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
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
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
which defines
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),
= 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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