A Diffusion-Based Solution Technique for Certain Schrödinger Equation Dynamical Systems

A particular class of Schrödinger initial value problems is considered, wherein a particle moves in a scalar field centered at the origin, and more specifically, the distribution associated to the solution of the Schrödinger equation has negligible mass in the neighborhood of the origin. The Schrödinger equation is converted to a stochastic control problem using the Maslov dequantization transform. A non-inertial frame centered along the trajectory of a classical particle is employed. A solution approximation as a series expansion in a small parameter is obtained through the use of complex-valued diffusion-process representations, where under a smoothness assumption, the expansion converges to the true solution. In the case of an expansion up through only the cubic terms in the space variable, there exist approximate solutions that are periodic with the period of a classical particle, but with an additional secular perturbation. The computations required for solution up to a finite order are purely analytical.