tailieunhanh - Báo cáo toán học: "A nuclear dissipative scattering theory "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một tiêu tán lý thuyết tán xạ hạt nhân. | J. OPERATOR THEORY 14 1985 57 66 Copyright by INCREST 1985 A NUCLEAR DISSIPATIVE SCATTERING THEORY HAGEN NEIDHARDT J. INTRODUCTION Let and 2 be two separable Hilbert spaces with the norms II 111 and II l2 and the scalar products 1 and 2 respectively. By J we denote a linear bounded operator from to 2 which is called the identification operator. Let Ifi and Hn be two maximal dissipative operators on the Hilbert spaces ố and Ộ respectively. We call an operator H on the Hilbert space a dissipative one if Im 7 n c 0 holds for every fe 3i H . A dissipative operator which has no proper dissipative extensions is called maximal dissipative. It is well known that H is maximal dissipative if and only if if is a generator of a one-parameter contraction semigroup on Ộ. The spectrum cr of a maximal dissipative operator is situated in the lower half plane . ff H c zeCJmz O . In the following we establish the existence of the wave operator w - s-lime je- Pac -Ị. CO under a trace condition. By Pac Hi we denote the projection onto the absolutely continuous subspace ac Wj of the maximal dissipative operator Hi. If e-i w t ặ 0 is an one-parameter contraction semigroup on Ộ there is a unique orthogonal decomposition - . where ộu and . are invariant and e iiH is unitary on ộu and completely non-- unitary on . . An one-parameter contraction semigroup T t on Ộ is called completely non-unitary if none of the nontrivial subspaces of reduces all the 58 HAGEN NEIDHARDT operators Tit 0 to unitary ones. In accordance with we have a decomposition H Hu . where Hu is a selfadjoint operator on ộu and . is a maximal dissipative operator on . generating a completely non-unitary one-parameter contraction semigroup. The subspace ộac W ac u . is called the absolutely continuous subspace of the maximal dissipative operator H. It is clear that .ộac f reduces the operator H. The subspace Ốs Ỉ ac H is called the singularly .