5 research outputs found
Fast Converging Path Integrals for Time-Dependent Potentials I: Recursive Calculation of Short-Time Expansion of the Propagator
In this and subsequent paper arXiv:1011.5185 we develop a recursive approach
for calculating the short-time expansion of the propagator for a general
quantum system in a time-dependent potential to orders that have not yet been
accessible before. To this end the propagator is expressed in terms of a
discretized effective potential, for which we derive and analytically solve a
set of efficient recursion relations. Such a discretized effective potential
can be used to substantially speed up numerical Monte Carlo simulations for
path integrals, or to set up various analytic approximation techniques to study
properties of quantum systems in time-dependent potentials. The analytically
derived results are numerically verified by treating several simple models.Comment: 29 pages, 5 figure
Brownian Motions on Metric Graphs
Brownian motions on a metric graph are defined. Their generators are
characterized as Laplace operators subject to Wentzell boundary at every
vertex. Conversely, given a set of Wentzell boundary conditions at the vertices
of a metric graph, a Brownian motion is constructed pathwise on this graph so
that its generator satisfies the given boundary conditions.Comment: 43 pages, 7 figures. 2nd revision of our article 1102.4937: The
introduction has been modified, several references were added. This article
will appear in the special issue of Journal of Mathematical Physics
celebrating Elliott Lieb's 80th birthda