Wave Functions and Energy Spectra in Rational Billiards Are Determined Completely by Their Periods

TL;DR

This paper demonstrates that the energy spectra and stationary solutions of quantum billiards with rational polygon shapes are fully determined by the periods of their associated Riemann surfaces, providing a new method for solving Schrödinger's equation in these systems.

Contribution

It introduces a novel approach linking the energy spectra of rational billiards to the periods of their Riemann surfaces, enabling explicit construction of solutions for certain classes of billiards.

Findings

Energy spectra depend on 2g independent periods of RBRS.

Stationary solutions can be extended to the entire RBRS and constructed from pre-solutions.

Explicit construction of solutions is possible for RB with decomposable EPPs.

Abstract

The rational billiards (RB) are classically pseudointegrable, i.e. their trajectories in the phase space lie on multi-tori. Each such a multi-torus can be unfolded into elementary polygon pattern (EPP). A rational billiards Riemann surface (RBRS) corresponding to each RB is then an infinite mosaic made by a periodic distribution of EPP. Periods of RBRS are directly related to periodic orbits of RB. It is shown that any stationary solutions (SS) to the Schr\"odinger equation (SE) in RB can be extended on the whole RBRS. The extended stationary wave functions (ESS) are then periodic on RBRS with its periods. Conversely, for each system of boundary conditions (i.e. the Dirichlet or the the Neumann ones or their mixture) consistent with EPP one can find so called stationary pre-solutions (SPS) of the Schr\"odinger equation defined on RBRS and respecting its periodic structure together with their energy spectra. Using SPS one can easily construct SS of RB for most boundary conditions on it by a trivial algebra over SPS. It proves therefore that the energy spectra defined by the boundary conditions for SS corresponding to each RB are totally determined by $2g$ independent periods of RBRS being homogeneous functions of these periods. RBRS can be constructed exclusively due to the rationality of the polygon billiards considered. Therefore the approach developed in the present paper can be seen as a new way in obtaining SS to SE in RB. SPS can be constructed explicitly for a class of RB which EPP can be decomposed into a set of periodic orbit channel (POC) parallel to each other (POCDRB). For such a class of RB the respective RBRS can be built as a standard multi-sheeted Riemann surface with a periodic structure. For POCDRB a discussion of the existence of the superscar states (SSS) can be done thoroughly.