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

Abstract

In this paper, we study the scattering theory for the cubic inhomogeneous Schrödinger equations with inverse square potential $i{u}_t+\Delta u-\frac{a}{|x{|}^2}u=\lambda |x{|}^{-b}|u{|}^{2}u$ with $a>-\frac{1}{4}$ 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 $H_a^1({\mathbb{R}}^3)$. 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.

arXiv comment15 pages. arXiv admin note: text overlap with arXiv:1808.02236 by other authors

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