Comment on the Breakdown of Van Vleck's Propagator and Its Implications for the Energy Domain Green's Function

Abstract

Van Vleck's propagator is perhaps one of the most fundamental objects of semiclassical theory. Its Laplace transform gives the energy Green's function G(x, x′, E), which is the starting point for the derivation of Gutzwiller trace formula. The validity of Van Vleck's formula for chaotic and mixed systems is therefore of crucial importance, especially since during the derivation of the trace formula it is assumed to be valid for infinite time. In the last two years there has been numerical evidence for the time-scale validity of the semiclassical propagator when the underlying dynamics is chaotic. The time-scales were longer that previously predicted by Tabor et al. An explanation for this long time validity based on the multidimensional stationary phase approximation was recently proposed by Sepúlveda-Tomsovic-Heller. In this contribution we would like to review these recent results and comment on their implications in the energy domain and particularly in the validity of the trace formula.