Non-Weyl Microwave Graphs

One of the most important characteristics of a quantum graph is the average density of resonances, $\rho =(\mathcal{L}/\pi )$, 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 that simulate quantum graphs we show that there exist graphs that do not adhere to this characteristic. Such graphs are 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.

Figure 1

Panels (a) and (b) show the schemes of Weyl and non-Weyl graphs, respectively. Both graphs are characterized by the same length $\mathcal{L}=\sum _{i=1}^7{\ell }_i$. The Weyl graph contains two unbalanced vertices containing infinite leads $L_1^{\infty }$ and $L_2^{\infty }$. The non-Weyl graph contains one balanced vertex with two attached leads $L_1^{\infty }$ and $L_2^{\infty }$. The leads are marked with red broken lines. Panels (c) and (d) show the corresponding Weyl and non-Weyl microwave networks constructed from microwave coaxial cables and joints. The optical lengths of the networks are the same and are equal to $\mathcal{L}$. The microwave networks are connected to the vector network analyzer (VNA) with the two elastic microwave cables, which is equivalent to attaching two infinite leads $L_1^{\infty }$ and $L_2^{\infty }$ to quantum graphs.

Figure 2

Panels (a) and (c) show the modulus of the determinant of the scattering matrix $|\det [\widehat{S}(\nu )]|$ of the experimentally studied Weyl networks ${\mathrm{W}}_1$ and ${\mathrm{W}}_2$ (full lines) with the lengths ${\mathcal{L}}_1=0.999\mathrm{m}$ and ${\mathcal{L}}_2=1.151\mathrm{m}$, respectively, containing two unbalanced vertices in the frequency range 0.3–2.2 GHz. Panels (b) and (d) show the modulus of the determinant of the scattering matrix $|\det [\widehat{S}(\nu )]|$ of the experimentally studied non-Weyl networks ${\mathrm{NW}}_1$ and ${\mathrm{NW}}_2$ with the same lengths as the networks ${\mathrm{W}}_1$ and ${\mathrm{W}}_2$ but possessing the effective sizes ${{\mathcal{L}}^{\prime }}_1=0.896\mathrm{m}$ and ${{\mathcal{L}}^{\prime }}_2=0.972\mathrm{m}$, respectively, in the same frequency range. The non-Weyl networks contained one balanced vertex. The theoretical positions of the expected resonances are marked with red arrows.