Variational principles for eigenvalues of self-adjoint operator functions

D. Eschwe, M. Langer

Research output: Contribution to journalArticle

31 Citations (Scopus)

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.
LanguageEnglish
Pages287-321
Number of pages35
JournalIntegral Equations and Operator Theory
Volume49
Issue number3
DOIs
Publication statusPublished - Jul 2004

Fingerprint

Self-adjoint Operator
Variational Principle
Eigenvalue
Zero
Operator Matrix
Quadratic Polynomial
Unbounded Operators
Essential Spectrum
Sturm-Liouville Problem
Block Matrix
Dirac Operator
Rayleigh
Value Function
Elliptic Problems
Eigenvalue Problem
Assign
Friction
Strings

Keywords

  • variational principle
  • operator function
  • Schur complement

Cite this

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

In: Integral Equations and Operator Theory, Vol. 49, No. 3, 07.2004, p. 287-321.

Research output: Contribution to journalArticle

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AU - Langer, M.

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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.

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KW - operator function

KW - Schur complement

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