### Abstract

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^{-DK}E^{+1/2} as h → 0. Here, K_{E} 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(K_{E}) 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(K_{E}) + 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 language | English (US) |
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Pages (from-to) | 295-329 |

Number of pages | 35 |

Journal | Journal of Computational Physics |

Volume | 176 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2002 |

Externally published | Yes |

### Keywords

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

### ASJC Scopus subject areas

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