A discrete Schrodinger equation via optimal transport on graphs

Abstract

In 1966, Edward Nelson presented an interesting derivation of the Schrodinger equation using Brownian motion. Recently, this derivation is linked to the theory of optimal transport, which shows that the Schrodinger equation is a Hamiltonian system on the probability density manifold equipped with the Wasserstein metric. In this paper, we consider similar matters on a finite graph. By using discrete optimal transport and its corresponding Nelson's approach, we derive a discrete Schrodinger equation on a finite graph. The proposed system is quite different from the commonly referred discretized Schrodinger equations. It is a system of nonlinear ordinary differential equations (ODEs) with many desirable properties. Several numerical examples are presented to illustrate the properties.