Obtaining Eigenvalues of the Schrodinger Equation by Stochastic Control Methods

This paper shows that each eigenvalue of the stationary Schrodinger equation can be characterized as the minimum value of a performance functional associated with a stochastic control problem. The stochastic control problem is defined for regions bounded by nodes of the solution to the Schrodinger equation. A set of admissible controls is defined and it is shown that the process defined by each control does not reach the boundary of the region being considered. It is also shown that there is a unique stationary distribution associated with each admissible control. The optimal control for this problem is defined in terms of the solution to the equation.