TAILIEUCHUNG - Báo cáo toán học: "Existance and completeness of the wave operators for dissipative hyperbolic systems"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài:Sự tồn tại và đầy đủ của các nhà khai thác sóng cho các hệ thống tiêu tán hyperbol. | J. OPERATOR THEORY 14 1985 291 - 310 C Copyright by INCREST 1985 EXISTENCE AND COMPLETENESS OF THE WAVE OPERATORS FOR DISSIPATIVE HYPERBOLIC SYSTEMS V. s. GEORGIEV 1. INTRODUCTION In the recent years scattering theory has been extensively studied both for the Schrodinger equation and the classical hyperbolic systems 1 4 10 23 . A new elegant method treating the Schrodinger equation has been developed by Enss in 2 4 On the other hand Lax and Phillips studied in 10 13 the scattering for the classical hyperbolic systems by using an abstract approach. In this theory a central role is played by two spaces Z and ZE with the properties i Uữ t D c. D_ for t ặ 0 1-1 ii n ư0 t Dị 0 t iii u Uo t M o t Here tzo z is a group of unitary operators acting on a Hilbert space .5 0. The perturbed system is described by a semigroup K í Í 0 of contraction operators acting on a Hilbert space ye cz yeữ. In order to build a scattering theory Lax and Phillips assumed in 12 the following properties ZE c JT tf0 - 0 V t d d for d E D t 0 U0 f V t d d for deD_ t 0. Let p denote the orthogonal projection onto the orthogonal complement of z in Jf. The condition limP F i 0 for e ye t- oo 292 V. s. GEORGÍEV is one of the basic assumptions in 12 and it leads to the main difficulties in the applications of the abstract theory. The proof of is given by Lax and Phillips in 12 only for the wave equation in the exterior of a bounded obstacle K c R n 2. The investigation of the case when n is even is rather technical. Recently Petkov considered in 17 the mixed problems for dissipative hyperbolic systems provided n odd. He proved the existence of the wave operator w introduced below without applying the abstract approach in 12 . However the proof in 17 is based essentially on the strong Huyghens principle fulfilled for odd space dimensions. Moreover in 17 some restrictions on the point spectrum of the generator G of Vịt has been imposed. The main purpose of this work is to prove the existence

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