Skeletons and superscars

TL;DR

This paper develops a semiclassical wave function framework based on skeletons in billiards, unifying integrable, pseudointegrable, and chaotic cases, and explains the nature of superscars as singular solutions.

Contribution

It introduces a skeleton-based approach to construct semiclassical wave functions in billiards, encompassing integrable, pseudointegrable, and chaotic systems, and clarifies the origin of superscars.

Findings

Skeletons can be closed or open, with different topologies.

Semiclassical wave functions are constructed on these skeletons.

Superscars are identified as singular solutions related to skeletons.

Abstract

Semiclassical wave functions in billiards based on the Maslov-Fedoriuk approach are constructed. They are defined on classical constructions called skeletons which are the billiards generalization of the Arnold tori. Skeletons in the rational polygon billiards considered in the phase space can be closed with a definite genus or can be open being a cylinder-like or Moebius-like bands. The skeleton formulation is applied to calculate semiclassical wave functions and the corresponding energy spectra for the integrable and pseudointegrable billiards as well as in the limiting forms in some cases of chaotic ones. The superscars of Bogomolny and Schmit are shown to be simply singular semiclassical solutions of the eigenvalue problem in the billiards well built on the singular skeletons in the billiards with flat boundaries in both the integrable and the pseudointegrable billiards as well as in the chaotic cases of such billiards.