# Stationary-Action for Conservative Systems and Diffusion-Based Representations for the Schrödinger Equation

Thursday, May 10, 2018 - 10:00am - 10:50am

Lind 305

William McEneaney (University of California, San Diego)

Stationary-action approaches to solution of two-point boundary problems (TPBVPs) for conservative systems and stationarity-based approaches to representations for solutions of Schrödinger initial value problems are closely related.

It is well known that the theory of stationary action for conservative dynamical systems provides an alternative approach from that of Newton's second law. More recently, it has been shown that this approach, coupled with concepts from max-plus algebraic methods for optimal control can be used to obtain fundamental solutions for TPBVPs. On sufficiently short time intervals, stationarity of action coincides with least action, and one may apply the same methods as used for control and games in the least-action approach to TPBVPs on short intervals. Appending various terminal payoffs to the action functional results in solutions of various TPBVPs. However, in order to extend the general approach to indefinite length time intervals, it is necessary to employ the more-general formulation of stationarity of action, rather than minimization of action. Under certain conditions, it has been demonstrated that the stationary value function satisfies the associated stationary-value Hamilton-Jacobi partial differential equation (HJ PDE).

Diffusion representations have long been a useful tool for solution of second-order HJ PDEs. The bulk of such results apply to real-valued HJ PDEs, that is, to HJ PDEs where the coefficients and solutions are real-valued. The Schrödinger equation is complex-valued, although generally defined over a real-valued space domain, which presents difficulties for the application of solution techniques based on stochastic control representations. Here, a Feynman-Kac approach will be taken to the dequantized form of the Schrödinger equation. However, the representation employs stationarity of the payoff rather than optimization, where the use of stationarity allows one to overcome the limited-duration constraints inherent in methods that use optimization. In the limit as the Planck constant goes to zero, one obtains the HJ PDE that is associated to the above stationary-action TPBVP.

It is well known that the theory of stationary action for conservative dynamical systems provides an alternative approach from that of Newton's second law. More recently, it has been shown that this approach, coupled with concepts from max-plus algebraic methods for optimal control can be used to obtain fundamental solutions for TPBVPs. On sufficiently short time intervals, stationarity of action coincides with least action, and one may apply the same methods as used for control and games in the least-action approach to TPBVPs on short intervals. Appending various terminal payoffs to the action functional results in solutions of various TPBVPs. However, in order to extend the general approach to indefinite length time intervals, it is necessary to employ the more-general formulation of stationarity of action, rather than minimization of action. Under certain conditions, it has been demonstrated that the stationary value function satisfies the associated stationary-value Hamilton-Jacobi partial differential equation (HJ PDE).

Diffusion representations have long been a useful tool for solution of second-order HJ PDEs. The bulk of such results apply to real-valued HJ PDEs, that is, to HJ PDEs where the coefficients and solutions are real-valued. The Schrödinger equation is complex-valued, although generally defined over a real-valued space domain, which presents difficulties for the application of solution techniques based on stochastic control representations. Here, a Feynman-Kac approach will be taken to the dequantized form of the Schrödinger equation. However, the representation employs stationarity of the payoff rather than optimization, where the use of stationarity allows one to overcome the limited-duration constraints inherent in methods that use optimization. In the limit as the Planck constant goes to zero, one obtains the HJ PDE that is associated to the above stationary-action TPBVP.