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We study one-parameter families of pseudodifferential operators whose Weylsymbols are obtained by dilation and a smooth deformation of a symbol in aweighted Sj\"ostrand class. We show that their spectral edges are Lipschitzcontinuous functions of the dilation or deformation parameter. Suitably localestimates hold also for the edges of every spectral gap. These statementsextend Bellissard's seminal results on the Lipschitz continuity of spectraledges for families of operators with periodic symbols to a large class ofsymbols with only mild regularity assumptions.  The abstract results are used to prove that the frame bounds of a family ofGabor systems $\mathcal{G}(g,\alpha\Lambda)$, where $\Lambda$ is a set ofnon-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 aboutthe precise blow-up rate of the condition number of Gabor frames near thecritical density.

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