Kuzhel S. On elements of the Lax-Phillips scheme in the scattering theory

The object is the Lax-Phillips scattering scheme; the aim is the more detail development of this scheme for abstract realizations of evolution systems, for which the Lax-Phillips scheme is used very often; the methods are the operator theory,the theory of positive boundary value spaces; the novelty consists in the following: for the first time the necessary and sufficient conditions are obtained, under which the Lax-Phillips scheme can be applied to the investigation of an abstract wave equation, for this equation, properties of operators, which determine unperturbed and perturbed evolution are studied, analogs of all principal results of the Lax-Phillips sheme to the case of Pontryagin spaces are obtained; the results are following: a new method of construction of scattering matrix is proposed, the inverse scattering problem are investigated, a new description of singularity points of the scattering matrix is obtained, the sphere of application is the investigation of scattering of evolution systems.