Asymptotics of resonances and eigenvalues for nonhomogeneous damped string

Abstract

We consider a semi-infinite and a finite string, both with a nonconstant density and subject to a positive viscous damping. In the first case it is assumed that both the density nonhomogeneity and the damping are concentrated on the finite interval [0,a]. Our main results are explicit asymptotic formulas for the resonances in the case of the semi-infinite string and for the eigenvalues in the case of the finite string. These formulas are obtained for three types of the density ρ(x): (i) ρ is bounded on [0,a] and (in the case of semi-infinite string) has a finite jump at x=a; (ii) ρ(a−0)=∞ (infinite condensation at x=a); (iii) ρ(a−0)=0 (infinite rarefaction at x=a). The result of this work is the first step in the spectral analysis of non-selfadjoint operator pencils generated by the equation of a nonhomogeneous damped string. Based on the results of this paper it will be shown in our forthcoming work that the systems of the quasimodes (of the infinite string) and the eigenmodes (of the finite string) have the Riesz-basis property.