One can hear the shape of ellipses of small eccentricity
Analysis of PDEs
2022-07-19 v2 Spectral Theory
Abstract
We show that if the eccentricity of an ellipse is sufficiently small then up to isometries it is spectrally unique among all smooth domains. We do not assume any symmetry, convexity, or closeness to the ellipse, on the class of domains. In the course of the proof we also show that for nearly circular domains, the lengths of periodic orbits that are shorter than the perimeter of the domain must belong to the singular support of the wave trace. As a result we also obtain a Laplace spectral rigidity result for the class of axially symmetric nearly circular domains.
Cite
@article{arxiv.1907.03882,
title = {One can hear the shape of ellipses of small eccentricity},
author = {Hamid Hezari and Steve Zelditch},
journal= {arXiv preprint arXiv:1907.03882},
year = {2022}
}
Comments
Accepted for publication in Annals of Mathematics, to be published in the January 2023 issue (Volume 197, no. 1)