[Paper Review] $C^{\infty} $ spectral rigidity of the ellipse

Hamid Hezari, Steve Zelditch|arXiv (Cornell University)|Jul 10, 2010

Mathematics and Applications21 references4 citations

TL;DR

This paper proves that ellipses are $C^{ iny ext{∞}}$ spectrally rigid among $C^{ iny ext{∞}}$ domains with $bZ_2\times\bbZ_2$ symmetry, meaning no non-trivial isospectral deformations exist. Using a novel application of the Hadamard variational formula for the wave trace, the authors show that any isospectral deformation must have a vanishing normal boundary variation at order zero, implying the deformation is flat at $\epsilon=0$. This establishes infinitesimal spectral rigidity and rules out real analytic isospectral deformations of the ellipse.

ABSTRACT

We prove that ellipses are infinitesimally spectrally rigid among $C^{\infty}$ domains with the symmetries of the ellipse.

Motivation & Objective

• To establish infinitesimal spectral rigidity of the ellipse among $C^\infty$ domains with the symmetries of an ellipse.
• To resolve a long-standing conjecture in inverse spectral theory regarding the uniqueness of the ellipse as a domain with completely integrable billiards.
• To develop and apply a new Hadamard variational formula for the wave trace on smooth Euclidean domains under boundary deformations.
• To show that any isospectral deformation of an ellipse must be flat at $\epsilon=0$, ruling out non-trivial real analytic deformations.
• To extend spectral rigidity results beyond real analytic domains to the $C^\infty$ setting, overcoming limitations of prior methods.

Proposed method

• Derives a new Hadamard variational formula for the wave trace on smooth domains under $C^1$ boundary deformations, valid under standard 'cleanliness' assumptions.
• Applies the formula to isospectral deformations $\Omega_\epsilon$ of an ellipse $\Omega_0$, relating the variation of the wave trace to the normal boundary variation $\dot{\rho}$.
• Uses the Poisson relation and stationary phase analysis to relate the wave trace's singular support to the length spectrum and periodic orbits.
• Analyzes the structure of the energy level sets $F_T$ for periodic orbits, showing that for certain $T$, $F_T$ consists of two symmetric invariant curves.
• Applies the Abel transform to the resulting integral equations involving $\dot{\rho}$, leveraging symmetry and density arguments to show $\dot{\rho} \equiv 0$.
• Reparameterizes the deformation family to isolate the first non-vanishing Taylor coefficient of $\rho_\epsilon$, leading to a contradiction if $\dot{\rho} \neq 0$.

Experimental results

Research questions

• RQ1 Can the ellipse be deformed isospectrally through $C^\infty$ domains with $\bbZ_2\times\bbZ_2$ symmetry without changing the spectrum?
• RQ2 Is the normal boundary variation $\dot{\rho}$ necessarily zero for any isospectral deformation of the ellipse under the given symmetry?
• RQ3 Does the absence of non-trivial real analytic isospectral deformations follow from infinitesimal spectral rigidity?
• RQ4 Can the Hadamard variational formula for the wave trace be used to derive spectral rigidity results in the $C^\infty$ category, beyond real analytic settings?
• RQ5 Are there cancellations in the wave trace trace formula due to multiplicity in the length spectrum that obstruct spectral rigidity, and how can they be resolved?

Key findings

• The normal boundary variation $\dot{\rho}$ must vanish identically for any isospectral deformation of the ellipse under $\bbZ_2\times\bbZ_2$ symmetry, proving infinitesimal spectral rigidity.
• All isospectral deformations of the ellipse are flat at $\epsilon = 0$, meaning the Taylor expansion of $\rho_\epsilon$ vanishes to all orders.
• There exist no non-trivial real analytic curves of $\bbZ_2\times\bbZ_2$ symmetric $C^\infty$ domains that are isospectral to the ellipse.
• The wave trace variational formula yields a non-trivial spectral invariant that detects the absence of boundary deformation under symmetry constraints.
• The method applies to $C^\infty$ domains, extending prior results that required real analyticity and overcoming limitations from length spectrum multiplicity.
• The proof relies on the Abel transform and symmetry properties to show that the only $\bbZ_2\times\bbZ_2$-invariant function satisfying the integral equations is $\dot{\rho} \equiv 0$.