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 $B^{\ast }E$ 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 $C^{\infty }$ domains with the symmetries of the ellipse.