On Selberg's trace formula: Chaos, resonances and time delays

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Abstract

The quantization of the chaotic geodesic motion on Riemann surfaces Σg,k of constant negative curvature with genus g and a finite number of points k infinitely far away (cusps) describing scattering channels is investigated. It is shown that terms in Selberg's trace formula describing scattering states can be expressed in terms of a renormalized time delay. This quantity is the time delay associated with the surface in question minus the time delay corresponding to the scattering problem on the Poincaré upper half-plane uniformizing our surface. Poles in these quantities give rise to resonances reflecting the chaos of the underlying classical dynamics. Our results are illustrated for the surfaces Σ1,1 (Gutzwiller's leaky torus), Σ0.3 (pants), and a class of Σg,2 surfaces. The generalization covering the inclusion of an integer B ≥ 2 magnetic field is also presented. It is shown that the renormalized time delay is not dependent on the magnetic field. This shows that the semiclassical dynamics with an integer magnetic field is the same as the free dynamics.

Original languageEnglish
Pages (from-to)4357-4376
Number of pages20
JournalJournal of Physics A: Mathematical and General
Volume33
Issue number23
DOIs
Publication statusPublished - Jun 16 2000

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ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)

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