D. Eschwé and M. Langer,
Variational principles for eigenvalues of self-adjoint operator functions,
Integral Equations Operator Theory 49 (2004), 287–321
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.