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3D quadratic NLS equation with electromagnetic perturbations. (English) Zbl 1455.35237

Summary: In this paper we study the asymptotic behavior of a quadratic Schrödinger equation with electromagnetic potentials. We prove that small solutions scatter. The proof builds on earlier work of the author for quadratic NLS with a non magnetic potential. The main novelty is the use of various smoothing estimates for the linear Schrödinger flow in place of boundedness of wave operators to deal with the loss of derivative. As a byproduct of the proof we obtain boundedness of the wave operator of the linear electromagnetic Schrödinger equation on an \(L^2\) weighted space for small potentials, as well as a dispersive estimate for the corresponding flow.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q41 Time-dependent Schrödinger equations and Dirac equations
78A45 Diffraction, scattering
78A60 Lasers, masers, optical bistability, nonlinear optics
35B40 Asymptotic behavior of solutions to PDEs
42B25 Maximal functions, Littlewood-Paley theory
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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References:

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