Semiclassical wave functions and energy spectra in polygon billiards

TL;DR

This paper develops a semiclassical quantization scheme for polygon billiards using wave functions, linking periodic structures on Riemann surfaces to energy spectra, and extends the approach to irrational polygons.

Contribution

It introduces a wave function formalism for semiclassical quantization in polygon billiards, including rational and irrational cases, with exact solutions for doubly rational polygons.

Findings

Semiclassical wave functions are exact for doubly rational polygons.

Periodic structures on Riemann surfaces determine energy spectra.

The formalism extends to irrational polygons via approximation.

Abstract

A consistent scheme of semiclassical quantization in polygon billiards by wave function formalism is presented. It is argued that it is in the spirit of the semiclassical wave function formalism to make necessary rationalization of respective quantities accompanied the procedure of the semiclassical quantization in polygon billiards. Unfolding rational polygon billiards (RPB) into corresponding Riemann surfaces (RS) periodic structures of the latter are demonstrated with 2g independent periods on the respective multitori with g as their genuses. However it is the two dimensional real space of the real linear combinations of these periods which is used for quantizing RPB. A class of doubly rational polygon billiards (DRPB) is distinguished for which these real linear relations are rational and their semiclassical quantization by wave function formalism is presented. It is shown that semiclassical quantization of both the classical momenta and the energy spectra are determined completely by periodic structure of the corresponding RS. Each RS is then reduced to elementary polygon patterns (EPP) as its basic periodic elements. Each such EPP can be glued to a torus of genus g. Semiclassical wave functions (SWF) are then constructed on EPP. The SWF for DRPB appear to be exact. They satisfy the Dirichlet, the Neumannn or the mixed boundary conditions. Not every mixing is allowed however and a respective incompleteness of SWF is discussed. Dens families of DRPB are used for approximate semiclassical quantization of RPB. General rational polygons are quantized by approximating them by DRPB. An extension of the formalism to irrational polygons is described as well. The semiclassical approximations constructed in the paper are controlled by general criteria of the eigenvalue theory. A relation between the superscar solutions and SWF constructed in the paper is also discussed.