Numerical study of quantum resonances in chaotic scattering

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36 Scopus citations


This paper presents numerical evidence that in quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy E scales like h-DKE+1/2 as h → 0. Here, KE denotes the subset of the energy surface {H = E} which stays bounded for all time under the flow generated by the classical Hamiltonian H and D(KE) denotes its fractal dimension. Since the number of bound states in a quantum system with n degrees of freedom scales like h-n, this suggests that the quantity (D(KE) + 1)/2 represents the effective number of degrees of freedom in chaotic scattering problems. The calculations were performed using a recursive refinement technique for estimating the dimension of fractal repellors in classical Hamiltonian scattering, in conjunction with tools from modern quantum chemistry and numerical linear algebra.

Original languageEnglish (US)
Pages (from-to)295-329
Number of pages35
JournalJournal of Computational Physics
Issue number2
StatePublished - Mar 1 2002


  • Chaotic trapping
  • Fractal dimensional
  • Scattering resonances
  • Semiclassical asymptotics

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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