1995/09/30 by Henrik Bruus, Niall D. Whelan · 50 citations
Mathematics · Physics and Astronomy · #Cold Atom Physics and Bose-Einstein Condensates #Combinatorics #Diffraction #Dynamical billiards #Formalism (music) #Geometry #Graph #Icosahedral symmetry #Mathematics #Nonlinear Photonic Systems #Physics #Quantum #Quantum chaos and dynamical systems #Quantum mechanics #Semiclassical physics #TRACE (psycholinguistics) #Vertex (graph theory) #chao-dyn #cond-mat #nlin.CD
paper · pdf · doi:10.1088/0951-7715/9/4/012
published in Nonlinearity 9(4), 1023-1047 (IOP Publishing) · 25 pages, 12 Postscript figures. Published version
openalex publication_date 1996/07/01 · arxiv created 1996/07/10 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the effect of edge diffraction on the semiclassical analysis of two-dimensional quantum systems by deriving a trace formula which incorporates paths hitting any number of vertices embedded in an arbitrary potential. This formula is used to study the cardioid billiard, which has a single vertex. The formula works well for most of the short orbits we analysed but fails for a few diffractive orbits due to a breakdown in the formalism for certain geometries. We extend the symbolic dynamics to account for diffractive orbits and use it to show that in the presence of parity symmetry the trace formula decomposes in an elegant manner such that for the cardioid billiard the diffractive orbits have no effect on the odd spectrum. Including diffractive orbits helps resolve peaks in the density of even states but does not appear to affect their positions. An analysis of the level statistics shows no significant difference between spectra with and without diffraction.