Scattering theory for 3d cubic inhomogeneous NLS with inverse square potential

TL;DR

This paper analyzes the scattering behavior of 3D cubic inhomogeneous nonlinear Schrödinger equations with inverse square potential, establishing global results in the defocusing case and scattering below ground state in the focusing case.

Contribution

It provides the first comprehensive scattering results for this class of equations, including both defocusing and focusing cases, using virial/Morawetz techniques without interaction Morawetz estimates.

Findings

Global well-posedness and scattering in the defocusing case.

Scattering below ground state in the focusing case.

Avoidance of interaction Morawetz estimate in proofs.

Abstract

In this paper, we study the scattering theory for the cubic inhomogeneous Schr\"odinger equations with inverse square potential $iu_t+\Delta u-\frac{a}{|x|^2}u=\lambda |x|^{-b}|u|^2u$ with $a>-\frac14$ and $0<b<1$ in dimension three. In the defocusing case (i.e. $\lambda=1$), we establish the global well-posedness and scattering for any initial data in the energy space $H^1_a(\mathbb R^3)$. While for the focusing case(i.e. $\lambda=-1$), we obtain the scattering for the initial data below the threshold of the ground state, by making use of the virial/Morawetz argument as in Dodson-Murphy [Proc. Amer. Math. Soc.,145(2017), 4859-4867.] and Campos-Cardoso [arXiv: 2101.08770v1.] that avoids the use of interaction Morawetz estimate.