Centrally symmetric analytic plane domains are spectrally determined in this class

Hezari, Hamid; Zelditch, Steve

doi:10.1090/tran/8889

Centrally symmetric analytic plane domains are spectrally determined in this class
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by Hamid Hezari and Steve Zelditch;

Trans. Amer. Math. Soc. 376 (2023), 7521-7553

DOI: https://doi.org/10.1090/tran/8889

Published electronically: August 29, 2023

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Abstract:

We prove that, under some generic non-degeneracy assumptions, real analytic centrally symmetric plane domains are determined by their Dirichlet (resp. Neumann) spectra. We prove that the conditions are open-dense for real analytic strictly convex domains. One step is to use a Maslov index calculation to show that the second derivative of the defining function of a centrally symmetric domain at the endpoints of a bouncing ball orbit is a spectral invariant. This is also true for up-down symmetric domains, removing an assumption from the proof in that case.

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