Spectral properties of chaotic microwave networks and quantum graphs under an edge swap transformation.

Abstract

We discuss the spectral properties of microwave networks and quantum graphs with preserved and partially violated T invariance (time-reversal invariance) under an edge swap transformation. A chaotic tetrahedral quantum graph was simulated by the tetrahedral microwave network in which the swap transformation was realized by replacing one pair of edges {e_{i}} and {e_{j}} adjacent to different vertices {v_{i}} and {v_{j}}. We show that the spectra {ν_{n}}_{n=1}^{∞} and {ν[over ̃]_{n}}_{n=1}^{∞} before and after an edge swap operation are level-2 interlaced. The experimental distribution P(ΔN) of the spectral shift ΔN=N(ν)-N[over ̃](ν), where N(ν) and N[over ̃](ν) are the counting functions for the original and swapped networks, was confirmed in the numerical calculations. For chaotic systems with partially violated T invariance, the cases of level-2 interlacing (ΔN=±2) appear less frequently than for the systems with time reversal symmetry. Furthermore, the widths of the overlapping areas with ΔN=±2 are getting narrower than for the system with preserved T invariance. The smaller susceptibility to the swap operation for networks and graphs with partially violated T invariance is likely caused by the stronger level repulsion in their spectra than in the systems with preserved T invariance leading to smaller rigidity of their spectra.