Variational principles and eigenvalue estimates for unbounded block operator matrices and applications
type: Item
instance of: scholarly article
title: Variational principles and eigenvalue estimates for unbounded block operator matrices and applications
zbMATH Open document ID: 1066.49030
author: Margarita Kraus
author: Matthias Langer
author: Christiane Tretter
published in: Journal of Computational and Applied Mathematics
publication date: 2004-08-10T00:00:00Z
review text: The authors consider three cases, the so-called ``top dominant'', the ``diagonal dominant'', and the ``off-diagonal dominant'' case depending on the position of the operators with smallest domain. In the case of differential operators, the dominating operators are those with the highest order. Examples for top dominant block operator matrices occur in magnetohydrodynamics or astrophysics, examples for the off-diagonal case appear in quantum mechanics (e.g., Dirac operators). They establish variational principles, eigenvalue estimates and asymptotic formulae for eigenvalues of these classes of unbounded block operator matrices. The results allow to characterise eigenvalues that are not necessarily located at the boundary spectrum. Applications to an example from magnetohydrodynamics and to Dirac operators on certain manifolds are given.
reviewed by: Bilender P. Allahverdiev
Mathematics Subject Classification ID: 49R50
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 76W05
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 47A75
zbMATH DE Number: 2085599
zbMATH Keywords: variational principle for eigenvalues
zbMATH Keywords: estimates for eigenvalues
zbMATH Keywords: asymptotic distribution of eigenvalues
zbMATH Keywords: quadratic numerical range
zbMATH Keywords: magnetohydrodynamics
zbMATH Keywords: warped product of spin manifolds
zbMATH Keywords: Dirac operator
MaRDI profile type: Publication
full work available at URL: j.cam.2004.01.024
OpenAlex ID: W2166959906
cites work: The Essential Spectrum of Some Matrix Operators
cites work: On the eigenvalues of operators with gaps. Application to Dirac operators
cites work: A Minimax Principle for the Eigenvalues in Spectral Gaps
cites work: Compact four-dimensional Einstein manifolds
cites work: Q4110164
cites work: Asymptotic estimates of Dirac and Laplace eigenvalues on warped products over \(S^1\)
cites work: A new method for eigenvalue estimates for Dirac operators on certain manifolds with \(S^{k}\)-symmetry
cites work: Variational principles for eigenvalues of block operator matrices
cites work: A new concept for block operator matrices: The quadratic numerical range
cites work: Q4529287
cites work: Q3995493
cites work: Q4040679
cites work: The spectrum of a linear magnetohydrodynamic model with cylindrical symmetry
cites work: Q4185363
cites work: Parallel spinors and parallel forms
cites work: On non-simply connected manifolds with non-trivial parallel spinors
cites work: Methods of intermediate problems for eigenvalues. Theory and ramifications
DOI: 10.1016/J.CAM.2004.01.024
description: scientific article; zbMATH DE number 2085599