Spectral statistics in the ensemble of quantum graphs with permuted edge -lengths and its random-matrix theory analogue

by | Nov 21, 2018 | Seminar Talks

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 \mathcal{G} (V, E; \mathcal{L}) with  V vertices  E edges, and a list of rationally independent edge lengths   \mathcal{L} = (L_1, ..., L_E), is defined topologically in terms of its  V dimensional adjacency matrix, and metrically by endowing the edges with the standard metric and edge lengths  \mathcal{L}. The associated Schrödinger operator consists of the one dimensional Lapalcian with appropriate boundary conditions. Its ordered spectrum   \big\{k_n\big\}^{\infty}_{n=1}, k_{n + 1} \geq k_n is specified by the counting function  \mathcal{N}_0 = \sharp \big\{\ k_n : k_n \leq k \big\}. Permute the lengths of  t \leq E edges in   \mathcal{L}. The spectrum is changed and its counting function is denoted by  \mathcal{N}_t(k). We measure the difference between the two spectra by the variance

\Delta t = \lim_{K \to \infty} \frac{1}{K} \int_{0}^{K[} dk [N_t(k) - N_0(k)]^2 while \lim_{K \to \infty} \frac{1}{K} \int_{0}^{K[} dk [N_t(k) - N_0(k)] = 0

We study the averaged variance over the different permutations of  t lengths,  \langle \Delta t \rangle, and its dependence on the connectivity of \mathcal{G} (V, E; \mathcal{L}). Similarly, given  N x N Hermitian matrix H, its ordered spectrum  \big\{x_n\big\}^{N}_{n=1}, x_{n + 1} \geq x_n and spectral counting function \mathcal{N}_0 = \sharp \big\{\ x_{n+1} : x_n \leq x \big\}. Permuting  t of its diagonal elements, the spectrum is changed, and its counting function is denoted by \mathcal{N}_t. Again the difference between the spectra is measured by the variance

\Delta t = \int dk [N_t(k) - N_0(k)]^2  - \Big( \int dk [N_t(k) - N_0(k)] \Big)^2

We study  \langle \Delta t \rangle: the averaged variance over the ensemble of matrices from which  H is chosen, and its dependence on t 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.