Lipschitz Continuity of Spectra of Pseudodifferential Operators in a Weighted Sj\"ostrand Class and Gabor Frame Bounds

TL;DR

This paper proves that spectral edges of certain pseudodifferential operators vary Lipschitz continuously with parameters, and applies these results to show Gabor frame bounds depend Lipschitz continuously on the density parameter.

Contribution

It extends Lipschitz continuity results for spectral edges from periodic symbols to a broader class with mild regularity, and applies this to Gabor frame bounds.

Findings

Spectral edges are Lipschitz continuous functions of dilation or deformation parameters.

Gabor frame bounds depend Lipschitz continuously on the density parameter.

Provides bounds on the blow-up rate of the Gabor frame condition number near critical density.

Abstract

We study one-parameter families of pseudodifferential operators whose Weyl symbols are obtained by dilation and a smooth deformation of a symbol in a weighted Sj\"ostrand class. We show that their spectral edges are Lipschitz continuous functions of the dilation or deformation parameter. Suitably local estimates hold also for the edges of every spectral gap. These statements extend Bellissard's seminal results on the Lipschitz continuity of spectral edges for families of operators with periodic symbols to a large class of symbols with only mild regularity assumptions. The abstract results are used to prove that the frame bounds of a family of Gabor systems $\mathcal{G}(g,\alpha\Lambda)$, where $\Lambda$ is a set of non-uniform time-frequency shifts, $\alpha>0$, and $g\in M^1_2(\mathbb{R}^d)$, are Lipschitz continuous functions in $\alpha$. This settles a question about the precise blow-up rate of the condition number of Gabor frames near the critical density.