An eigenvalue problem for the Schrödinger-Maxwell equations
type: Item
instance of: scholarly article
title: An eigenvalue problem for the Schrödinger-Maxwell equations
zbMATH Open document ID: 0926.35125
DOI: 10.12775/TMNA.1998.019
author: Donato Fortunato
author: Vieri Benci
published in: Topological Methods in Nonlinear Analysis
publication date: 1999-03-28T00:00:00Z
review text: The authors study the eigenvalue problem for the Schrödinger operator coupled with the electromagnetic field \(E\), \(H\). The authors do not assume that the electromagnetic field is assigned, then they study a system of equations whose unknowns are the wave function \(\psi(x,t)\) and the gauge potentials \(A(x,t)\), \(\varphi(x,t)\) related to \(E\), \(H\). They investigate the case in which \(A\) and \(\varphi\) do not depend on time \(t\) and \(\psi(x,t)= u(x)e^{iwt}\), \(u\) a real function and \(\omega\) a real number. The problem is reduced to studying the existence of a real number \(\omega\) and real functions \(u\), \(\psi\) satisfying the system \[ -\textstyle{{1\over 2}}\Delta u- \varphi u=\omega u,\quad \Delta\varphi= 4\pi u^2\quad\text{in }\Omega,\quad u(x)= 0,\quad \varphi(x)= g\quad\text{on }\partial\Omega,\tag{1} \] \(\| u\|_{L_2}= 1\). Here \(g\) is an assigned function and \(\partial\Omega\) is the boundary of an open bounded set \(\Omega\subset \mathbb{R}^3\). The authors prove that there exists a sequence \((\omega_n,u_n,\varphi_n)\) with \(\omega_n\in \mathbb{R}\), \(\omega_n\to\infty\) and \(u_n\), \(\varphi_n\) real functions, solving (1).
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 1270104
zbMATH Keywords: Schrödinger-Maxwell equations
zbMATH Keywords: variational methods
zbMATH Keywords: electromagnetic field
zbMATH Keywords: wave function
zbMATH Keywords: gauge potentials
MaRDI profile type: Publication
full work available at URL: tmna.1998.019
OpenAlex ID: W2603154707
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description: scientific article; zbMATH DE number 1270104