Centrally symmetric analytic plane domains are spectrally determined in this class

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.