Lu Junjie, Hofmann Tobias, Kuhl Ulrich, Stöckmann Hans-Jürgen
Institut de Physique de Nice, CNRS, Université Côte d'Azur, 06108 Nice, France.
Fachbereich Physik, Philipps-Universität Marburg, 35032 Marburg, Germany.
Entropy (Basel). 2023 Jan 4;25(1):109. doi: 10.3390/e25010109.
Quantum graphs are ideally suited to studying the spectral statistics of chaotic systems. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The latter ones correspond to totally disassembled graphs with a spectrum being the superposition of the spectra of the individual bonds. According to the interlacing theorem, Neumann and Dirichlet eigenvalues on average alternate as a function of the wave number, with the consequence that the Neumann spectral statistics deviate from random matrix predictions. There is, e.g., a strict upper bound for the spacing of neighboring Neumann eigenvalues given by the number of bonds (in units of the mean level spacing). Here, we present analytic expressions for level spacing distribution and number variance for ensemble averaged spectra of Dirichlet graphs in dependence of the bond number, and compare them with numerical results. For a number of small Neumann graphs, numerical results for the same quantities are shown, and their deviations from random matrix predictions are discussed.
量子图非常适合用于研究混沌系统的谱统计。根据顶点处的边界条件，存在诺伊曼图和狄利克雷图。后者对应于完全拆解的图，其谱是各个键的谱的叠加。根据交错定理，诺伊曼和狄利克雷特征值平均而言作为波数的函数交替出现，结果是诺伊曼谱统计偏离随机矩阵预测。例如，相邻诺伊曼特征值的间距存在一个由键的数量给出的严格上限（以平均能级间距为单位）。在此，我们给出了狄利克雷图的系综平均谱的能级间距分布和数方差作为键数函数的解析表达式，并将它们与数值结果进行比较。对于一些小的诺伊曼图，展示了相同量的数值结果，并讨论了它们与随机矩阵预测的偏差。
