Posted by: matheuscmss | May 3, 2021

Spectral rigidity of almost circular ellipses (after Hezari–Zelditch)

In 1966, M. Kac wrote a famous article asking whether Can one hear the shape of drum?: mathematically speaking, one wants to reconstruct (up to isometries) a domain from the knowledge of the spectrum of its Laplacian.

In his article, M. Kac showed that one can hear the shape of a disk {\mathbb{D}(0,R)=\{z\in\mathbb{R}^2:|z|\leq R\}} because of the following two facts:

  • the area {A_{\Omega}} and the perimeter {L_{\partial\Omega}} of a smooth domain {\Omega\subset\mathbb{R}^2} are determined by the eigenvalues {0<\lambda_1(\Omega)<\lambda_2(\Omega)\leq \dots} of its Laplacian {\Delta_{\Omega}} via the asymptotics of the trace of the heat operator:\displaystyle \textrm{tr}(e^{-t\Delta_{\Omega}}) = \sum\limits_{n=1}^{\infty} e^{-t\lambda_n(\Omega)} \sim \frac{1}{t}\left(\frac{A_{\Omega}}{2\pi} - \frac{L_{\partial\Omega}}{4\sqrt{2\pi}}\sqrt{t}\right) \quad \textrm{ as } t\rightarrow 0^+;
  • the isoperimetric inequality says that we can recognise the disk from its perimeter and area: indeed, any smooth domain {\Omega} satisfies {A_{\Omega}\leq \frac{1}{4\pi}L_{\partial\Omega}^2}, and the equality holds if and only if {\Omega} is isometric to a disk of radius {L_{\partial\Omega}/2\pi}.

(In particular, a smooth domain {\Omega\subset \mathbb{R}^2} with the same Laplace eigenvalues of {\mathbb{D}(0,R)} has area {\pi R^2} and perimeter {2\pi R}, so that the isoperimetric inequality ensures that {\Omega} and {\mathbb{D}(0,R)} are isometric.)

In a preprint from 2019, Hezari and Zelditch showed that one can also hear the shape of an ellipse {E_{\varepsilon}=\{(x,y):x^2+\frac{y^2}{1-\varepsilon^2}=1\}} of small eccentricity {0\leq \varepsilon\leq\varepsilon_0 < 1}. As it is explained by Zelditch in this video here, a first-order approximation to their basic strategy to hear ellipses of small eccentricities is to replace “trace of the heat operator” and “isoperimetric inequality” in Kac’s argument by “trace of the wave operator” and “dynamical rigidity of ellipses”.

More concretely, Hezari and Zelditch considered a smooth domain {\Omega} which is isospectral to an ellipse {E_{\varepsilon}} (i.e., {\lambda_n(\Omega) = \lambda_n(E_{\varepsilon})} for all {n\in\mathbb{N}}) of small eccentricity {\varepsilon}, and they took the following steps:

  • in Section 2 of their paper, it is shown that {\Omega} is necessarily close to {E_{\varepsilon}}; in fact, after parametrising {\partial\Omega} with an arc-length parameter {s}, we can control the {L^2}-norms of the derivatives of the curvature {\kappa} of {\partial\Omega} from the asymptotics of the trace of the heat operator: indeed,\displaystyle \textrm{tr}(e^{-t\Delta_{\Omega}}) = \sum\limits_{n=1}^{\infty} e^{-t\lambda_n(\Omega)} \sim (4\pi t)^{-1}\sum_{k\geq 0}b_k(\Omega) t^{k/2} \quad \textrm{ as } t\rightarrow0^+,where\displaystyle b_3(\Omega) = \frac{\sqrt{\pi}}{64}\int_{\partial\Omega}\kappa^2\,ds, \quad b_5(\Omega) = \frac{37\sqrt{\pi}}{2^{13}}\int_{\partial\Omega}\kappa^4\,ds - \frac{\sqrt{\pi}}{2^{10}}\int_{\partial\Omega}(\kappa')^2\,ds,and, in general, {b_{2n+3}(\Omega) = c_{2n+3}\int_{\partial\Omega}\kappa_n^2+ \int_{\partial\Omega} Q_n(\kappa,\dots, \kappa_{n-1})\,ds} for a certain constant {c_{2n+3}\neq0} and an adequate “universal” polynomial {Q_n} (here, {\kappa_j} is the {j}-th derivative of {\kappa} with respect to {s}); in particular, since {\Omega} and {E_{\varepsilon}} are isospectral, {b_k(\Omega) = b_k(E_{\varepsilon})} for all {k\in\mathbb{N}}, and Melrose explored this fact to get a pre-compactness bound {\|\kappa_n(\Omega)\|_{L^2} = O_n(1)} for all {n\in\mathbb{N}} (via a bootstrap argument where the Poincaré inequality and the Sobolev embedding theorem are employed to convert {L^2} bounds on {\kappa}, {\dots}, {\kappa_{j+1}} into {C^0} bounds on {\kappa}, {\dots}, {\kappa_j}); in other terms, after Melrose, the shape of any {\Omega} isospectral to {E_{\varepsilon}} is bounded; by reworking Melrose’s argument, Hezari and Zelditch actually show that {\Omega} is almost circular in the sense that {\|\kappa_n\|_{C^0}=O_n(\sqrt{\varepsilon})} for all {n\geq 1};
  • in view of a theorem of Avila, de Simoi and Kaloshin, an almost circular domain {\Omega} is isometric to an ellipse provided {\Omega} is rationally integrable, i.e., for each {q>2}, the periodic trajectories with rotation number {1/q} in the billiard table determined by {\Omega} are tangent to a smooth convex curve (usually called caustic);
  • in Sections 3 to 6 of their paper, it is shown that the portion between {5} and {L_{\partial\Omega}} of the singular support of the trace of the wave operator {\textrm{tr}(\cos (t\sqrt{\Delta_{\Omega}}))} coincides with the set {\mathcal{L}} of lengths of periodic trajectories with rotation number {1/q}, {q\geq 3}, in an almost circular billiard table {\Omega}; in particular, if {\Omega} is isospectral to {E_{\varepsilon}}, one concludes (in Section 7 of their paper) that, for each {q\geq 3}, all periodic trajectories in {\Omega} with rotation number {1/q} have the same length {t_q(\varepsilon)} and they form a caustic, so that {\Omega} is rationally integrable (and, a fortiori, isometric to {E_{\varepsilon}} after Avila–de Simoi–Kaloshin).

A natural question raised by Hezari–Zelditch work is to determine the magnitude of the upper bound {\varepsilon_0} on the eccentricities of the ellipses which can be heard from their methods.

In this direction, I would like to conclude this short post by noticing that I asked a group of 6 undergraduate students (in their 2nd year) at \’Ecole Polytechnique to follow closely the articles by Avila–de Simoi–Kaloshin and Hezari–Zelditch (while trying to explicitly compute as many implied constants as possible), and, after 6 months of work, they produced this report here (in French) concluding that {\varepsilon_0} is not bigger than {10^{-112}}. (Of course, there is plenty of room for tiny improvements here, but one will probably need some new ideas before reaching a “normal size” {\varepsilon_0} [e.g., {\varepsilon_0\sim 10^{-10}}].)

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