Limiting Absorption Principles and Wave-Operators on L1(μ) Spaces: Applications to Transport Theory
✍ Scribed by M. Mokhtarkharroubi
- Publisher
- Elsevier Science
- Year
- 1993
- Tongue
- English
- Weight
- 687 KB
- Volume
- 115
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
Let (X=L^{1}(\mu)) and let ({S(t) ; t \in \mathbb{R}}) be a positive and bounded (\left(c_{0}\right))-group of linear operators on (X) with generator (T). Let (B \in \mathscr{L}(D(T) ; X)) be a positive operator and let ({V(t) ; t \in \mathbb{R}}) be the (\left(c_{0}\right))-group with generator (T+B). We prove that the boundedness of ({V(t) ; t \geqslant 0},{V(t), t \leqslant 0}) as well as the existence of the wave operators (\lim _{t \rightarrow \pm x} V(t) S(-t), \lim {t{\rightarrow+\infty}} S(-t) V(t)) are intimately connected to the existence of the strong limits (\lim _{\lambda \rightarrow 0 \pm} B(\lambda-T)^{-1}) and to their spectral radii. An optimal theory is given. Applications to scattering theory for transport operators are also given. 1993 Academic Press, Inc.