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4) -+ is a better (and in our case good enough) approximation of Vl (x) than to set it equal to zero. g. with the stationary phase method. 1. Similarly the range of n~ consists of those vectors ~ with 57 GEOMETRIC METHODS IN SPECTRAL AND SCATTERING THEORY lim -+ T ~[e-iHt _ U(t + T,T)] e-iHT~~ sup t 00 = O. 9) holds for all ~ E ~ont then we say that asymptotic completeness holds. The analogous statements for negative times correspond to the incoming modified wave operator ~~ The crucial tool to prove existence and completeness is again Cook's estimate.
We call these vectors asymptotic configurations because <1> represents a "boundary condition at t = + 00" for which there exists a solution of the interacting Schrodinger equation. 3) =~_ -+ defined on those vectors where the limit exists. 1. 2) holds. 1) holds. Proof. 2). 1) is equivalent to convergence of V. 7) 00 and this is equivalent to lim e iHt e -iH 0 t t + . e. 8) o For any holds which fulfills lim t++ otherwise. 2), and '¥ e -iHt '¥ _ e -iHot <1>11 o. 9) oo In this sense is an asymptotic (here outgoing) configuration.
1. 5) is satisfied. e. 7) ¢. We will give two different proofs, but we start with some arguments common to both of them. Proof. We consider the outgoing wave operator ~_, the proof for ~+ is analogous. Since Ran ~- is closed it is sufficient to show that a set of vectors dense in ~ont lies in the range of ~_. A convenient dense set consists of the states with compact ener y support which does not include zero. 8) ~. Since Ran ~- is time translation invariant it is sufficient to show that exp(-iHt)~ is arbitrarily close to Ran ~- for some time t.