Spectral Statistics of "Cellular" Billiards

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.

18 pages, 4 figures

References in corpus (5)

• Semiclassical Foundation of Universality in Quantum Chaos
• Periodic-Orbit Theory of Universality in Quantum Chaos
• Hearing shapes of drums - mathematical and physical aspects of isospectrality
• Expanded boundary integral method and chaotic time-reversal doublets in quantum billiards
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Cited by in corpus (3)

• GSE statistics without spin
• Semiclassical approach to discrete symmetries in quantum chaos
• GSE spectra in uni-directional quantum systems