Abstract
We extend the classical concept of transformation operators to the one‐dimensional wave equation with spatially nonhomogeneous coefficients containing the first order damping term. The equation governs the vibrations of a damped string. Our results hold in the cases of an infinite, semi‐infinite or a finite string. Transformation operators were introduced in the fifties by I.M. Gelfand, B.M. Levitan and V.A. Marchenko in connection with the inverse scattering problem for the one‐dimensional Schrödinger equation. In the classical case, the transformation operator maps the exponential function (stationary wave function of a free particle) into the so‐called Jost solution of the perturbed Schrödinger equation. In our case, it is natural to introduce two transformation operators, which we call outgoing and incoming transformation operators respectively. (The terminology is motivated by an analog with the Lax–Phillips scattering theory.) The first of them is related to the nonselfadjoint quadratic operator pencil generated by the original problem, and the second one is related to the adjoint pencil. We introduce a pair of asymptotically exponential solutions for each of the pencils and show that our transformation operators map certain exponential type functions to these solutions. Our main results are the proof of the existence of transformation operators (which have the forms of the identity operator plus certain Volterra integral operators) and estimates for their kernels. To obtain these results, we derive a pair of integral equations for the kernels of the transformation operators. These equations are the generalizations of the corresponding classical equation which is valid in the case of a wave equation without damping term. One of possible applications of the method developed in this paper is given in our forthcoming work. In that work, we use the transformation operators to prove the fact that the dynamics generator of a finite string with damping both in the equation and in the boundary conditions is a Riesz spectral operator. The latter result provides a class of nontrivial examples of nonselfadjoint operators which admit an analog of the spectral decomposition. The result also has significant applications in the control theory of distributed parameter systems.