H. Langer, M. Langer and C. Tretter,
Variational principles for eigenvalues of block operator matrices,
Indiana Univ. Math. J. 51 (2002), 1427–1459
Abstract:
In this paper variational principles for block operator matrices are established
which are based on functionals associated with the quadratic numerical range.
These principles allow to characterize, e.g., eigenvalues in gaps of the
essential spectrum and to derive two-sided eigenvalue estimates in terms of the
spectral characteristics of the entries of the block operator matrix.
The results are applied to a second order partial differential equation
depending on the spectral parameter nonlinearly.