Abstract. We prove global-in-time Strichartz estimates without loss of derivatives for the solution of the Schrödinger equation on a class of non-trapping asymptotically conic manifolds. We obtain estimates for the full set of admissible indices, including the endpoint, in both the homogeneous and inhomogeneous cases. This result improves on the results by Tao, Wunsch and the first author in [23] and [34], which are local in time, as well as the results of the second author in [41], which are global in time but with a loss of angular derivatives. In addition, the endpoint inhomogeneous estimate is a strengthened version of the uniform Sobolev estimate recently proved by Guillarmou and the first author [16].

We consider the defocusing energy-critical nonlinear Schrödinger equation with inverse-square potential iut = −∆u + a|x| −2 u + |u| 4 u in three space dimensions. We prove global well-posedness and scattering for a > − 1 4 + 1 25 . We also carry out the variational analysis needed to treat the focusing case.

We study the long-time behavior of solutions to nonlinear Schrödinger equations with some critical rough potential of a|x| −2 type. The new ingredients are the interaction Morawetz-type inequalities and Sobolev norm property associated with Pa = −∆ + 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 H 1 (R n ).