Centrally symmetric analytic plane domains are spectrally determined in this class

TL;DR

This paper demonstrates that, under generic conditions, real analytic, centrally symmetric plane domains are uniquely determined by their spectral data, extending previous results to a broader class of symmetric domains.

Contribution

It establishes spectral determinacy for centrally symmetric real analytic convex domains using Maslov index calculations, removing previous assumptions.

Findings

Centrally symmetric analytic domains are spectrally determined under generic conditions.

The second derivative of the defining function at bouncing ball orbit endpoints is a spectral invariant.

Conditions for spectral determination are open-dense in the space of real analytic convex domains.

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 convex domains. The proof is parallel to the proof that up/down symmetric domains are spectrally determined. 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.