One can hear the shape of ellipses of small eccentricity

Hamid Hezari; Steve Zelditch

doi:https://doi.org/10.4007/annals.2022.196.3.4

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 using a similar result of De Simoi, Kaloshin, and Wei concerning the length spectrum of such domains.

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@INCOLLECTION{Z, author = {Zelditch, Steve}, title = {The inverse spectral problem}, booktitle = {Surveys in Differential Geometry. {V}ol. {IX}}, series = {Surv. Differ. Geom.}, volume = {9}, pages = {401--467}, note = {with an appendix by Johannes Sjöstrand and Maciej Zworski}, publisher = {Int. Press, Somerville, MA}, year = {2004}, mrclass = {58J53 (35P20 35R30 58J50)}, mrnumber = {2195415}, mrreviewer = {David Borthwick}, doi = {10.4310/SDG.2004.v9.n1.a12}, url = {https://doi.org/10.4310/SDG.2004.v9.n1.a12}, zblnumber = {1061.58029}, }
[Ze] $Go to document$ S. Zelditch, "Inverse spectral problem for analytic domains. II. $\Bbb Z_2$-symmetric domains," Ann. of Math. (2), vol. 170, iss. 1, pp. 205-269, 2009.

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@ARTICLE{Ze, author = {Zelditch, Steve}, title = {Inverse spectral problem for analytic domains. {II}. {$\Bbb Z_2$}-symmetric domains}, journal = {Ann. of Math. (2)}, fjournal = {Annals of Mathematics. Second Series}, volume = {170}, year = {2009}, number = {1}, pages = {205--269}, issn = {0003-486X}, mrclass = {58J53 (35P20 35R30 37D50)}, mrnumber = {2521115}, mrreviewer = {David Borthwick}, doi = {10.4007/annals.2009.170.205}, url = {https://doi.org/10.4007/annals.2009.170.205}, zblnumber = {1196.58016}, }
[Z14] $Go to document$ S. Zelditch, "Survey on the inverse spectral problem," ICCM Not., vol. 2, iss. 2, pp. 1-20, 2014.

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@ARTICLE{Z14, author = {Zelditch, Steve}, title = {Survey on the inverse spectral problem}, journal = {ICCM Not.}, fjournal = {ICCM Notices. Notices of the International Congress of Chinese Mathematicians}, volume = {2}, year = {2014}, number = {2}, pages = {1--20}, issn = {2326-4810}, mrclass = {58J50 (58J53)}, mrnumber = {3314780}, mrreviewer = {Ruth Gornet}, doi = {10.4310/ICCM.2014.v2.n2.a1}, url = {https://doi.org/10.4310/ICCM.2014.v2.n2.a1}, zblnumber = {}, }

One can hear the shape of ellipses of small eccentricity

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