# Non-Weyl Microwave Graphs

## TL;DR

This paper demonstrates experimentally that certain microwave networks, modeled as quantum graphs, can deviate from the classical Weyl law for resonance density, especially when balanced vertices are introduced, confirming theoretical predictions.

## Contribution

It provides experimental evidence of non-Weyl quantum graphs and identifies the role of balanced vertices in this deviation, extending the understanding of resonance behavior in quantum graph models.

## Key findings

- Existence of non-Weyl graphs confirmed experimentally.
- Transition from Weyl to non-Weyl occurs with balanced vertices.
- Experimental results align with theoretical predictions and numerical calculations.

## 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.