Periodic Orbit Theory and Spectral Statistics for Quantum Graphs

Details Periodic Orbit Theory and Spectral Statistics for Quantum Graphs Periodic Orbit Theory and Spectral Statistics for Quantum Graphs

1998-11-29

arxiv.org/abs/chao-dyn/9812005v2

Physics

Condensed Matter

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

Scientific paper

10.1006/aphy.1999.5904

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.

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