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We consider the final state problem for the inhomogeneous nonlinearSchr\"odinger equation with a critical long-range nonlinearity. Given aprescribed asymptotic profile, which has a logarithmic phase correctioncompared with the free evolution, we construct a unique global solution whichconverges to the profile. As a consequence, the existence of modified waveoperators for localized small scattering data is obtained. We also study thesame problem for the case with the critical inverse-square potential under theradial symmetry. In particular, we construct the modified wave operators forthe long-range nonlinear Schr\"odinger equation with the criticalinverse-square potential in three space dimensions, under the radial symmetry.

Modified scattering for inhomogeneous nonlinear Schr\"odinger equations with and without inverse-square potential