From K.A.M. Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables

Abstract

This article is a part of a project investigating the relationship between the dynamics of completely integrable or "close" to completely integrable billiard tables, the integral geometry on them, and the spectrum of the corresponding Laplace-Beltrami operators. It is concerned with new isospectral invariants and with the spectral rigidity problem for the Laplace-Beltrami operators ∆ t , t ∈ [0, 1], with Dirichlet, Neumann or Robin boundary conditions, associated with C 1 families of billiard tables (X, g t ). We introduce a notion of weak isospectrality for such deformations.The main dynamical assumption on (X, g 0 ) is that the corresponding billiard ball map B 0 or an iterate P 0 = B m 0 of it posses a Kronecker invariant torus with a Diophantine frequency ω 0 and that the corresponding Birkhoff Normal Form is nondegenerate in Kolmogorov sense. Then we prove that there exists δ 0 > 0 and a set Ξ of Diophantine frequencies containing ω 0 and of full Lebesgue measure around ω 0 such that for each ω ∈ Ξ and 0 < δ < δ 0 there exists a C 1 family of Kronecker tori Λ t (ω) of P t for t ∈ [0, δ]. If the family ∆ t , t ∈ [0, 1], satisfies the weak isospectral condition we prove that the average action β t (ω) on Λ t (ω) and the Birkhoff Normal Form of P t at Λ t (ω) are independent of t ∈ [0, δ] for each ω ∈ Ξ.As an application we obtain infinitesimal spectral rigidity for Liouville billiard tables in dimensions 2 and 3. In particular infinitesimal spectral rigidity for the ellipse and the ellipsoid is obtained under the weak isospectral condition. Applications are obtained also for strictly convex billiard tables in R 2 as well as in the case when (X, g 0 ) admits an elliptic periodic billiard trajectory with no resonances of order ≤ 4.In particular we obtain spectral rigidity (under the weak isospectral condition) of elliptical billiard tables in the class of analytic and Z 2 × Z 2 symmetric billiard tables in R 2 . We prove also that billiard tables with boundaries close to ellipses are spectrally rigid in this class.The results are based on a construction of C 1 families of quasi-modes associated with the Kronecker tori Λ t (ω) and on suitable KAM theorems for C 1 families of Hamiltonians. We propose a new iteration schema (a modified iterative lemma) in the proof of the KAM theorem with parameters, which avoids the Whitney extension theorem for C ∞ jets and allows one to obtain global estimates of the corresponding canonical transformations and Hamiltonians in the scale of all Hölder norms. The classical and quantum Birkhoff Normal Forms for C 1 or analytic families of symplectic mappings (Hamiltonians) obtained here can be used as well in order to investigate problems related to the quantum non-ergodicity of C ∞ -smooth KAM systems.

One can hear the shape of ellipses of small eccentricity

“…In fact, there do not even exist prior 'local spectral determination' results, which would say that an ellipse (or any other domain) is determined by its spectrum among domains which lie in a sufficiently small C k neighborhood of the ellipse. The only prior positive result specific to the inverse Laplace spectral problem for ellipses is [HeZe12] (see also [PoTo16] for ellipsoids), which says that ellipses are infinitesimally spectrally rigid among C ∞ domains with the same left-right and up-down symmetries. The progress in that article is to allow competing domains to be C ∞ and not real-analytic.…”

“…The results of the present article, by comparison, do not make any symmetry assumptions and allow the competing domains to be general C ∞ domains. There also exists a sequence of results of PoTo12,PoTo16] using the KAM structure of convex smooth plane domains to deduce spectral rigidity results for Liouville billiards (including ellipses) with two commuting reflection symmetries, and for analytic domains that are sufficiently close to an ellipse and possess the two reflection symmetries of the ellipse.…”

Eigenfunction asymptotics and spectral Rigidity of the ellipse

“…There is also a series of articles of G. Popov and P. Topalov (see e.g. [PoTo03,PT16]) on the use of KAM quasi-modes to study Laplace inverse spectral problems. In particular, in [PT16], Popov-Topalov also give a new proof of the rigidity result of [HeZe12] and extend it to other settings.…”

“…[PoTo03,PT16]) on the use of KAM quasi-modes to study Laplace inverse spectral problems. In particular, in [PT16], Popov-Topalov also give a new proof of the rigidity result of [HeZe12] and extend it to other settings. The approach in this article is closely related to theirs, although it does not seem that the authors directly studied Cauchy data of eigenfunctions of an ellipse.…”

Robin spectral rigidity of nearly circular domains with a reflectional symmetry

“…In [HZ10], it was shown that real analytic domains in R n with reflectional symmetries about all coordinate axes are spectrally determined among the same domains. In [HZ10], ellipses were shown to be to infinitesimally spectrally rigid among smooth domains with the symmetries of the ellipse (see [PT03,PT12,PT16] for results in the context of completely integrable tables other than ellipses). Guillemin-Melrose [GM79a] showed Robin spectral rigidity of ellipses when the Robin functions preserve both reflectional symmetries.…”