# Variational principles for eigenvalues of self-adjoint operator functions

Eschwe, D. and Langer, M.
(2004)
*Variational principles for eigenvalues of self-adjoint operator functions.*
Integral Equations and Operator Theory, 49 (3).
pp. 287-321.
ISSN 0378-620X

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## Abstract

Variational principles for eigenvalues of certain functions whose values are possibly unbounded self-adjoint operators T(λ) are proved. A generalised Rayleigh functional is used that assigns to a vector x a zero of the function (T(λ)x, x), where it is assumed that there exists at most one zero. Since there need not exist a zero for all x, an index shift may occur. Using this variational principle, eigenvalues of linear and quadratic polynomials and eigenvalues of block operator matrices in a gap of the essential spectrum are characterised. Moreover, applications are given to an elliptic eigenvalue problem with degenerate weight, Dirac operators, strings in a medium with a viscous friction, and a Sturm-Liouville problem that is rational in the eigenvalue parameter.

Item type: | Article |
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ID code: | 2176 |

Keywords: | variational principle, operator function, Schur complement, Mathematics, Algebra and Number Theory, Analysis |

Subjects: | Science > Mathematics |

Department: | Faculty of Science > Mathematics and Statistics |

Depositing user: | Strathprints Administrator |

Date Deposited: | 04 Jan 2007 |

Last modified: | 06 Jan 2017 03:35 |

URI: | http://strathprints.strath.ac.uk/id/eprint/2176 |

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