Eigenfunction asymptotics and spectral Rigidity of the ellipse

TL;DR

This paper investigates the asymptotic behavior of eigenfunctions in elliptical domains, revealing how their boundary data concentrates on invariant curves and establishing the ellipse's spectral rigidity among symmetric smooth domains.

Contribution

It provides a new proof of the spectral rigidity of ellipses by analyzing eigenfunction Cauchy data and their microlocal defect measures, leveraging classical ellipse eigenfunction results.

Findings

Eigenfunction Cauchy data concentrates on invariant curves

Ellipses are infinitesimally spectrally rigid among symmetric smooth domains

New proof of spectral rigidity using microlocal analysis

Abstract

This paper is part of a series concerning the isospectral problem for an ellipse. In this paper, we study Cauchy data of eigenfunctions of the ellipse with Dirichlet or Neumann boundary conditions. Using many classical results on ellipse eigenfunctions, we determine the microlocal defect measures of the Cauchy 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 of eigenfunctions whose Cauchy data concentrates on $C$. 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.