Time mapping in power series expansions for the time evolution operator

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Abstract
Two formally equivalent methods for systematically evaluating either the propagator or the average of dynamical variables are developed by expanding these quantities in a power series in a given function τ(t). The expansion coefficients are analytically determined by recursion relations. The methods are an extension of our power series expansion formalism [Phys. Rev. Lett. 75, 4342 (1995)] to a general Fokker-Planck-Schrödinger process. The role of the time transformation in accelerating the series convergence is emphasized and the generalization to an arbitrary conformal time mapping τ(t) is presented. An appropriate truncation scheme is suggested to eliminate the openness of the series representations. We also develop a regular procedure to minimize the truncation error. The formalism thus constructed provides a basis for an efficient error controlled treatment of simple or complex systems with any number of degrees of freedom. The application to a well-known problem of the decay of an unstable state driven by exponentially correlated Gaussian noise shows that an accurate description for arbitrarily large t is attained with a few terms of the present expansions and their utility is rather insensitive with respect to the noise strength. This is in contrast to the various available approximate solutions of the problem that are all asymptotic in the noise strength.

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