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 $\hbar^{-\frac{D(K_E)+1}{2}}$ as $\hbar\to{0}$. 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 $\hbar^{-n}$, this suggests that the quantity $\frac{D(K_E)+1}{2}$ represents the effective number of degrees of freedom in scattering problems.