Solitary charged waves interacting with the electrostatic field
type: Item
instance of: scholarly article
title: Solitary charged waves interacting with the electrostatic field
zbMATH Open document ID: 1091.35074
author: Teresa D'Aprile
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-04-06T00:00:00Z
review text: The Schrödinger-Maxwell equation is solved for the case of a charged particle interacting with its own electromagnetic field. The semiclassical limit \(\hbar\to{0^{+}}\) with unknown potential \(\Phi\) and magnetic potential \(A=0\) is considered for the system \[ -\hbar^{2}\Delta{v}+e\Phi{v}=\lambda{v}, \qquad -\Delta\Phi=4\pi{e}{v}^{2}. \] in the unit ball \(B_{1}\). The main theorem is a statement that this system for every \(\hbar\) has a solution \((v_{\hbar},\Phi_{\hbar},\lambda_{\hbar})\) which belongs to the Sobolov space \(H^{1}(B_{1})\). Each \((v_{\hbar},\Phi_{\hbar})\) is obtained as a stationary point of a suitable functional having \(\lambda=\lambda_{\hbar}\) as a Lagrangian multiplier for the constraint \(\| v\| _{L_{2}}=1\).
reviewed by: Thomas Ernst
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 78A35
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 81Q20
zbMATH DE Number: 5017398
zbMATH Keywords: charged particle
zbMATH Keywords: semiclassical limit
zbMATH Keywords: Lagrangian multiplier
zbMATH Keywords: Sobolev space
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.06.036
OpenAlex ID: W2147415801
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DOI: 10.1016/J.JMAA.2005.06.036
description: scientific article; zbMATH DE number 5017398