Eigenfunction asymptotics and spectral rigidity of the ellipse

Hamid Hezari; Steve Zelditch

doi:10.4171/JST/393

Eigenfunction asymptotics and spectral rigidity of the ellipse

Hamid Hezari, Steve Zelditch

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Microlocal defect measures for Cauchy data of Dirichlet, resp. Neumann, eigenfunctions of an ellipse E are determined. We prove that, for any invariant curve for the billiard map on the boundary phase space B*E of an ellipse, there exists a sequence of eigenfunctions whose Cauchy data concentrates on the invariant curve. We use this result to give a new proof that ellipses are infinitesimally spectrally rigid among C¹ domains with the symmetries of the ellipse.

Original language	English (US)
Pages (from-to)	23-52
Number of pages	30
Journal	Journal of Spectral Theory
Volume	12
Issue number	1
DOIs	https://doi.org/10.4171/JST/393
State	Published - 2022

Keywords

Cauchy data
Laplacian
Spectral rigidity
billiards
ellipse

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
Geometry and Topology

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10.4171/JST/393

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title = "Eigenfunction asymptotics and spectral rigidity of the ellipse",

abstract = "Microlocal defect measures for Cauchy data of Dirichlet, resp. Neumann, eigenfunctions of an ellipse E are determined. We prove that, for any invariant curve for the billiard map on the boundary phase space B*E of an ellipse, there exists a sequence of eigenfunctions whose Cauchy data concentrates on the invariant curve. We use this result to give a new proof that ellipses are infinitesimally spectrally rigid among C1 domains with the symmetries of the ellipse.",

keywords = "Cauchy data, Laplacian, Spectral rigidity, billiards, ellipse",

author = "Hamid Hezari and Steve Zelditch",

note = "Funding Information: Funding. Hamid Hezari is supported by the Simons Collaborations Grants for Mathematicians 638398. Steve Zelditch is partially supported by NSF grant DMS-1810747. Publisher Copyright: {\textcopyright} 2022 European Mathematical Society",

year = "2022",

doi = "10.4171/JST/393",

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AU - Hezari, Hamid

AU - Zelditch, Steve

N1 - Funding Information: Funding. Hamid Hezari is supported by the Simons Collaborations Grants for Mathematicians 638398. Steve Zelditch is partially supported by NSF grant DMS-1810747. Publisher Copyright: © 2022 European Mathematical Society

PY - 2022

Y1 - 2022

N2 - Microlocal defect measures for Cauchy data of Dirichlet, resp. Neumann, eigenfunctions of an ellipse E are determined. We prove that, for any invariant curve for the billiard map on the boundary phase space B*E of an ellipse, there exists a sequence of eigenfunctions whose Cauchy data concentrates on the invariant curve. We use this result to give a new proof that ellipses are infinitesimally spectrally rigid among C1 domains with the symmetries of the ellipse.

AB - Microlocal defect measures for Cauchy data of Dirichlet, resp. Neumann, eigenfunctions of an ellipse E are determined. We prove that, for any invariant curve for the billiard map on the boundary phase space B*E of an ellipse, there exists a sequence of eigenfunctions whose Cauchy data concentrates on the invariant curve. We use this result to give a new proof that ellipses are infinitesimally spectrally rigid among C1 domains with the symmetries of the ellipse.

KW - Cauchy data

KW - Laplacian

KW - Spectral rigidity

KW - billiards

KW - ellipse

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Eigenfunction asymptotics and spectral rigidity of the ellipse

Abstract

Keywords

ASJC Scopus subject areas

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