Scattering for nonlinear Schrodinger equation under partial harmonic confinement

Paolo Antonelli (1) , Rémi Carles (2) , Jorge Drumond Silva (3)

1 SNS - Scuola Normale Superiore di Pisa
2 I3M - Institut de Mathématiques et de Modélisation de Montpellier
3 CAMGSD - Center for Mathematical Analysis, Geometry and Dynamical Systems

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.

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Paolo Antonelli, Rémi Carles, Jorge Drumond Silva. Scattering for nonlinear Schrodinger equation under partial harmonic confinement. Communications in Mathematical Physics, 2015, 334 (1), pp.367-396. ⟨10.1007/s00220-014-2166-y⟩. ⟨hal-00870043v2⟩