Scattering theory for dissipative hyperbolic systems

Abstract

An abstract theory of scattering is developed for dissipative hyperbolic systems, a typical example is the wave equation: u tt = Δu in an exterior domain with lossy boundary conditions: u n + αu t = 0, α ⩾ 0. In this theory, as in an earlier theory developed by the authors for conservative systems, a central role is played by two distinguished subspaces of data common to both the perturbed and unperturbed problems. Associated with each subspace is a translation representation of the unperturbed system. When these representations coincide they provide a convenient tool for extending the data so as to include a large class of generalized eigenfunctions for both the perturbed and unperturbed generators. The scattering matrix is characterized in terms of these generalized eigenfunctions; it is shown to be meromorphic in the whole complex plane and holomorphic in the lower half plane. The zeroes and poles of the scattering matrix correspond, respectively, to incoming and outgoing generalized eigenfunctions of the perturbed generator. In the second part of the paper the assumptions introduced in the abstract theory are verified for the wave equation problem cited above.