The Schrödinger-Poisson eigenmatrix problem
type: Item
instance of: scholarly article
title: The Schrödinger-Poisson eigenmatrix problem
zbMATH Open document ID: 0893.35120
author: B. Toomire
author: Horst Lange
author: Paul F. Zweifel
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-12-09T00:00:00Z
review text: The existence of at least one stationary solution of the Schrödinger-Poisson system (modeling some semiconductor devices) is proved. After some transformations the nonlinear system to be solved is as follows: \(m\in {\mathbf N}\); \({\mathbf x}\in Q_{L}\) where \(Q_{L}\) is a cube in \({\mathbb{R}}^{d}\), \(d=1,2,3\); \(\Phi ={\phi_{m}({\mathbf x})}_{m\in {\mathbf N}}\), \(\phi_{m}({\mathbf x})\) satisfy periodic boundary conditions and: \[ -{\textstyle \frac{1}{2}} \Delta \phi _{m}({\mathbf x}) + (V_{\Phi}({\mathbf x}) +\widetilde{V} ({\mathbf x})) \phi_{m}({\mathbf x})=\mu_{m} \phi_{m}({\mathbf x}), \] \[ -\Delta V_{\Phi}({\mathbf x}) =n_{\Phi }({\mathbf x}) -n_{D}({\mathbf x}),\quad n_{\Phi}({\mathbf x})=\sum_{m}\lambda_{m}(\mu_{m}) |\phi_{m}({\mathbf x})|^{2}. \] The data are \(n_{D}({\mathbf x})\) and the functions \(\lambda_{m}(t)\). The existence of a solution \(\Phi\) is proved by a fixed point argument. In addition, bounds and an iteration scheme for computing \(\mu_{m}\) are given.
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 46N50
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 944355
zbMATH Keywords: stationary solution of the Schrödinger-Poisson system
zbMATH Keywords: fixed point argument
zbMATH Keywords: existence
zbMATH Keywords: semiconductor devices
zbMATH Keywords: iteration scheme
reviewed by: Gheorghe Nenciu
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0381
OpenAlex ID: W2085846193
DOI: 10.1006/JMAA.1996.0381
description: scientific article; zbMATH DE number 944355