One can hear the shape of ellipses of small eccentricity

• 成果类型: Article
• 署名作者: Hezari, Hamid; Zelditch, Steve
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2022.196.3.4
• 发表日期: 2022
• 页码: 1083-1134
• 关键词: inverse spectral problem; INVARIANTS; domains; RIGIDITY

摘要

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 sym-metric nearly circular domains using a similar result of De Simoi, Kaloshin, and Wei concerning the length spectrum of such domains.