Lipschitz continuity of spectra of pseudodifferential operators in a weighted Sjöstrand class and Gabor frame bounds
Karlheinz Gröchenig
Universität Wien, AustriaJosé Luis Romero
Universität Wien, AustriaMichael Speckbacher
Universität Wien, Austria
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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 Sjöstrand 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.
Cite this article
Karlheinz Gröchenig, José Luis Romero, Michael Speckbacher, Lipschitz continuity of spectra of pseudodifferential operators in a weighted Sjöstrand class and Gabor frame bounds. J. Spectr. Theory 13 (2023), no. 3, pp. 805–839
DOI 10.4171/JST/465