Uniform bounds and weak solutions to an open Schrödinger-Poisson system. (English) Zbl 1135.35078

Summary: This paper is concerned with the derivation of uniform bounds with respect to the scaled Planck constant \(\varepsilon\) for solutions to the open transient Schrödinger-Poisson system introduced by N. Ben Abdallah, F. Méhats and O. Pinaud [Math. Models Methods Appl. Sci. 15, No. 5, 667–688 (2005; Zbl 1076.22017)]. The uniform estimates are obtained from a careful analysis of the non-local in time transparent boundary conditions which allow to restrict the original problem posed on an unbounded domain to a bounded domain of interest. These bounds can be used to obtain the semi-classical limit of the system. The paper also gives existence and uniqueness result for weak solutions while they were previously defined in a strong sense.


35Q55 NLS equations (nonlinear Schrödinger equations)
35B45 A priori estimates in context of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
82D37 Statistical mechanics of semiconductors


Zbl 1076.22017
Full Text: DOI