Periodic orbit theory and the statistical analysis of scaling quantum graph spectra

The explicit solution to the spectral problem of quantum graphs found recently by Dabaghian and Blümel [Phys. Rev. E 68, 055201(R) (2003); 70, 046206 (2004); JETP Lett. 77, 530 (2003)] is used to produce an exact periodic orbit theory description for the probability distributions of spectral statistics, including the distribution for the nearest neighbor separations $s_n=k_n-k_{n-1}$, and the distribution of the spectral oscillations around the average, $\delta k_n=k_n-{\overline{k}}_n$.

Figure 1

(Color online) Continuous function $f\left(k\right)$ that pierces every stair step of the spectral staircase $N\left(k\right)$ of a four-vertex linear chain graph. The intersection points ${\widehat{k}}_n^{\left(1\right)}$ separate the momentum eigenvalues ${\widehat{k}}_n^{\left(0\right)}$.

Figure 2

(Color online) Histogram of the fluctuations of the roots of the six-star graph of irregularity degree 5 and of the following five separating sequences ${\delta }_n^{\left(0\right)},\dots ,{\delta }_n^{\left(5\right)}$. The initial spread of the fluctuations ${\delta }_n^{\left(0\right)}$ becomes progressively narrower for higher values of $j$.

Figure 3

(Color online) Bootstrapping of the spectral staircase (dotted line) by the $N^{\left(1\right)}$ staircase (dash-dotted line), bootstrapped in turn by the $N^{\left(2\right)}$ staircase (solid line) intersected by the Weyl average (straight line). Note that the $N^{\left(2\right)}$ staircase does not bootstrap the $N^{\left(0\right)}$. These graphs were obtained for the dressed four-vertex linear chain graph [18].

Figure 4

(Color online) Histogram of the nearest neighbor separations $s_n^{\left(0\right)}$ (bottom distribution), $s_n^{\left(1\right)}$ and $s_n^{\left(2\right)}$ (top distribution) for the fully connected quadrangle graph (top right corner) of irregularity degree 3, obtained for 80 000 roots of the corresponding spectral equations. The maximal nearest neighbor separation for $j=0$ in this case is $s_{\mathit{max}}=8.68$, while the regular cell size is $\pi ∕S_0=2.28$.

Figure 5

(Color online) Histogram of the nearest neighbor average for the even levels of the hierarchy of the fully connected quadrangle graph of irregularity degree 7, obtained for 80 000 roots of the corresponding spectral equations. The solid line in the background of each histogram represents a Gaussian fit with $\Delta$ given by the corresponding sum (75).