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)),