Asymptotics of spectrum and eigenfunctions for nonselfadjoint operators generated by radial nonhomogeneous damped wave equations

Article type: Research Article

Authors: Shubov, Marianna A.

Affiliations: Texas Tech University, Department of Mathematics, P.O. Box 41042, Lubbock, TX 79409‐1042, USA Tel.: +1 806 742 2566 (office), +1 806 745 5238 (home); E‐mail: mshubov@ttmath.ttu.edu

Abstract: We consider an infinite sequence of radial wave equations obtained by the separation of variables in the spherical coordiantes from the 3‐dimensional damped wave equation with spacially nonhomogeneous spherically symmetric coefficients. The nonconservative boundary conditions are given on the sphere |x|=a. Our main objects of interest are the nonselfadjoint operators in the energy space of 2‐component initial data, which are the dynamics generators for the systems governed by the aforementioned equations and boundary conditions. Our main results are precise asymptotic formulas for the complex eigenvalues and eigenfunctions of these operators and the corresponding nonselfadjoint quadratic operator pencils. Based on the asymptotic results of the present work, we will show in a forthcoming paper that the sets of root vectors of the above operators and of the dynamics generator corresponding to the full 3‐dimensional damped wave equation form Riesz bases in the appropriate energy spaces. Therefore, we will show that all these operators are spectral in the sense of Dunford.

Journal: Asymptotic Analysis, vol. 16, no. 3-4, pp. 245-272, 1998