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)u_o in θY ∩{u ∈H∣∣u∣≤R}. We assume that for every υ_o ∈ $\bar \sum$, u(t) − υ(t) (where υ(t) = S(t)υ_o) does not converge to 0 as t → ∞. Thus for υ_o ∈ $\bar \sum$ 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))