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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.