Spectral Statistics of "Cellular" Billiards

TL;DR

This paper investigates the spectral properties of quantum billiards formed by patching multiple copies of a domain with flat boundary pieces, revealing how classical chaos and symmetry influence quantum energy level statistics.

Contribution

It introduces a framework linking classical periodic orbit degeneracies to quantum spectral components, deriving semiclassical trace formulas and classifying spectral statistics by symmetry type.

Findings

Spectral degeneracies are governed by a matrix group G.

Quantum spectra split into uncorrelated subspectra corresponding to group representations.

Spectral statistics follow standard Random Matrix ensembles depending on symmetry.

Abstract

For a bounded planar domain $\Omega^0$ whose boundary contains a number of flat pieces $\Gamma_i$ we consider a family of non-symmetric billiards $\Omega$ constructed by patching several copies of $\Omega^0$ along $\Gamma_i$'s. It is demonstrated that the length spectrum of the periodic orbits in $\Omega$ is degenerate with the multiplicities determined by a matrix group $G$. We study the energy spectrum of the corresponding quantum billiard problem in $\Omega$ and show that it can be split in a number of uncorrelated subspectra corresponding to a set of irreducible representations $\alpha$ of $G$. Assuming that the classical dynamics in $\Omega^0$ are chaotic, we derive a semiclassical trace formula for each spectral component and show that their energy level statistics are the same as in standard Random Matrix ensembles. Depending on whether ${\alpha}$ is real, pseudo-real or complex, the spectrum has either Gaussian Orthogonal, Gaussian Symplectic or Gaussian Unitary types of statistics, respectively.