Circuits,signals,and systems
Numerical computational of the scattering frequencies for acoustic wave equations
Journal of Computational Physics
Pseudospectra of Linear Operators
SIAM Review
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
Structure and interpretation of classical mechanics
Structure and interpretation of classical mechanics
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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-(D(KE)+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.