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

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.