Abstract
Since the publication of the very first paperson quantum mechanics the theory of self-adjointoperators in Hilbert space has been a basic tool ofquantum theories. It turns out that the description of the irreversible dynamics of complex systemsrequires the development of the spectral theory ofnon-self-adjoint operators as well. In this paper weconsider the Hilbert space version of the theory ofdissipative operators, which appear as generators of theevolution reduced to a properly selected observationsubspace. The spectral analysis of these operators isbased on ideas of the functional model and dilation theory rather than on traditional resolventanalysis and Riesz integrals. The role of the parameterof the functional model is played by an analyticfunction — the characteristic function —which is interpreted and calculated as a scattering matrix for therelevant scattering problem. Thus the most importantobject of the spectral analysis of dissipative operatorsappears as an element of spectral analysis of a self-adjoint spectral problem. This paper isintended both as an introduction and a sort of bilingualtext for specialists in harmonic analysis and operatortheory who are interested in mathematical problems of the description of irreversible dynamics.The last part describes original results of the authorpublished in different journals during the lastdecade.
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Pavlov, B. Irreversibility, Lax-Phillips Approach to Resonance Scattering and Spectral Analysis of Non-Self-Adjoint Operators in Hilbert Space. International Journal of Theoretical Physics 38, 21–45 (1999). https://doi.org/10.1023/A:1026624905808
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DOI: https://doi.org/10.1023/A:1026624905808