Periodic Orbit Theory and Spectral Statistics for Quantum Graphs

1998/12/28 by Tsampikos Kottos, Uzy Smilansky · 32 citations
Computer Science · Physics and Astronomy · #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems #Quantum many-body systems #chao-dyn #cond-mat #nlin.CD

paper · pdf · doi:10.1006/aphy.1999.5904

37 pages, 20 figures, other comments, accepted for publication in the Annals of Physics

arxiv created 1998/12/28 · openalex publication_date 1999/05/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We quantize graphs (networks) which consist of a finite number of bonds and vertices. We show that the spectral statistics of fully connected graphs is well reproduced by random matrix theory. We also define a classical phase space for the graphs, where the dynamics is mixing and the periodic orbits proliferate exponentially. An exact trace formula for the quantum spectrum is developed in terms of the same periodic orbits, and it is used to investigate the origin of the connection between random matrix theory and the underlying chaotic classical dynamics. Being an exact theory, and due to its relative simplicity, it offers new insights into this problem which is at the fore-front of the research in Quantum Chaos and related fields.

Citations

• Semiclassical theory of spectral rigidity

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