In this paper we study the asymptotic behavior of a quadratic Schrodinger 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 Schrodinger 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 Schrodinger equation on an $L^2$ weighted space for small potentials, as well as a dispersive estimate for the corresponding flow.

中文翻译：

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具有电磁扰动的 3D 二次 NLS 方程

在本文中，我们研究了具有电磁势的二次薛定谔方程的渐近行为。我们证明小解是分散的。证明建立在作者早期的非磁势二次 NLS 工作的基础上。主要的新颖之处在于使用线性薛定谔流的各种平滑估计代替波算子的有界性来处理导数的损失。作为证明的副产品，我们获得了线性电磁薛定谔方程的波算子在 $L^2$ 加权空间上对于小势能的有界性，以及对相应流的色散估计。