LINEAR VS. NONLINEAR EFFECTS FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH POTENTIAL

2004/05/21 by Remi Carles, RÉMI CARLES · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #math-ph #math.AP #math.MP #msc:35A05 #msc:35B30 #msc:35B35 #msc:35Q55

paper · pdf · doi:10.1142/s0219199705001829

published as Commun. Contemp. Math., Vol. 7, No. 4 (2005) 483-508 · 20 pages; survey of previous results

arxiv created 2004/05/21 · openalex publication_date 2005/08/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We review some recent results on nonlinear Schrödinger equations with potential, with emphasis on the case where the potential is a second order polynomial, for which the interaction between the linear dynamics caused by the potential, and the nonlinear effects, can be described quite precisely. This includes semi-classical régimes, as well as finite time blow-up and scattering issues. We present the tools used for these problems, as well as their limitations, and outline the arguments of the proofs.

Citations

• Spectral properties of Schrödinger operators and time-decay of the wave functions
• Time decay for solutions of Schr�dinger equations with rough and time-dependent potentials
• Uniform decay estimates and the lorentz invariance of the classical wave equation
• Nonlinear Schr�dinger equations and sharp interpolation estimates

Cited by

• Semi-classical limit of Schrödinger–Poisson equations in space dimension n⩾3

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