Inside-outside duality for planar billiards: A numerical study

Inside-outside duality for planar billiards: A numerical study.

This paper reports the results of extensive numerical studies related to spectral properties of the Laplacian and the scattering matrix for planar domains (called billiards). There is a close connection between eigenvalues of the billiardLaplacian and the scattering phases, basically that every energy at which a scattering phase is 2 corresponds to an eigenenergy of the Laplacian. Interesting p...

Spectral Duality for Planar Billiards

ao -d yn /9 40 50 01 2 M ay 1 99 4 Spectral Duality for Planar Billiards J.-P. Eckmann1;2 and C.-A. Pillet1 Dépt. de Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland Section de Mathématiques, Université de Genève, CH-1211 Genève 4, Switzerland Abstract. For a bounded open domain with connected complement in R2 and piecewise smooth boundary, we consider the Dirichlet Lapla...

Billiards inside a Cusp

Duality relations, correspondences and numerical results for planar elastic composites

This paper addresses three topics relating to planar elastic composites with anisotropic phases. First, the duality relations of Berdichevski are generalized to a wider class of planar elastic media. These yield phase interchange relations for the e ective compliance tensors of certain two phase media. Second, a simple derivation is given of the correspondence between a speci c class of planar ...

The Frequency Map for Billiards inside Ellipsoids

The billiard motion inside an ellipsoid Q ⊂ R is completely integrable. Its phase space is a symplectic manifold of dimension 2n, which is mostly foliated with Liouville tori of dimension n. The motion on each Liouville torus becomes just a parallel translation with some frequency ω that varies with the torus. Besides, any billiard trajectory inside Q is tangent to n caustics Qλ1 , . . . , Qλn ...