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Local existence and WKB approximation of solutions to Schrödinger-Poisson system in the two-dimensional whole space. (English) Zbl 1232.35155
Summary: We consider the Schrödinger-Poisson system in the two-dimensional whole space. A new formula of solutions to the Poisson equation is used. Although the potential term solving the Poisson equation may grow at the spatial infinity, we show the unique existence of a time-local solution for data in the Sobolev spaces by an analysis of a quantum hydrodynamical system via a modified Madelung transform. This method has been used to justify the WKB approximation of solutions to several classes of nonlinear Schrödinger equation in the semiclassical limit.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35B25 Singular perturbations in context of PDEs
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