The Schr{\"o}dinger equation with spatial white noise potential

TL;DR

This paper studies the Schrödinger equation with spatial white noise potential in two dimensions, establishing existence and uniqueness of solutions through a novel change of variables and conserved quantities.

Contribution

It introduces a new approach to handle the Schrödinger equation with white noise potential, proving well-posedness in 2D.

Findings

Existence of solutions for the Schrödinger equation with white noise potential.

Uniqueness of solutions in the considered setting.

Conservation laws are established for these solutions.

Abstract

We consider the linear and nonlinear Schr{\"o}dinger equation with a spatial white noise as a potential in dimension 2. We prove existence and uniqueness of solutions thanks to a change of unknown originally used in [8] and conserved quantities.