We investigate the semiclassical scattering amplitude for systems, where the classical dynamics is non-hyperbolic, i.e. where islands of KAM trajectories exist in an otherwise chaotic phase space. With the help of semiclassical calculations for the three-disk billiard in an external magnetic field, in which a hyperbolic-non-hyperbolic transition is observed as a function of the field strength, we show that the "stickiness" of the KAM tori leads to a much slower decrease of the survival probability, as compared with the hyperbolic case. This is reflected by a much narrower shape of the energy correlation function. However, we also find that the algebraic asymptotic decay of the survival probability in the non-hyperbolic case is not important for the quantum fluctuations. (2) The conclusion in ref. [9] might have been caused by erroneously identifying the survival probability N II (E, t) with P II (E, t).
We study the classical and semiclassical scattering behavior of electrons in an open three-disk billard in the presence of a homogeneous magnetic field, which is confined to the inner part of the scattering region. As the magnetic field is increased the phase space of the invariant set of the classical scattering trajectories changes from hyperbolic (fully chaotic) to a mixed situation, where KAM tori are present. The "stickiness" of the stable trajectories leads to a much slower decay of the survival probability of trajectories as compared to the hyperbolic case. We show that this effect influences strongly the quantum fluctuations of the scattering amplitude and cross sections.
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