On spectra of a certain class of quadratic operator pencils with one-dimensional linear part

Abstract

We consider a class of quadratic operator pencils that occur in many problems of physics. The part of such a pencil linear with respect to the spectral parameter describes the viscous friction in problems of small vibrations of strings and beams. Patterns in location of eigenvalues of such pencils are established. If the viscous friction (damping) is pointwise, then the operator in the linear part of the pencil is one-dimensional. For this case, rules in the location of the purely imaginary eigenvalues are found.