Select Page

# 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.