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

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.

arXiv comment23 pages, no figure

References in corpus (1)

• Bound states of two-dimensional Schrödinger-Newton equations

Cited by in corpus (2)

• A short note on simplified pseudospectral methods for computing ground state and dynamics of spherically symmetric Schrodinger--Poisson--Slater system
• Energy solution to Schrödinger-Poisson system in the two-dimensional whole space