Partner orbits and action differences on compact factors of the hyperbolic plane. Part I: Sieber-Richter pairs

Abstract

Physicists have argued that periodic orbit bunching leads to universal spectral fluctuations for chaotic quantum systems. To establish a more detailed mathematical understanding of this fact, it is first necessary to look more closely at the classical side of the problem and determine orbit pairs consisting of orbits which have similar actions. In this paper we specialize to the geodesic flow on compact factors of the hyperbolic plane as a classical chaotic system. We prove the existence of a periodic partner orbit for a given periodic orbit which has a small-angle self-crossing in configuration space which is a `2-encounter'; such configurations are called `Sieber-Richter pairs' in the physics literature. Furthermore, we derive an estimate for the action difference of the partners. In the second part of this paper [13], an inductive argument is provided to deal with higher-order encounters.

to appear on Nonlinearity

References in corpus (4)

• Semiclassical Foundation of Universality in Quantum Chaos
• Periodic-Orbit Theory of Universality in Quantum Chaos
• Semiclassical form factor for spectral and matrix element fluctuations of multi-dimensional chaotic systems
• Statistics of Self-Crossings and Avoided Crossings of Periodic Orbits in the Hadamard-Gutzwiller Model