Calculation of the quantum-mechanical propagator at large real times

The propagator K(x,x′; t c ) = 〈x| exp(- iHt c / h ̵ )|x′〉 is widely used in quantum statistical mechanics for for the calculation of both equilibrium and non-equilibrium properties. In non-equilibrium applications the propagator is required at a complex time t c = t − iβ h ̵ ; cancellation effects make the computation of K( x, x′; t c) numerically challenging for t ⪆ βħ. Accurate calculation of the propagator at large real times is thus a subject of continuing interest. Sethia, Sanyal and Singh have recently proposed a method in which the short-time propagator is diagonalized, yielding the eigenvalues and eigenvectors of H; these may be used to reconstruct K( x, x′; t c). In this paper we analyze the errors inherent in such an approach. We compare Sethia et al.'s algorithm with a method recently suggested by Dunn and Grieves, which diagonalizes the Hamiltonian. The latter method is found to be superior in terms of the accuracy which may be achieved and the computational effort which must be expended. It can yield accurate values for the propagator at t = 10 6β h ̵ and beyond.