# The nonlinear Schr\"{o}dinger equation with white noise dispersion on quantum graphs

## TL;DR

This paper establishes the global well-posedness of the nonlinear Schrödinger equation with white noise dispersion on quantum graphs under certain decay conditions, and explores its properties and limits.

## Contribution

It proves well-posedness in $L^2$ for NLSE with white noise dispersion on quantum graphs and analyzes the solution's behavior and scaling limits.

## Key findings

- Global well-posedness in $L^2$ under decay conditions
- Solution with white noise dispersion is a scaling limit
- Analysis of well-posedness in energy domain

## Abstract

We show that the nonlinear Schr\"{o}dinger equation (NLSE) with white noise dispersion on quantum graphs is globally well-posed in $L^2$ once the free deterministic Schr\"{o}dinger group satisfies a natural $L^1-L^{\infty}$ decay, which is verified in many examples. Also, we investigate the well-posedness in the energy domain in general and in concrete situations, as well as the fact that the solution with white noise dispersion is the scaling limit of the solution to the NLSE with random dispersion.