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.

We introduce eigenfunctions of the Reed-Muller transform. Eigenfunctions are functions whose canonical sumof-products expression and PPRM (positive polarity ReedMuller expression) are isomorphic. In the case of symmetric functions, the eigenfunction can be viewed as a function whose reduced truth vector is identical to the reduced ReedMuller spectrum. We show that the number of symmetric (ordin...