Convergence to SPDE of the Schrodinger equation with large, random potential

TL;DR

This paper investigates the asymptotic behavior of solutions to the Schrödinger equation with large, random potential, showing convergence to a stochastic differential equation in lower dimensions and to a deterministic solution in higher dimensions.

Contribution

It constructs the heterogeneous solution via Duhamel expansion and proves convergence to a stochastic PDE in certain dimensions, extending understanding of Schrödinger equations with random potentials.

Findings

Convergence to stochastic differential equation in dimension d<𝔪

Solution converges to deterministic Schrödinger in dimension d>𝔪

Preservation of mass ensures uniqueness of the limiting solution

Abstract

We study the asymptotic behavior of solutions to the Schr{\"o}dinger equation with large-amplitude, highly oscillatory, random potential. In dimension $d<\mathfrak{m}$, where $\mathfrak{m}$ is the order of the leading operator in the Schr\"odinger equation, we construct the heterogeneous solution by using a Duhamel expansion and prove that it converges in distribution, as the correlation length $\varepsilon$ goes to 0, to the solution of a stochastic differential equation, whose solution is represented as a sum of iterated Stratonovich integral, over the space $C([0,+\infty),\mathcal{S}')$. The uniqueness of the limiting solution in a dense space of $L^2(\Omega\times\mathbb{R}^d)$ is shown by verifying the property of conservation of mass for the Schr\"odinger equation. In dimension $d>\mathfrak{m}$, the solution to the Schr{\"o}dinger equation is shown to converge in $L^2(\Omega\times\mathbb{R}^d)$ to a deterministic Schr{\"o}dinger solution in \cite{ZB-12}.