Scattering theory for nonlinear Schrödinger equations with inverse-square potential

We study the long-time behavior of solutions to nonlinear Schrödinger equations with some critical rough potential of a|x|⁻²$a{|x|}^{-2}$ type. The new ingredients are the interaction Morawetz-type inequalities and Sobolev norm property associated with _Pa=−Δ+a|x|⁻²$P_a=-\mathrm{\Delta }+a{|x|}^{-2}$. We use such properties to obtain the scattering theory for the defocusing energy-subcritical nonlinear Schrödinger equation with inverse square potential in energy space H¹(Rⁿ)$H^1({\mathbb{R}}^n)$.