Rotating points for the conformal NLS scattering operator

TL;DR

This paper demonstrates that for the mass-critical nonlinear Schrödinger equation, the scattering operator can act as a rotation by any angle on infinitely many functions, using a lens transform and constrained minimization.

Contribution

It introduces a novel approach to construct solutions where the scattering operator acts as a rotation, extending understanding of scattering behavior in nonlinear Schrödinger equations.

Findings

Existence of solutions with scattering operator acting as arbitrary angle rotations.

Use of lens transform to relate scattering problem to harmonic potential Schrödinger equation.

Infinite solutions constructed via constrained minimization.

Abstract

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.