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 phenomena appear when the shape of the domain does not allow an extension of the eigenfunction to the exterior. In this paper these phenomena are studied and illustrated from several points of view. We consider quantum billiards, i.e., the Laplacian in a bounded domain Ω, with Dirichlet (zero) conditions on the boundary Γ. The billiard will be looked at from two different points of view, which define two seemingly independent problems. The interior problem is the more commonly studied aspect of the billiard dynamics, and the main objective in that case is to calculate the spectrum, i.e., the eigenvalues E of the problem ∆ = E , where vanishes on the boundary. In the exterior problem one considers scattering from the obstacle defined by the billiard boundary, with the same boundary conditions. It was suggested [5] that there is a strong link between these two problems, which in a crude form states that an energy E = k2 is an eigenenergy of the interior problem if and only if the on-shell scattering matrix S(k) of the exterior problem has an eigenvalue 1. This statement is exact for the circular and elliptic billiards [3]. Using a truncated matrix, numerical calculation for the square [4] gives excellent agreement. In the semiclassical limit, it is justified by observing that the semiclassical spectral density it predicts [5] coincides with the Gutzwiller trace formula [7]. As we shall see below, the conjecture implies that at an eigenenergy En the obstacle is “transparent” for a well-chosen wave function. In the interior of the billiard it equals the eigenfunction of the interior problem, and in the exterior of the billiard it is the wave function corresponding to the eigenvalue 1 of the scattering matrix. The conjecture therefore leads to the result that the eigenfunction of the interior problem can be continued to a single valued bounded function in the plane. Billiards whose eigenfunctions may not be continued to the whole plane, due to branch points, are easily constructed [6]. The “cake” billiard discussed in that paper is one of the examples, and other examples without corners are also given. These examples show that the conjecture cannot hold in the form stated above, but a rephrasing of the basic idea leads to the following relation between the inside problem and the scattering problem. We consider domains which are simply connected1, with a boundary which is piecewise In fact, the domain can also have several pieces, provided that the complement of Ω is connected.