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 (15)

• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Experimental simulation of quantum graphs by microwave networks
• Quantum Graphs: A simple model for Chaotic Scattering
• 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
• A microwave realization of the Gaussian symplectic ensemble
• Experimental investigation of Wigner's reaction matrix for irregular graphs with absorption
• Non-universality in the spectral properties of time-reversal invariant microwave networks and quantum graphs
• Explicitly solvable cases of one-dimensional quantum chaos
• Universal Quantum Graphs
• Finite pseudo orbit expansions for spectral quantities of quantum graphs
• Non-Weyl asymptotics for quantum graphs with general coupling conditions
• Non-Weyl resonance asymptotics for quantum graphs in a magnetic field
• On the Effective Size of a Non-Weyl Graph

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• The edge switch transformation in microwave networks
• Narrow peaks of full transmission in simple quantum graphs
• Simple quantum graphs proposal for quantum devices
• How T-invariance violation leads to an enhanced backscattering with increasing openness of a wave-chaotic system
• A spectral duality in graphs and microwave networks
• Experimental study of closed and open microwave waveguide graphs with preserved and partially violated time-reversal invariance
• Quantum graphs and microwave networks as narrow band filters for quantum and microwave devices
• Missing level statistics in a dissipative microwave resonator with partially violated time-reversal invariance
• Closed and open superconducting microwave waveguide networks as a model for quantum graphs
• Application of topological resonances in experimental investigation of a Fermi golden rule in microwave networks
• Coupled unidirectional chaotic microwave graphs
• Stochastic thermal feedback in switching measurements of superconducting nanobridge caused by overheated electrons and phonons
• Investigation of the enhancement factor in the regime of semi-Poisson statistics in a singular microwave cavity
• Families of isospectral and isoscattering quantum graphs
• Dynamical processes on metric networks
• Experimental study of the distributions of off-diagonal scattering-matrix elements of quantum graphs with symplectic symmetry