The 1D Schr\"odinger equation with a spacetime white noise: the average wave function

TL;DR

This paper demonstrates that the average wave function of a 1D Schrödinger equation with mollified spacetime white noise converges to a deterministic Schrödinger equation with an effective potential after renormalization.

Contribution

It introduces a renormalization approach showing convergence of the average wave function in a stochastic Schrödinger equation with white noise.

Findings

Average wave function converges to a deterministic solution

Effective potential emerges after renormalization

Provides a rigorous framework for stochastic Schrödinger equations

Abstract

For the 1D Schr\"odinger equation with a mollified spacetime white noise, we show that the average wave function converges to the Schr\"odinger equation with an effective potential after an appropriate renormalization.