Applications of the Karhunen-Loeve decomposition to quantum dynamical problems

The Karhunen-Loeve (KL)-decomposition is a standard method to analyze spatiotemporal patterns which are characteristic for non-linear chemical dynamics on surfaces. We apply the KL-decomposition to quantum dynamical problems. Using a variational principle it is shown how to arrive at the KL-expansion of an arbitrary complex valued function. In the case of an one-dimensional bound state motion, the KL-decomposition yields the eigenstates of the system in a straightforward manner. The time-resolved spectroscopy of an atom-molecule collision serves as another application. We demonstrate that the orthonormal decomposition into KL-modes provides an efficient tool to calculate long-time pump-probe signals.