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 $a{\left|x\right|}^{-2}$ type. The new ingredients are the interaction Morawetz-type inequalities and Sobolev norm property associated with $P_a=-\Delta +a{\left|x\right|}^{-2}$. We use such properties to obtain the scattering theory for the defocusing energy-subcritical nonlinear Schrödinger equation with inverse square potential in energy space $H^1\left(R^n\right)$.