Stability of Gabor frames under small time Hamiltonian evolutions

TL;DR

This paper proves that Gabor frames remain stable under small Hamiltonian deformations, where the window and phase-space points evolve according to Schrödinger and Hamiltonian flows, respectively, for certain classes of Hamiltonians.

Contribution

It establishes the stability of Gabor frames under small Hamiltonian evolutions, extending previous results to include potentials in the Sj"ostrand class with bounded second derivatives.

Findings

Frame property remains stable for small times.

Stability holds for Hamiltonians with quadratic plus Sj"ostrand class potentials.

Answers a previously posed question on Gabor frame stability under Hamiltonian deformations.

Abstract

We consider Hamiltonian deformations of Gabor systems, where the window evolves according to the action of a Schr\"odinger propagator and the phase-space nodes evolve according to the corresponding Hamiltonian flow. We prove the stability of the frame property for small times and Hamiltonians consisting of a quadratic polynomial plus a potential in the Sj\"ostrand class with bounded second order derivatives. This answers a question raised in [de Gosson, M. Symplectic and Hamiltonian Deformations of Gabor Frames. Appl. Comput. Harmon. Anal. Vol. 38 No.2, (2015) p.196--221.]