Asymptotic Completeness: Proof of Theorem 12.1

Abstract

In the previous chapter, Steps 1 to 3 do not depend on the validity of the asymptotic completeness property. In this chapter we assume that u(t) never belongs to \(\bar \sum \) since otherwise the result is obvious, and we argue by contradiction, assuming that Theorem 12.1 is not valid for u(t). All further steps in this chapter will hinge on the negation of Theorem 12.1. Without loss of generality we can consider a trajectory u(t) = S(t)uo in θY ∩{u ∈H∣∣u∣≤R}. We assume that for every υo ∈ \(\bar \sum \), u(t) − υ(t) (where υ(t) = S(t)υo) does not converge to 0 as t → ∞. Thus for υo ∈ \(\bar \sum \) fixed, there exists ε > 0 and a sequence t j → ∞ such that$$|u\left( {{t_j}} \right) - \upsilon \left( {{t_j}} \right)| \geqslant \varepsilon > 0,{\text{ for all }}j{\text{.}}$$ ((13.1)) KeywordsOpen NeighborhoodVertical DistanceElementary ComputationCell ComplexStringent ConditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.