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This paper is part of a series concerning the isospectral problem for anellipse. In this paper, we study Cauchy data of eigenfunctions of the ellipsewith Dirichlet or Neumann boundary conditions. Using many classical results onellipse eigenfunctions, we determine the microlocal defect measures of theCauchy data of the eigenfunctions. The ellipse has integrable billiards, i.e.the boundary phase space is foliated by invariant curves of the billiard map.We prove that, for any invariant curve $C$, there exists a sequence ofeigenfunctions whose Cauchy data concentrates on $C$. We use this result togive a new proof that ellipses are infinitesimally spectrally rigid among$C^{\infty}$ domains with the symmetries of the ellipse.

Eigenfunction asymptotics and spectral Rigidity of the ellipse