On the Characters and the Plancherel Formula of Nilpotent Groups ...

SCATTERING THEORY WITH TWO HILBERT SPACES 353
According to Theorem 6.2, it follows that the wave operators W,
(which are identical for all J, J, and p) are partially isometric if they
exist. The unitarity of p follows easily from the following identity, in
which we set 4(x) = &x) = &(~)-r/~ Ei’s$(x):
= s 4(x)* E;“E,(x)-~‘~ E,(x) E,(x)-“’ E;‘2rj(x) dx
= s (b(x)* J%+(X) dx = II 4 1/12-
The equivalence of 9 with J and J can be proved in the same way
as above by noting that E2(~)lj2 -+ Ei/2 as 1 x 1 -+ co.
As was remarked in Section 1, the existence of a unitary indentifica-
tion operator f equivalent to J makes it possible to reduce the problem
to that of Schrodinger type, according to
f-lW* = s-lim 02(- t) U1(t) Pr , 02(t) = f1U2(t) 1. (7.7)
We have not proved the existence, let alone the completeness, of
W, . The reduction (7.7) to the wave operators of Schrodinger type
raises the hope that some of the criteria deduced for such wave opera-
tors might be applicable. But this does not seem to be easy, for the
selfadjoint generator of the group 02(t) is given by
fi2 = j-1H2j = E;1’2E2(x)-1’2 c A
(k rc 4 E,(x)-~‘~ E:12, (7.8)
a rather complicated expression.
It should be noted that the existence of W, has been proved in [S]
under rather mild assumptions, but the question of the completeness
of W, is open.
8. ABSTRACT DIFFERENTIAL EQUATIONS OF SECOND ORDER
As the second application, we consider the scattering problem
associated with abstract differential equations of the form
$+Au=O, -co