Spectral Statistics of "Cellular" Billiards

Abstract

For a bounded planar domain $Ω^0$ whose boundary contains a number of flat pieces $Γ_i$ we consider a family of non-symmetric billiards $Ω$ constructed by patching several copies of $Ω^0$ along $Γ_i$'s. It is demonstrated that the length spectrum of the periodic orbits in $Ω$ is degenerate with the multiplicities determined by a matrix group $G$. We study the energy spectrum of the corresponding quantum billiard problem in $Ω$ and show that it can be split in a number of uncorrelated subspectra corresponding to a set of irreducible representations $α$ of $G$. Assuming that the classical dynamics in $Ω^0$ 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 $α$ is real, pseudo-real or complex, the spectrum has either Gaussian Orthogonal, Gaussian Symplectic or Gaussian Unitary types of statistics, respectively.