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Resonances in the rigged Hilbert space and Lax-Phillips scattering theory. (English) Zbl 1050.81066

The aim of the article is to uncover certain relations between the Lax-Phillips scattering theory, developed originally for acoustic or electromagnetic systems, and the rigged Hilbert space formalism. To this end the essentials of the Lax-Phillips scattering theory are presented in order to put the theory in a more general framework. The construction fits into the theory of B. Sz.-Nagy and C. Foias [Harmonic analysis of operators on Hilbert spaces. Amsterdam: North-Holland (1970; Zbl 0201.45003)] on contraction operators in Hilbert spaces.
The last section is devoted to the development of a rigged Hilbert space formalism for resonances in the framework of the Lax-Phillips scattering theory. Considering operators of the form \(H = H_{0} +V\) the assumptions on the spectrum of \(H\) given in [A. Bohm and M. Gadella, Dirac kets, Gamov vectors and Gel’fand triplets. Berlin: Springer (1989; Zbl 0691.46051)] are imposed here as well. Additional assumptions are made in order to be able to use the theory of Hardy spaces.

MSC:

81U05 \(2\)-body potential quantum scattering theory
81U99 Quantum scattering theory
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
47A40 Scattering theory of linear operators
47N50 Applications of operator theory in the physical sciences
46E15 Banach spaces of continuous, differentiable or analytic functions
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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