Quantum resonances in chaotic scattering

This Letter summarizes numerical results from [J. Comp. Phys. (to appear)] which show that in quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy E scales like ℏ−(DKE+1)/2. Here, KE denotes the subset of the classical energy surface H=E which stays bounded for all time under the flow of H and DKE denotes its fractal dimension. Since the number of bound states in a quantum system with n degrees of freedom scales like ℏ−n, this suggests that the quantity (DKE+1)/2 represents the effective number of degrees of freedom in chaotic scattering problems.