This paper presents numerical evidence that for quantum systems with chaotic
classical dynamics, the number of scattering resonances near an energy E
scales like ℏ−2D(KE)+1 as ℏ→0. Here, KE 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(KE) 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 2D(KE)+1 represents the effective number of degrees of
freedom in scattering problems.Comment: 24 pages, including 44 figure