The Schr\"odinger equation with spatial white noise: the average wave function

TL;DR

This paper derives a representation for the average wave function of the Schrödinger equation with spatial white noise in one and two dimensions, linking it to the renormalized self-intersection local time of Brownian motion.

Contribution

It introduces a novel representation connecting the average wave function to Brownian motion's self-intersection local time, advancing understanding of stochastic Schrödinger equations.

Findings

Representation of average wave function in terms of Brownian motion

Connection established between Schrödinger equation with white noise and self-intersection local time

Provides mathematical framework for analyzing stochastic quantum systems

Abstract

We prove a representation for the average wave function of the Schr\"odinger equation with a white noise potential in $d=1,2$, in terms of the renormalized self-intersection local time of a Brownian motion.