Local existence and WKB approximation of solutions to Schrödinger-Poisson system in the two-dimensional whole space

2009/12/08 by Satoshi Masaki, Masaki, Satoshi · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics #math-ph #math.AP #math.MP #msc:35Q55

paper · pdf · doi:10.48550/arxiv.0912.1388

23 pages, no figure

arxiv created 2009/12/08 · arxiv updated 2010/01/07

Abstract

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.

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• Energy solution to Schrödinger-Poisson system in the two-dimensional whole space

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