Liouville billiard tables and an inverse spectral result

Abstract

We consider a class of billiard tables (X, g), where X is a smooth compact manifold of dimension two with smooth boundary ∂X and g is a smooth Riemannian metric on X, the billiard flow of which is completely integrable. The billiard table (X, g) is defined by means of a special double cover with two branched points and it admits a group of isometries G ∼ = Z 2 × Z 2 . Its boundary can be characterized by the string property; namely, the sum of distances from any point of ∂X to the branched points is constant. We provide examples of such billiard tables in the plane (elliptical regions), on the sphere S 2 , on the hyperbolic space H 2 , and on quadrics. The main result is that the spectrum of the corresponding Laplace-Beltrami operator with Robin boundary conditions involving a smooth function K on ∂X uniquely determines the function K, provided that K is invariant under the action of G.

C∞spectral rigidity of the ellipse

“…The densities dµ on the fixed point sets of β and its powers are very similar to the canonical densities defined in Lemma 4.2 of [Duistermaat and Guillemin 1975], and further discussed in [Guillemin and Melrose 1979a;Popov and Topalov 2003;. The constants C are explicit and depend on the boundary conditions.…”

“…Abel transform. The remainder of the proof of Theorem 1 is identical to that of Theorem 4.5 of [Guillemin and Melrose 1979a] (see also [Popov and Topalov 2003]). For the sake of completeness, we sketch the proof.…”

“…In [Zelditch 2009; it is shown that analytic domains with one symmetry are spectrally determined if the length of the minimal bouncing ball orbit and one iterate is a -isospectral invariant. The prior results on -isospectral deformations that we are aware of are contained in the articles [Guillemin and Melrose 1979a;Popov and Topalov 2003; and concern deformations of boundary conditions. To our knowledge, the only prior results on -isospectral deformations of the domain are contained in [Marvizi and Melrose 1982].…”

The inverse spectral problem

“…An isospectral deformation of a Riemannian manifold (possibly with boundary) is one-parameter family of metrics satisfying Spec(M, g t ) = Spec(M, g 0 ) for each t. Similarly, an isospectral deformation of a domain with a fixed background metric g 0 and boundary conditions B is a family Ω t with Spec B (Ω t ) = Spec B (Ω). One could also pose the inverse spectral problems for boundary conditions (while holding the other data fixed) as in [GM2,PT]. The inverse spectral and isospectral deformation problems are difficult because the map Spec is highly nonlinear.…”

Discrete analog of the projective equivalence and integrable billiard tables

“…This is part of a series of papers [10][11][12] concerned with the integrability and spectral rigidity of compact Liouville billiard tables of dimensions n ≥ 2. In this paper we define a class of discrete systems whose integrability follows from a simple construction that we call the geodesic equivalence principle.…”