10 INTRODUCTION
initial conditions satisfy the cohomological equations defined in [10] and [11]. In particular
(after trivial transformations) the second order terms Ug , U2 and 0 + must satisfy
6il% = C
2
,
6R^2=0,
(1.24)
where 6 is the coboundary operator defined by the representation (p, A) i— UgA2(®2Ug-i)
of the Poincare group Vo on bilinear symmetric maps A2 from E^ E^ and where
C2 = R(+)*-R2 is the cocycle defined by R2g = U^U^^U^) and R{+)2 = U^2 (U]^®
Ug_i), p G ?o- (For a n explicit definition of the coboundary operator 6 see §1.6 of this
introduction). Equation (1.24) shows that the cochain R^2 has to be a cocycle and
then that the cocycle C2 has to be a coboundary. This is equivalent to the existence of
a solution U^+\ H
+
of equation (1.14) modulo terms of order at least three. There are
similar equations for higher order terms:
Sill = C n , MJ+n - 0, (1.25)
where Cn and R^n are functions of ft2.,..., Ct1^1 and U^2,..., U^n.
In a previous article [14], we proved that there exists a modified wave operator and
global solutions of the M-D equations for a set of scattering data (/, / , a) , which is a subset
of the spaces E%£ introduced in the present paper, satisfying /M(fc) = /M(A:) 0 for k in a
neighbourhood of zero, i.e. /M and /^ have no low frequencies component. It follows from
that paper that the usual wave operator (i.e. U^ = U1, si 0 in (1.17c)) does not
exist, even in this case where there are no low frequencies. In fact there is an obstruction
to the existence of such a solution of equation (1.14) for n = 3 due to the self-coupling of I/J
with the electromagnetic potential created by the current ip^f^ip. However, as was proved
in [14] (see also Lemma 3.2 of the present paper) there exists a modified wave operator
satisfying (1.25) for n 2, with [/(+) = U1. Moreover, as we have already pointed out,
the phase function si is given by formula (1.18) (see formulas (1.5), (3.28) and (3.33) of
[14])-
Thus in the absence of low frequencies in the scattering data, for the electromagnetic
potential, we can choose ft+ such that it intertwines the linear representation U1 and the
nonlinear representation U of Vo- Now if (/, / ) 6 M£f, 1/2 p 1, have nontrivial low
frequency part, as is the case for the Coulomb potential (/M(/c) ~ \k\~2) and which is nec-
essary to assume in order to have asymptotic completeness, then there is a cohomological
obstruction already for n = 2 if we want to obtain U^2 = 0. The essential point now is
that the cocycle R2 can be split into a trivializable part C2 (a coboundary) and a non-
trivializable part
R^2,
which defines
U^2
and therefore the whole representation U^\
We shall call this nontrivial part the infrared cocycle. According to the definition of R^2
and formula (1.23a) it follows that Rg+)2 = 0 for g e SL(2, C), Rg+)2 = {Rg+)2M", Rg+)2D),
jR(+)2M
= 0 a n d
{R^2D{u1^u2)T{k) (1.26)
= \ E (tf°°(£i+)1(-) - ^ i + ) 1 ( - - a), (w(fc), -ek))P£(k)a2(k)
+ tf°°(i4+)1(.) - B ^ \ - - a), (*(*), -ek))Pe(*)&i(fc)),
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