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Scattering for Nonlinear Schrödinger Equation Under Partial Harmonic Confinement

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Abstract

We consider the nonlinear Schrödinger 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.

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Authors and Affiliations

  1. Gran Sasso Science Institute, viale F. Crispi, 7, 67100, L’Aquila, Italy

    Paolo Antonelli

  2. CNRS and University Montpellier, Mathématiques CC 051, 34095, Montpellier, France

    Rémi Carles

  3. Center for Mathematical Analysis, Geometry and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Jorge Drumond Silva

Authors
  1. Paolo Antonelli
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  2. Rémi Carles
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  3. Jorge Drumond Silva
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Correspondence to Rémi Carles.

Additional information

Communicated by W. Schlag

This work was supported by the French ANR projects SchEq (ANR-12-JS01-0005-01) and BECASIM (ANR-12-MONU-0007-04). J. Drumond Silva was partially funded by FCT/Portugal through Project PEst-OE/EEI/LA0009/2013 and Grants PTDC/MAT114397/2009, UTA_CMU/MAT/0007/2009.

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Antonelli, P., Carles, R. & Silva, J.D. Scattering for Nonlinear Schrödinger Equation Under Partial Harmonic Confinement. Commun. Math. Phys. 334, 367–396 (2015). https://doi.org/10.1007/s00220-014-2166-y

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  • DOI: https://doi.org/10.1007/s00220-014-2166-y

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