### Abstract

Language | English |
---|---|

Pages | 287-321 |

Number of pages | 35 |

Journal | Integral Equations and Operator Theory |

Volume | 49 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jul 2004 |

### Fingerprint

### Keywords

- variational principle
- operator function
- Schur complement

### Cite this

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*Integral Equations and Operator Theory*, vol. 49, no. 3, pp. 287-321. https://doi.org/10.1007/s00020-002-1209-5

**Variational principles for eigenvalues of self-adjoint operator functions.** / Eschwe, D.; Langer, M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Variational principles for eigenvalues of self-adjoint operator functions

AU - Eschwe, D.

AU - Langer, M.

PY - 2004/7

Y1 - 2004/7

N2 - 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.

AB - 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.

KW - variational principle

KW - operator function

KW - Schur complement

U2 - 10.1007/s00020-002-1209-5

DO - 10.1007/s00020-002-1209-5

M3 - Article

VL - 49

SP - 287

EP - 321

JO - Integral Equations and Operator Theory

T2 - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

SN - 0378-620X

IS - 3

ER -