Hul Oleh, Sirko Leszek
Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Jun;83(6 Pt 2):066204. doi: 10.1103/PhysRevE.83.066204. Epub 2011 Jun 10.
The parameter-dependent spectral statistics of totally connected quantum graphs with n = 4-30 vertices, such as the parametric velocities correlation functions and the distribution of curvatures, are studied. The inverse participation ratio (IPR), an important measure of localization effects, was also numerically investigated. In the calculations, we successfully used two different theoretical approaches. The first approach was based on the graphs' eigenenergies and wave functions calculations, while the second one used the eigenphases and the eigenvectors of the bond scattering matrix S(k). We considered graphs with Neumann and circular orthogonal ensemble (COE) boundary conditions. We show that in contrast to large Neumann graphs, for which the departure of many parameter-dependent spectral statistics from the random matrix theory (RMT) predictions is observed, for large COE graphs, the spectral statistics and IPR are in good agreement with the RMT predictions.
研究了具有(n = 4 - 30)个顶点的完全连通量子图的参数相关谱统计，如参数速度相关函数和曲率分布。还对作为局域化效应重要度量的逆参与率（IPR）进行了数值研究。在计算中，我们成功使用了两种不同的理论方法。第一种方法基于图的本征能量和波函数计算，而第二种方法使用键散射矩阵(S(k))的本征相位和本征向量。我们考虑了具有诺伊曼边界条件和圆正交系综（COE）边界条件的图。我们表明，与观察到许多参数相关谱统计偏离随机矩阵理论（RMT）预测的大型诺伊曼图不同，对于大型COE图，谱统计和IPR与RMT预测高度吻合。
