Eigenfunction asymptotics and spectral rigidity of the ellipse

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