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|−2a|x|−2 type. The new ingredients are the interaction Morawetz-type inequalities and Sobolev norm property associated with Pa=−Δ+a|x|−2Pa=−Δ+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)H1(Rn).