Spectral statistics for quantum graphs: periodic orbits and combinatorics

ISSN： 1463-6417

Source： Philosophical Magazine B, Vol.80, Iss.12, 2000-12, pp. : 1999-2021

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Abstract

We consider the Schrödinger operator on graphs and study the spectral statistics of a unitary operator which represents the quantum evolution, or a quantum map on the graph. This operator is the quantum analogue of the classical evolution operator of the corresponding classical dynamics on the same graph. We derive a trace formula, which expresses the spectral density of the quantum operator in terms of periodic orbits on the graph, and show that one can reduce the computation of the two-point spectral correlation function to a well defined combinatorial problem. We illustrate this approach by considering an ensemble of simple graphs. We prove by a direct computation that the two-point correlation function coincides with the circular unitary ensemble expression for 2 x 2 matrices. We derive the same result using the periodic orbit approach in its combinatorial guise. This involves the use of advanced combinatorial techniques which we explain.