The Riesz basis property of the system of root vectors for the equation of a nonhomogeneous damped string: Transformation operators method

We prove the spectral decomposition theorem for a class of nonselfadjoint operators in a Hilbert space.These operators are the dynamics generators for the systems governed by 1-dimensional hyperbolic equations with spacially nonhomogeneous coefficients containing first order damping terms and subject to linear nonself adjoint boundary conditions.These equations and boundary conditions describe, in particular, a spacially nonhomogeneous string subject to a distributed viscous damping and damped at the boundary points.Our main result leading to the spectral decomposition is the fact that the root vectors (eigenvectors and associated vectors together) of the above operators form Riesz bases in the corresponding energy spaces.Our proofs are based on the transformation operator method suggested in our previous work [1].In that work, we extended the classical concept of transformation operators to the equation of a damped string.Originally, this concept was developed by I.M. Gelfand, B.M. Levitan and V.A. Marchenko for 1-dimensional Schrodinger equation in connection with the inverse scattering problem.Using the transformation operator method, we reduce the problem of Riesz basis property for the root vectors to the similar problem for a system of nonharmonic exponentials whose complex frequencies are precisely the eigenvalues of our operators.The latter problem is solved based on the spectral asymptotics and known facts about the families of nonharmonic exponentials.