Non-Weyl Microwave Graphs

Abstract

One of the most important characteristics of a quantum graph is the average density of resonances, $\rho =\frac{\mathcal{L}}{\pi }$, where $\mathcal{L}$ denotes the length of the graph. This is a very robust measure. It does not depend on the number of vertices in a graph and holds also for most of the boundary conditions at the vertices. Graphs obeying this characteristic are called Weyl graphs. Using microwave networks which simulate quantum graphs we show that there exist graphs which do not adhere to this characteristic. Such graphs will be called non-Weyl graphs. For standard coupling conditions we demonstrate that the transition from a Weyl graph to a non-Weyl graph occurs if we introduce a balanced vertex. A vertex of a graph is called balanced if the numbers of infinite leads and internal edges meeting at a vertex are the same. Our experimental results confirm the theoretical predictions of [E. B. Davies and A. Pushnitski, Analysis and PDE 4, 729 (2011)] and are in excellent agreement with the numerical calculations yielding the resonances of the networks.

12 pages, 2 figures

References in corpus (4)

• Are Scattering Properties of Graphs Uniquely Connected to Their Shapes?
• Power spectrum analysis and missing level statistics of microwave graphs with violated time reversal invariance
• Experimental and numerical investigation of the reflection coefficient and the distributions of Wigner's reaction matrix for irregular graphs with absorption
• Experimental investigation of Wigner's reaction matrix for irregular graphs with absorption

Cited by in corpus (1)

• Hearing Euler characteristic of graphs