Quantum chaos and quantum graphs

2018 ◽  
Author(s):  
Jac Verbaarschot
Keyword(s):  
Periodic Orbit ◽  
Quantum Chaos ◽  
Sigma Model ◽  
Chaotic Systems ◽  
Semiclassical Approximation ◽  
Quantum Evolution ◽  
Quantum Graphs ◽  
Periodic Orbit Theory ◽  
Spectral Statistics ◽  
Orbit Theory

This article examines the origins of the universality of the spectral statistics of quantum chaotic systems in the context of periodic orbit theory. It also considers interesting analogies between periodic orbit theory and the sigma model, along with related work on quantum graphs. The article first reviews some facts and definitions for classically chaotic systems in order to elucidate their quantum behaviour, focusing on systems with two degrees of freedom: one characterized by ergodicity and another by hyperbolicity. It then describes two semiclassical approximation techniques — Gutzwiller’s periodic orbit theory and a refined approach incorporating the unitarity of the quantum evolution — and highlights their importance in understanding universal spectral statistics, and how they are related to the sigma model. This is followed by an analysis of parallel developments for quantum graphs, which are relevant to quantum chaos.

scholarly journals Periodic-orbit theory of universal level correlations in quantum chaos

2009 ◽  
Vol 11 (10) ◽  
pp. 103025 ◽  
Author(s):  
Sebastian Müller ◽  
Stefan Heusler ◽  
Alexander Altland ◽  
Petr Braun ◽  
Fritz Haake
Keyword(s):  
Periodic Orbit ◽  
Quantum Chaos ◽  
Periodic Orbit Theory ◽  
Orbit Theory

QUANTIZATION RULES FOR STRONGLY CHAOTIC SYSTEMS

1992 ◽  
Vol 06 (27) ◽  
pp. 1691-1719 ◽  
Author(s):  
R. AURICH ◽  
J. BOLTE
Keyword(s):  
Periodic Orbit ◽  
Chaotic Systems ◽  
Level Density ◽  
Quantization Condition ◽  
Periodic Orbit Theory ◽  
Symmetry Classes ◽  
Dynamical Zeta Function ◽  
The Mean ◽  
Orbit Theory ◽  
Quantization Rules

We discuss the quantization of strongly chaotic systems and apply several quantization rules to a model system given by the unconstrained motion of a particle on a compact surface of constant negative Gaussian curvature. We study the periodic-orbit theory for distinct symmetry classes corresponding to a parity operation which is always present when such a surface has genus two. Recently, several quantization rules based on periodic orbit theory have been introduced. We compare quantizations using the dynamical zeta function Z(s) with the quantization condition [Formula: see text] where a periodic-orbit expression for the spectral staircase N(E) is used. A general discussion of the efficiency of periodic-orbit quantization then allows us to compare the different methods. The system dependence of the efficiency, which is determined by the topological entropy τ and the mean level density [Formula: see text], is emphasized.

scholarly journals Periodic-orbit theory of universality in quantum chaos

2005 ◽  
Vol 72 (4) ◽  
Author(s):  
Sebastian Müller ◽  
Stefan Heusler ◽  
Petr Braun ◽  
Fritz Haake ◽  
Alexander Altland
Keyword(s):  
Periodic Orbit ◽  
Quantum Chaos ◽  
Periodic Orbit Theory ◽  
Orbit Theory
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