Properties of time delay and S-matrix for chaotic scattering on a leaky surface of constant negative curvature

Abstract

Properties of a model quantum scattering system displaying hard classical chaos-'elastic' scattering on a leaky surface of constant negative curvature-are analysed theoretically and serve to interpret previously obtained numerical results. The low energy scattering behaviour is shown to be influenced, in the usual fashion, by a bound state just below the scattering continuum threshold. A connection between the widths of the infinite number of simple poles of the S-matrix and the Lyapunov exponent for classical trajectories is analysed. At high energies, the scattering is characterized by fluctuations in the S-matrix (via its phase) and the time delay. Analytic expressions for the autocorrelation function of the S-matrix (via its phase) and the time delay. Analytic expressions for the autocorrelation function of the S-matrix and of the time delay are obtained using Montgomery's conjecture for the pair correlation function of the celebrated Riemann zeros whose values correspond to the positions of the S-matrix poles in momentum space. The autocorrelation function for the S-matrix is found to be Lorentzian asymptotically (at large energy differences Delta E), that is, to decrease as Delta E-2, but that for the time delay is not. The distribution of fluctuations of S-matrix phases is likely to be Gaussian.