Numerical Study of Quantum Resonances in Chaotic Scattering

Abstract

This paper presents numerical evidence that for quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy $E$ scales like ${\hslash }^{-\frac{D(K_E)+1}{2}}$ as $\hslash \to 0$. Here, $K_E$ denotes the subset of the classical energy surface $\{H=E\}$ which stays bounded for all time under the flow generated by the 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 ${\hslash }^{-n}$, this suggests that the quantity $\frac{D(K_E)+1}{2}$ represents the effective number of degrees of freedom in scattering problems.

24 pages, including 44 figures

References in corpus (1)

• Quantum Mechanics and Semiclassics of Hyperbolic n-Disk Scattering Systems

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