2000/01/19 by Tsampikos Kottos, Holger Schanz
Mathematics · Physics and Astronomy · #Matrix (chemical analysis) #Operator (biology) #Periodic boundary conditions #Quantum #Quantum chaos #Quantum chaos and dynamical systems #Quantum graph #Quantum many-body systems #Random matrix #Scattering #Simple (philosophy) #Spectral Theory in Mathematical Physics #nlin.CD
paper · pdf · doi:10.1016/s1386-9477(00)00257-5
published as Physica E 9 (2001) 523-530 · 8 pages, 3 figures. Contribution to the conference on Dynamics of Complex Systems, Dresden (1999)
arxiv created 2000/01/19 · openalex publication_date 2001/03/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We study the statistical properties of the scattering matrix associated with generic quantum graphs. The scattering matrix is the quantum analogue of the classical evolution operator on the graph. For the energy-averaged spectral form factor of the scattering matrix we have recently derived an exact combinatorial expression. It is based on a sum over families of periodic orbits which so far could only be performed in special graphs. Here we present a simple algorithm implementing this summation for any graph. Our results are in excellent agreement with direct numerical simulations for various graphs. Moreover we extend our previous notion of an ensemble of graphs by considering ensemble averages over random boundary conditions imposed at the vertices. We show numerically that the corresponding form factor follows the predictions of random-matrix theory when the number of vertices is large---even when all bond lengths are degenerate. The corresponding combinatorial sum has a structure similar to the one obtained previously by performing an energy average under the assumption of incommensurate bond lengths.