Trace formulae for three-dimensional hyperbolic lattices and application to a strongly chaotic tetrahedral billiard

Abstract

This paper is devoted to the quantum chaology of three-dimensional systems. A trace formula is derived for compact polyhedral billiards which tessellate the three-dimensional hyperbolic space of constant negative curvature. The exact trace formula is compared with Gutzwiller's semiclassical periodic-orbit theory in three dimensions, and applied to a tetrahedral billiard being strongly chaotic. Geometric properties as well as the conjugacy classes of the defining group are discussed. The length spectrum and the quantal level spectrum are numerically computed allowing the evaluation of the trace formula as is demonstrated in the case of the spectral staircase N(E), which in turn is successfully applied in a quantization condition.

32 pages, compressed with gzip / uuencode

Cited by in corpus (15)

• CMB Anisotropy of the Poincare Dodecahedron
• Eigenmodes of 3-dimensional spherical spaces and their application to cosmology
• Simulating Cosmic Microwave Background maps in multi-connected spaces
• CMB Anisotropy in Compact Hyperbolic Universes I: Computing Correlation Functions
• Symbolic Dynamics and Periodic Orbits for the Cardioid Billiard
• The Cosmic Microwave Background for a Nearly Flat Compact Hyperbolic Universe
• Billiard Systems in Three Dimensions: The Boundary Integral Equation and the Trace Formula
• The Fluctuations of the Cosmic Microwave Background for a Compact Hyperbolic Universe
• On the Accuracy of the Semiclassical Trace Formula
• Numerical Study of Length Spectra and Low-lying Eigenvalue Spectra of Compact Hyperbolic 3-manifolds
• Topology of the universe from COBE-DMR; a wavelet approach
• Quantum cat maps with spin 1/2
• Classical and quantum ergodicity on orbifolds
• Distributions of length multiplicities for negatively curved locally symmetric Riemannian manifolds
• The quantum three-dimensional Sinai billiard - a semiclassical analysis