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

Boyko O., Martynyuk O., Pivovarchik V.

Opuscula Mathematica

2018

Vol. 38, no. 4

483--500

The spectrum of a selfadjoint quadratic operator pencil of the form [formula] is investigated where M ≥ 0, G ≥ 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: q

The Sturm-Liouville inverse spectral problem with boundary conditions depending on the spectral parameter

Van der Mee C., Pivovarchik V.

Opuscula Mathematica

2005

Vol. 25, no. 2

243-260

We present the complete version including proofs of the results announced in [1]. Namely, for the problem of small transversal vibrations of a damped string of nonuniform stiffness with one end fixed we give the description of the spectrum and solve the inverse problem: find the conditions which should be satisfied by a sequence of complex numbers to be the spectrum of a damped string.