Author: Pavlov B.
Publisher: Springer Publishing Company
ISSN: 0020-7748
Source: International Journal of Theoretical Physics, Vol.38, Iss.1, 1999-01, pp. : 21-45
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Abstract
Since the publication of the very first papers on quantum mechanics the theory of self-adjoint operators in Hilbert space has been a basic tool of quantum theories. It turns out that the description of the irreversible dynamics of complex systems requires the development of the spectral theory of non-self-adjoint operators as well. In this paper we consider the Hilbert space version of the theory of dissipative operators, which appear as generators of the evolution reduced to a properly selected observation subspace. The spectral analysis of these operators is based on ideas of the functional model and dilation theory rather than on traditional resolvent analysis and Riesz integrals. The role of the parameter of the functional model is played by an analytic function — the characteristic function —which is interpreted and calculated as a scattering matrix for the relevant scattering problem. Thus the most important object of the spectral analysis of dissipative operators appears as an element of spectral analysis of a self-adjoint spectral problem. This paper is intended both as an introduction and a sort of bilingual text for specialists in harmonic analysis and operator theory who are interested in mathematical problems of the description of irreversible dynamics. The last part describes original results of the author published in different journals during the last decade.