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

TL;DR

This paper analyzes the spectral properties of quadratic operator pencils with a rank-one gyroscopic linear part, revealing eigenvalue types and their locations, with applications to Sturm-Liouville operators.

Contribution

It characterizes the eigenvalues of quadratic operator pencils with rank-one gyroscopic parts, including their independence from certain operators and their spectral location.

Findings

Eigenvalues are of two types, with one type independent of G.

Locations of eigenvalues are explicitly described.

Examples include Sturm-Liouville operators.

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.