A spectral duality in graphs and microwave networks

TL;DR

This paper explores the spectral properties of quantum graphs and microwave networks, revealing a duality and interlacing phenomena between Neumann and Dirichlet boundary conditions, with experimental and numerical validation.

Contribution

It uncovers a spectral duality and interlacing between Neumann and Dirichlet spectra in quantum graphs and microwave networks, supported by experiments and numerical analysis.

Findings

Neumann and Dirichlet eigenvalues alternate with wave number

Neumann spectrum shows local random matrix theory behavior

Green's function exhibits interlacing spectral features

Abstract

Quantum graphs and their experimental counterparts, microwave networks, are ideally suited to study the spectral statistics of chaotic systems. The graph spectrum is obtained from the zeros of a secular determinant derived from energy and charge conservation. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The first ones are realized in experiments, since the standard junctions connecting the bonds obey Neumann boundary conditions due to current conservation. On average, the corresponding Neumann and Dirichlet eigenvalues alternate as a function of the wave number, with the consequence that the Neumann spectrum is described by random matrix theory only locally, but adopts features of the interlacing Dirichlet spectrum for long-range correlations. Another spectral interlacing is found for the Green's function, which in contrast to the secular determinant is experimentally accessible. This is illustrated by microwave studies and numerics.