  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.
