symplectic and Hamiltonian Deformations of Gabor Frames

TL;DR

This paper explores symplectic and Hamiltonian deformations of Gabor frames, providing a unified framework that simplifies known results and introduces a new deformation scheme based on Hamiltonian flows, with practical implementation methods.

Contribution

It introduces a general deformation scheme for Gabor frames using Hamiltonian isotopies, extending previous symplectic deformation approaches with a detailed, implementable method.

Findings

Simplified derivation of known results using covariance properties

Development of a Hamiltonian deformation scheme for Gabor frames

Implementation of the method via symplectic integrators

Abstract

We study symplectic deformations of Gabor frames using the covariance properties of the Heisenberg operators. This allows us to recover in a very simple way known results. We thereafter propose a general deformation scheme by Hamiltonian isotopies, which are paths of Hamiltonian flows. We define and study in detail a weak notion of Hamiltonian deformations, using ideas from semiclassical analysis due to Heller and Hagedorn. This method can be easily implemented using symplectic integrators.