Uniform bounds and weak solutions to an open Schrödinger-Poisson system

This paper is concerned with the derivation of uniform bounds with respect to the scaled Planck constant ${\cal E}$ for solutions to the open transient Schrödinger-Poisson system introduced by Ben Abdallah et al in [Math.Meth.Mod. in App. Sci., 15, 667-688, 2005]. The uniform estimates stem 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 an existence and uniqueness result for weak solutions while they were previously defined in a strong sense.

Keywords

semiconductors, non-linear Schrödinger equation, open boundary conditions, uniform estimates

2010 Mathematics Subject Classification

35Q40, 35Q55

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Published 1 January 2007