The spectra of two-parameter quadratic operator pencils

Abstract

This paper is devoted to a variational theory of the eigenvalue spectra of two-parameter, unbounded operator pencils, of the so-called waveguide type, which have important physical applications. It is known that the classical, nonlinear variational theory of these spectra is only applicable to eigenvalues with a definite type (positive or negative), provided ± type eigenvalues are separated. However, the classical theory has recently been extended to mixed-type eigenvalues: first for linear matrix and operator pencils of the form L ( λ ) = λ A − B , and later for self-adjoint operators in a Krein space (see Binding et al. (2005) [17] and references therein). The real eigenvalues of the operator pencils studied in this paper can be split into five intervals according to type, the central range containing mixed-type eigenvalues (see Fig. 3). The main goal of this paper is to investigate the variational principles for real eigenvalues in this zone. In addition, this paper studies neutral (resonance) pairs and their existence criteria. We provide a complete analysis of the spectra of two-parameter, waveguide-type operator pencils. We use a variational approach for the spectrum of non-overdamped pencils, relying on Krein space techniques and the Ljusternik-Schnirelman theory of critical points of nonlinear functionals. By using the Ljusternik-Schnirelman theory we investigate neutral eigenvalues in a mixed spectral zone.