Quantum Graphology

T. Kottos Max-Planck-Institute for Dynamics and Self-Organization, 37073 Göttingen, Germany

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Received: 19 05 2005;

We review quantum chaos on graphs. We construct a unitary operator which represents the quantum evolution on the graph and study its spectral and wave function statistics. This operator is the analogue of the classical evolution operator on the graph. It allows us to establish a connection between the corresponding periodic orbits and the statistical properties of eigenvalues and eigenfunctions. Specifically, for the energy-averaged spectral form factor we derived an exact combinatorial expression which illustrate the role of correlations between families of isometric orbits. We also show that enhanced wave function localization due to the presence of short unstable periodic orbits and strong scarring can rely on completely different mechanisms

DOI: 10.12693/APhysPolA.109.7 PACS numbers:05.45.Mt, 03.65.Sq