Large time WKB approximation for multi-dimensional semiclassical Schr\"odinger-Poisson system

TL;DR

This paper extends the WKB approximation for the multi-dimensional semiclassical Schr"odinger-Poisson system, demonstrating its validity over large time intervals for specific large, radially symmetric initial data in the focusing case.

Contribution

It provides a new example of large initial data where the WKB approximation remains valid over arbitrarily large times in higher dimensions.

Findings

WKB approximation justified for large data with slow-decaying phase.

Analysis of Euler-Poisson equations for initial data construction.

Extension of previous results to larger time intervals and data sizes.

Abstract

We consider the semiclassical Schr\"odinger-Poisson system with a special initial data of WKB type such that the solution of the limiting hydrodynamical equation becomes time-global in dimensions at least three. We give an example of such initial data in the focusing case via the analysis of the compressible Euler-Poisson equations. This example is a large data with radial symmetry, and is beyond the reach of the previous results because the phase part decays too slowly. Extending previous results in this direction, we justify the WKB approximation of the solution with this data for an arbitrarily large interval of $\R_+$.