Invariant imbedding solution of driven (inhomogeneous) and homogeneous Schrödinger equations

The method of invariant imbedding is generalized and extended to treat the solution of the inhomogeneous (driven) multichannel Schrödinger equation arising from two potential problems with one of the potentials being weak (as in photodissociation, photoionization, etc.). The method directly propagates the physical transition amplitudes of interest and is consequently extremely stable for situations where a number of other integration methods for the driven equations are found to be numerically unstable. We also present a simplified derivation of the invariant imbedding equations for the homogeneous Schrödinger scattering equations. A transformation is introduced enabling the propagation of symmetric matrices only. A simple analytically solvable two-channel driven equations model is introduced to exhibit the origins of numerical instabilities in a number of other integration methods for the driven equations in situations when an asymptotically open channel is badly closed in the Franck-Condon region.