Scattering for nonlinear Schrodinger equation under partial harmonic confinement

Abstract

We consider the nonlinear Schrodinger equation under a partial quadratic confinement. We show that the global dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates, from which we infer the existence of wave operators thanks to suitable vector-fields. Conversely, given an initial Cauchy datum, the solution is global in time and asymptotically free, provided that confinement affects one spatial direction only. This stems from anisotropic Morawetz estimates, involving a marginal of the position density.

26 pages. Some typos fixed, especially in Section 6

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• Global Well-posedness and scattering for fourth-order Schrödinger equations on waveguide manifolds
• Scattering for the nonlinear Schrodinger equation with a general one-dimensional confinement
• Scattering and blow up for nonlinear Schrödinger equation with the averaged nonlinearity
• On the decay property of the cubic fourth-order Schrödinger equation
• Multiplicity, asymptotics and stability of standing waves for nonlinear Schrödinger equation with rotation
• Uniqueness and orbital stability of standing waves for the nonlinear Schrodinger equation with a partial confinement