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- W2017576275 abstract "Let H be a selfadjoint operator on a separable Hilbert space H. Then the set of all bounded operators X such that $${{text{X}}_{text{t}}}{text{u: = }}{{text{e}}^{{text{itH}}}}{text{X }}{{text{e}}^{{text{ - itH}}}}{text{u , t }}{mathbf{}}{text{ }}{mathbf{R}},$$ (1) tend “very fast” to limits X± u as t → ± ∞ for vectors u ε D ± (special linear manifolds) is called the set of smooth asymptotic constants for H (see e.g. [3] and [8]). In particular if there is a strongly continuous unitary representation U(g) of the restricted Poincaré group P + ↑ g on H such that e-itH is the corresponding subrepresentation of the time translation group of P + ↑ э and the limits X± of the smooth asymptotic constant X commute with U(g), g ε P + ↑ , then we call X a smooth asymptotic constant for the Poincaré group. Such operators were studied in [3], [4] and were used for the construction of special quantum fields in [4]. The aim of this paper is to extend some results of [3], [4], and [5] with the help of the statements in [8] and [9]. In particular the following question is solved affirmatively in the present paper: Let H be the symmetric Fock space and U(g) the usual representation of P + ↑ on H. Further let A, B be two unitary operators on H which commute with U(g), g ε P + ↑ , and satisfy a certain smoothness property explained later. Does there exist a smooth asymptotic constant X such that X is unitary, XU(g) = U(g)X for g ε Λ + ↑ (restricted Lorentz group), and the limits X+ and X− coincide with the given operators A and B, respectively?" @default.
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- W2017576275 date "1988-01-01" @default.
- W2017576275 modified "2023-10-18" @default.
- W2017576275 title "On Smooth Asymptotic Constants for the Poincare Group. II" @default.
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- W2017576275 doi "https://doi.org/10.1007/978-3-0348-9164-6_21" @default.
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