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The scattering problem for a noncommutative nonlinear Schrödinger equation. (English) Zbl 1219.35278

Summary: We investigate scattering properties of a Moyal deformed version of the nonlinear Schrödinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general initial data has a unique globally defined solution, and also has solitary wave solutions if the interaction potential is suitably chosen. We demonstrate how to set up a scattering framework for equations of this type, including appropriate decay estimates of the free time evolution and the construction of wave operators defined for small scattering data in the general case and for arbitrary scattering data in the rotationally symmetric case.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
58B34 Noncommutative geometry (à la Connes)
53D55 Deformation quantization, star products
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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