In this talk, we consider one-parameter families of pseudodifferential

operators whose Weyl symbols are obtained by dilation and a smooth

deformation of a symbol in a weighted Sjöstrand class. We show that

their spectral edges are Lipschitz/Hölder 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.

If time permits, we will also discuss how these results can be used to

study the behavior of the frame bounds of Gabor systems generated from

a dilated non-uniform set of time-frequency shifts.