Spectral statistics of nearly unidirectional quantum graphs
Abstract
The energy levels of a quantum graph with time reversal symmetry and unidirectional classical dynamics are doubly degenerate and obey the spectral statistics of the Gaussian Unitary Ensemble. These degeneracies, however, are lifted when the unidirectionality is broken in one of the graph's vertices by a singular perturbation. Based on a Random Matrix model we derive an analytic expression for the nearest neighbour distribution between energy levels of such systems. As we demonstrate the result agrees excellently with the actual statistics for graphs with a uniform distribution of eigenfunctions. Yet, it exhibits quite substantial deviations for classes of graphs which show strong scarring.
Cite
@article{arxiv.1503.01342,
title = {Spectral statistics of nearly unidirectional quantum graphs},
author = {Maram Akila and Boris Gutkin},
journal= {arXiv preprint arXiv:1503.01342},
year = {2015}
}
Comments
23 pages, 13 figures (changes: more details on the derivation of $F_{k,l}$ added, minor corrections)