Microlocal defect measures for Cauchy data of Dirichlet, resp. Neumann, eigenfunctions of an ellipse are determined. We prove that, for any invariant curve for the billiard map on the boundary phase space 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 domains with the symmetries of the ellipse.
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Hamid Hezari, Steve Zelditch, Eigenfunction asymptotics and spectral rigidity of the ellipse. J. Spectr. Theory 12 (2022), no. 1, pp. 23–52DOI 10.4171/JST/393