Local Existence and WKB Approximation of Solutions to Schrödinger–Poisson System in the Two-Dimensional Whole Space
Local Existence and WKB Approximation of Solutions to Schrödinger–Poisson System in the Two-Dimensional Whole Space
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DOI：
10.1080/03605301003717142
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发表时间：
2009-12
影响因子：
1.9
通讯作者：
Satoshi Masaki
中科院分区：
文献类型：
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作者：
Satoshi Masaki
我们考虑二维整体空间中的Schrödinger-Poisson系统。使用了泊松方程的一个新的解公式。尽管泊松方程的潜在解项可能在空间无穷大处增长，但我们通过修正马德隆变换对量子流体动力系统进行分析，证明了Sobolev空间中数据的时间局部解的唯一存在性。该方法已用于证明几种非线性Schrödinger方程在半经典极限下解的WKB近似。
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.