# Spacetime estimates and scattering theory for quasilinear Schr\"{o}dinger equations in arbitrary space dimension

## TL;DR

This paper develops spacetime estimates and scattering theory for quasilinear Schrödinger equations in any dimension, introducing new methods for proving global existence, conservation laws, and asymptotic behavior.

## Contribution

It presents novel techniques for establishing scattering in arbitrary dimensions for quasilinear Schrödinger equations, including direct proofs using tailored Strichartz estimates.

## Key findings

- Established sufficient conditions for global existence.
- Derived pseudoconformal conservation law and Morawetz estimates.
- Provided simple proofs of scattering in $L^2$ and $\,\Sigma$ spaces.

## Abstract

In this paper, we consider the following Cauchy problem of \begin{equation*} \left\{ \begin{array}{lll} iu_t=\Delta u+2\delta_huh'(|u|^2)\Delta h(|u|^2)+V(x)u+F(|u|^2)u+(W*|u|^2)u,\ x\in \mathbb{R}^N,\ t>0\\ u(x,0)=u_0(x),\quad x\in \mathbb{R}^N. \end{array}\right. \end{equation*} Here $\delta_h$ is a constant, $N\geq 1$, $h(s)$, $F(s)$, $V(x)$ and $W(x)$ are some real functions, $W(x)$ is even. Besides obtaining some sufficient conditions on global existence of the solution, we establish pseudoconformal conservation law and give Morawetz type estimates, spacetime bounds and asymptotic behaviors for the global solution. We bring two ideas to establish scattering theory, one is that we take different admissible pairs in Strichartz estimates for different terms on the right side of Duhamel's formula in order to keep each term independent, another is that we factitiously let a continuous function be the sum of two piecewise functions and chose different admissible pairs in Strichartz estimates for the terms containing these functions. Basing on the two ideas, we provide the direct and simple proofs of classic scattering theories in $L^2(\mathbb{R}^N)$ and $\Sigma$ for any space dimension($N\geq 1$) under certain assumptions. Here $$ \Sigma=\{u\in H^1(\mathbb{R}^N),\quad |xu|\in L^2(\mathbb{R}^N)\}. $$