Regular Couplings of Dissipative and Anti-Dissipative Unbounded Operators, Asymptotics of the Corresponding Non-Dissipative Processes and the Scattering Theory

摘要

In this paper a triangular model of a class of unbounded non-selfadjoint K r -operators A presented as a coupling of dissipative and anti-dissipative operators in a Hilbert space with real absolutely continuous spectra and with different domains of A and A * is considered. The asymptotic behaviour of the corresponding non-dissipative processes T t f = e itA f, generated from the semigroups T t with generators iA, as t → ± ∞ are obtained. The strong wave operators, the scattering operator for the couple (A *, A) and the similarity of A and the operator of multiplication by the independent variable are obtained explicitly. The considerations are based on the triangular models and characteristic functions of A. Kuzhel for unbounded operators and the limit values of the multiplicative integrals, describing the characteristic function of the considered model.