Numerical study of quantum resonances in chaotic scattering

Kevin K. Lin

doi:10.1006/jcph.2001.6986

Numerical study of quantum resonances in chaotic scattering

Kevin K. Lin

Research output: Contribution to journal › Article › peer-review

37 Scopus citations

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^-DKE^+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)
Pages (from-to)	295-329
Number of pages	35
Journal	Journal of Computational Physics
Volume	176
Issue number	2
DOIs	https://doi.org/10.1006/jcph.2001.6986
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)
General Physics and Astronomy
Computer Science Applications
Computational Mathematics
Applied Mathematics

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10.1006/jcph.2001.6986

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title = "Numerical study of quantum resonances in chaotic scattering",

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-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.",

keywords = "Chaotic trapping, Fractal dimensional, Scattering resonances, Semiclassical asymptotics",

author = "Lin, {Kevin K.}",

note = "Funding Information: I am grateful to J. Demmel, B. Parlett, and Z. Bai for crucial advice on matrix computations, and to X. S. Li and C. Yang for help with ARPACK. I also thank R. Littlejohn, M. Cargo, and W. H. Miller for suggestions on bases, F. Bonetto for teaching me how to compute fractal dimensions, and C. Pugh and J. Harrison for helpful conversations. In writing this paper, I benefited greatly from the comments of A. J. Chorin and S. Zelditch. Finally, I thank M. Zworski for inspiring this project. This work was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under Contract DE-AC03-76-SF00098. Computational resources were provided by the Mathematics Department at Lawrence Berkeley National Laboratory, the National Energy Research Scientific Computing Center (NERSC), and the Millennium Project at U. C. Berkeley.",

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doi = "10.1006/jcph.2001.6986",

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PY - 2002/3/1

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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.

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KW - Semiclassical asymptotics

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Numerical study of quantum resonances in chaotic scattering

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