Formation of superscar waves in plane polygonal billiards

TL;DR

This paper investigates the formation of superscar waves in polygonal billiards, revealing how singular vertices cause quantum wave functions to vanish along certain lines, leading to quasi-modes called superscars that are nearly exact at high energies.

Contribution

It introduces the concept of superscars in polygonal billiards, demonstrating their formation due to singular scattering and developing methods to detect and analyze these quasi-modes.

Findings

Superscars form along periodic orbit pencils in polygonal billiards.

Overlap between superscar waves and eigenfunctions follows Breit-Wigner distribution.

Eigenfunctions in Veech polygons exhibit fractal structures in Fourier space.

Abstract

Polygonal billiards constitute a special class of models. Though they have zero Lyapunov exponent their classical and quantum properties are involved due to scattering on singular vertices. It is demonstrated that in the semiclassical limit multiple singular scattering on such vertices when optical boundaries of many scatters overlap leads to vanishing of quantum wave functions along straight lines built by these scatters. This phenomenon has an especially important consequence for polygonal billiards where periodic orbits (when they exist) form pencils of parallel rays restricted from the both sides by singular vertices. Due to singular scattering on boundary vertices, waves propagated inside periodic orbit pencils in the semiclassical limit tend 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. Many pictures of eigenfunctions for a triangular billiard and a barrier billiard which have clear superscar structures are presented in the paper. Special attention is given to the development of quantitative methods of detecting and analysing such superscars. In particular, it is demonstrated that the overlap between superscar waves associated with a fixed periodic orbit and eigenfunctions of a barrier billiard is distributed according to the Breit-Wigner distribution typical for weakly interacting quasi-modes (or doorway states). For special sub-class of rational polygonal billiards called Veech polygons where all periodic orbits can be calculated analytically it is argued and checked numerically that their eigenfunctions are fractal in the Fourier space.