# 3D quadratic NLS equation with electromagnetic perturbations

## TL;DR

This paper investigates the long-term behavior of a quadratic nonlinear Schr"{o}dinger equation influenced by electromagnetic potentials, demonstrating scattering of small solutions and introducing novel smoothing estimate techniques.

## Contribution

It introduces new smoothing estimates for the linear electromagnetic Schr"{o}dinger flow, replacing wave operator boundedness to handle derivative loss, extending previous work to electromagnetic contexts.

## Key findings

- Small solutions scatter asymptotically.
- Wave operator boundedness on weighted L^2 spaces established.
- Dispersive estimates for the electromagnetic Schr"{o}dinger flow obtained.

## Abstract

In this paper we study the asymptotic behavior of a quadratic Schr\"{o}dinger equation with electromagnetic potentials. We prove that small solutions scatter. The proof builds on earlier work of the author for quadratic NLS with a non magnetic potential. The main novelty is the use of various smoothing estimates for the linear Schr\"{o}dinger flow in place of boundedness of wave operators to deal with the loss of derivative. As a byproduct of the proof we obtain boundedness of the wave operator of the linear electromagnetic Schr\"{o}dinger equation on an $L^2$ weighted space for small potentials, as well as a dispersive estimate for the corresponding flow.