Spectral statistics in the ensemble of quantum graphs with permuted edge -lengths and its random-matrix theory analogue
Speaker: Uzy Smilansky– (Weizmann Institute of Science)
Title: Spectral statistics in the ensemble of quantum graphs with permuted edge-lengths and its random-matrix theory analogue
Abstract: A quantum graph with vertices edges, and a list of rationally independent edge lengths , is defined topologically in terms of its dimensional adjacency matrix, and metrically by endowing the edges with the standard metric and edge lengths . The associated Schrödinger operator consists of the one dimensional Lapalcian with appropriate boundary conditions. Its ordered spectrum is specified by the counting function . Permute the lengths of edges in . The spectrum is changed and its counting function is denoted by . We measure the difference between the two spectra by the variance
We study the averaged variance over the different permutations of lengths, , and its dependence on the connectivity of . Similarly, given Hermitian matrix H, its ordered spectrum and spectral counting function . Permuting of its diagonal elements, the spectrum is changed, and its counting function is denoted by . Again the difference between the spectra is measured by the variance
We study : the averaged variance over the ensemble of matrices from which is chosen, and its dependence on for various matrix ensembles.
Finally, we compare the mean variances for the graphs and the matrix ensembles and discuss the case where they are not in agreement.