By Smirnov V.I.

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We may therefore define analogous operators Sa and Za. The change in the scattering operator results from the shift in the origin of -a in the incoming translation representation and of +a in the outgoing translation representation. 36) where P! are projections which remove the 5)! and ~a components. Za forms a semigroup of operators on the subspace The subspace Ka is slightly larger than K; this difference gives us extra room for maneuvering. ,b denote the projection onto ~! e 5)Z. It follows from this formula that if the resolvent of Bb can be- conti~ued from the right half plane into some open set Po' then so can the resolvent of B a for a < b.

In the outgoing, respectively incoming, translation representation these operators act as truncation above, respectively below, zero. 16) and so they can be extended by continuity to For an element f of H H. e. 18) for all e = (~,s*e) and ~ in L2 (R). CLAIM. 18) is true for all ~ in L 21oc (R) and all P in L 2 (R+). s*e PROOF. 5\ that belongs to L 2 (R+). 18) depends continuously on ~ in all the semi- norms. 18) depends continuously on ~ in all the seminorms. 18) for all ~ in L 21oc (R) follows by continuity .

19)+ e- izs m , §2. AN ABSTRACT SCA TTERING THEORY m some element of :n; 23 m is a function of nand z. Clearly m is a linear function of n, so we can write it in the form m = S(z) n . 14). 20) are the basis of our study of the analytic properties of S(z). We start by showing that S(z) is analytic in the lower half plane. 20) that sK = . e-1ZSS(z)n. s