Quasinormal mode expansion and late-time behavior of leaking systems.

Abstract

The wave function of a particle escaping from a barrier, in general, would have its quasinormal modes, which decay like exp(-\ensuremath{\gamma}t/2), dominated by anomalous ${\mathit{t}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$ terms at asymptotically late times. One implication of the existence of these anomalous terms is that the wave function cannot be expressed as a sum of quasinormal modes. We show that these anomalous terms are related to classical motion linking the initial point (x',t=0) to the final point (x,t). We then show that when the potential is unbounded from below, such terms do not appear, and more importantly, for such potential, the time development of a wave function initially concentrated in the finite region can be expressed in terms of a summation over quasinormal modes.