Correlations between Periodic Orbits and their R?le in Spectral Statistics

2001/01/01 by Martin Sieber, Klaus Richter · 329 citations
Chemistry · Mathematics · Physics and Astronomy · #Chaotic #Chaotic systems #Classical mechanics #Diagonal #Eigenvalues and eigenvectors #Function (biology) #Mathematics #Matrix (chemical analysis) #Molecular spectroscopy and chirality #Order (exchange) #Periodic orbits #Physics #Quantum #Quantum chaos #Quantum chaos and dynamical systems #Quantum dynamics #Quantum mechanics #Random Matrices and Applications #Random matrix #Semiclassical physics #Statistical physics

paper · doi:10.1238/physica.topical.090a00128

published in Physica Scripta T90(1), 128 (IOP Publishing)

openalex publication_date 2001/01/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06

Abstract

We consider off-diagonal contributions to double sums over periodic orbits that arise in semiclassical approximations for spectral statistics of classically chaotic quantum systems. We identify pairs of periodic orbits whose actions are strongly correlated. For a class of systems with uniformly hyperbolic dynamics, we demonstrate that these pairs of orbits give rise to a τ 2 contribution to the spectral form factor K (τ) which agrees with random matrix theory. Most interestingly, this contribution has its origin in a next-to-leading-order behaviour of a classical distribution function for long times.

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