# On the scattering problem for the nonlinear Schr\"{o}dinger equation with a potential in 2D

## TL;DR

This paper studies the scattering problem for the 2D nonlinear Schrödinger equation with a potential, establishing resolvent estimates, operator equivalences, and proving global existence, decay, and scattering of solutions for small initial data.

## Contribution

It provides new resolvent estimates and operator equivalences for the 2D nonlinear Schrödinger equation with potential, leading to results on global existence and scattering.

## Key findings

- Proved resolvent estimates for the Schrödinger operator with potential.
- Established the equivalence of fractional powers of operators in L^2 norm.
- Demonstrated global existence, decay, and scattering for small initial data.

## Abstract

We consider the scattering problem for the nonlinear Schr\"{o}dinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator $A(s)$ appearing in commutator relations. The equivalence between the operators $\left(-\Delta_{V}\right)^{\frac{s}{2}}$ and $\left(-\Delta \right)^{\frac{s}{2}}$ in $L^{2}$ norm sense for $0\leq s <1$ is investigated by using free resolvent estimates and Gaussian estimates for the heat kernel of the Schr\"{o}dinger operator $-\Delta_{V}$. Our main result guarantees the global existence of solutions and time decay of the solutions assuming initial data have small weighted Sobolev norms. Moreover, the global solutions obtained in the main result scatter.