Implications of Spectral Interlacing for Quantum Graphs

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.

Automated summary

Quantum graphs are useful for studying the spectral statistics of chaotic systems. Neumann and Dirichlet graphs have different boundary conditions at the vertices. The Neumann spectral statistics deviate from random matrix predictions due to the interlacing theorem. We provide analytic expressions for level spacing distribution and number variance of ensemble averaged spectra of Dirichlet graphs, and compare them with numerical results. The deviations of numerical results for small Neumann graphs from random matrix predictions are also discussed.