Formation of superscar waves in plane polygonal billiards

Abstract

Polygonal billiards constitute a special class ofmodels. Though they have zero Lyapunov exponent their classical and quantum properties are involved due to scattering on singular verticeswith angles¹p n with integer n. It is demonstrated that in the semiclassical limit the multiple singular scattering on such vertices, when optical boundaries ofmany scatters overlap, leads to the vanishing of quantumwave functions along straight lines built by these scatters. This phenomenon has an especially important consequence for polygonal billiards where periodic orbits (when they exist) formpencils of parallel rays restricted from the both sides by singular vertices.Due to the singular scattering on boundary vertices,waves propagated inside a periodic orbit pencil tend in the semiclassical limit to zero along pencil boundaries thus forming weakly interacting quasi-modes.Contrary to scars in chaotic systems, the discussed quasi-modes in polygonal billiards become almost exact for high-excited states and for brevity they are designated as superscars. Manypictures of eigenfunctions for a triangular billiard and a barrier billiardwhich have clear superscar structures are presented in the paper. Special attention is given to the development of quantitativemethods of detecting and analysing such superscars. In particular, it is demonstrated that the overlap between superscarwaves associated with afixed periodic orbit and eigenfunctions of a barrier billiard is distributed according to the Breit-Wigner distribution typical for weakly interacting quasi-modes. For special subclass of rational polygonal billiards called Veech polygonswhere all periodic orbits can be calculated analytically it is argued and checked numerically that their eigenfunctions are fractal in the Fourier space.