8th International Conference on
Computational Harmonic Analysis

September 12-16, 2022

Ingolstadt, Germany

"Lipschitz Continuity of Spectra of Pseudodifferential Operators in a Weighted Sjöstrand Class"

Speckbacher, Michael

We consider pseudodifferential operators with real-valued symbols in a weighted Sjöstrand class depending on a parameter $\delta$. We show that under certain regularity assumptions, the spectral edges of such operators are Lipschitz continuous in $\delta$. Our main theorem allows us to obtain a result for non-uniform Gabor families: Let $\Lambda\in\mathbb{R}^{2d}$ be an arbitrary discrete set and $\alpha>0$. If $g\in M^1_2(\mathbb{R}^d)$, then the optimal frame bounds of the Gabor system $G(g,\alpha\Lambda)$ are Lipschitz functions in the dilation parameter $\alpha$.

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