On spectra of quadratic operator pencils with rank one gyroscopic linear part

Abstract

The spectrum of a selfadjoint quadratic operator pencil of the form \(\lambda^2M-\lambda G-A\) is investigated where \(M\geq 0\), \(G\geq 0\) are bounded operators and \(A\) is selfadjoint bounded below is investigated. It is shown that in the case of rank one operator \(G\) the eigenvalues of such a pencil are of two types. The eigenvalues of one of these types are independent of the operator \(G\). Location of the eigenvalues of both types is described. Examples for the case of the Sturm-Liouville operators \(A\) are given.

Keywords

• quadratic operator pencil
• gyroscopic force
• eigenvalues
• algebraic multiplicity