Barrier billiard and random matrices

TL;DR

This paper investigates the quantum spectral properties of barrier billiards, showing that their statistics are insensitive to barrier height and follow semi-Poisson distributions, using a novel random matrix approach.

Contribution

It introduces a new random matrix model for barrier billiards and demonstrates its spectral statistics are universal and described by semi-Poisson distributions.

Findings

Spectral statistics are insensitive to barrier height.

Spectral distributions follow semi-Poisson statistics.

A new random matrix model effectively describes the quantum properties.

Abstract

The barrier billiard is the simplest example of pseudo-integrable models with interesting and intricate classical and quantum properties. Using the Wiener-Hopf method it is demonstrated that quantum mechanics of a rectangular billiard with a barrier in the centre can be reduced to the investigation of a certain unitary matrix. Under heuristic assumptions this matrix is substituted by a special low-complexity random unitary matrix of independent interest. The main results of the paper are (i) spectral statistics of such billiards is insensitive to the barrier height and (ii) it is well described by the semi-Poisson distributions.