Rotating points for the conformal NLS scattering operator

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We consider the nonlinear Schrodinger equation, with mass-critical nonlinearity, focusing or defocusing. For any given angle, we establish the existence of infinitely many functions on which the scattering operator acts as a rotation of this angle. Using a lens transform, we reduce the problem to the existence of a solution to a nonlinear Schrodinger equation with harmonic potential, satisfying suitable periodicity properties. The existence of infinitely many such solutions is proved thanks to a constrained minimization problem.