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In this paper, we consider the final state problem for the nonlinearSchr\"odinger equation with a homogeneous nonlinearity of the critical orderwhich is not necessarily a polynomial. In [10], the first and the secondauthors consider one- and two-dimensional cases and gave a sufficient conditionon the nonlinearity for that the corresponding equation admits a solution thatbehaves like a free solution with or without a logarithmic phase correction.The present paper is devoted to the study of the three-dimensional case, inwhich it is required that a solution converges to a given asymptotic profile ina faster rate than in the lower dimensional cases. To obtain the necessaryconvergence rate, we employ the end-point Strichartz estimate and modify atime-dependent regularizing operator, introduced in [10]. Moreover, we presenta candidate of the second asymptotic profile to the solution.

Long range scattering for nonlinear Schr\"odinger equations with critical homogeneous nonlinearity in three space dimensions