### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy(all)

### Cite this

**Numerical study of quantum resonances in chaotic scattering.** / Lin, Kevin.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 176, no. 2, pp. 295-329. https://doi.org/10.1006/jcph.2001.6986

}

TY - JOUR

T1 - Numerical study of quantum resonances in chaotic scattering

AU - Lin, Kevin

PY - 2002/3/1

Y1 - 2002/3/1

N2 - 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.

AB - 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.

KW - Chaotic trapping

KW - Fractal dimensional

KW - Scattering resonances

KW - Semiclassical asymptotics

UR - http://www.scopus.com/inward/record.url?scp=0036501082&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036501082&partnerID=8YFLogxK

U2 - 10.1006/jcph.2001.6986

DO - 10.1006/jcph.2001.6986

M3 - Article

AN - SCOPUS:0036501082

VL - 176

SP - 295

EP - 329

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 2

ER -