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

SCATTERING THEORY WITH TWO HILBERT SPACES 343
be defined by (1.1). In more general cases, however, (1.1) does not
make sense and must be replaced by a different definition like
where J E B(ljl,-tF2). (F or any two Banach spaces X, 9, we denote
by B(x, 9,) the set of all bounded linear operators on X to 9 and we
write B(X) for B(x, w).) We shall call J the identification operator,
though we do not always assume that J is bijective nor even injective.
We shall call (1.1) with the single space $j wave operators of Schriidinger
type to distinguish them from the more general ones (1.2) with two
spaces.
In what follows we propose to study general properties of the wave
operators (1.2) an d su ffi cient conditions for their existence and completeness.
In general IV, are no longer partially isometric, unlike the
wave operators of Schrodinger type, but it will be seen that in many
cases of practical importance they are. Later we shall consider more
specific problems, with applications to the scattering for the wave
equations and Maxwell equations. In particular we shall deduce some
of the results found in [5]-[S].
It seems to the author that one thing has not been properly
recognized, or at least not mentioned explicitly, regarding the identification
operator J. Namely, J is not necessarily uniquely determined
on the physical basis in individual problems. This fact has been
obscured by the definition (l.l), which is usually used under the tacit
assumption that the identification by the identity in the common
space 2 is natural and unique.
But a little reflection reveals that this is not so. To illustrate the
point, let us take the Maxwell equations, which are a special case of
the systems considered in [S]. Here it suffices to note that the unperturbed
and perturbed electromagnetic fields are described by unitary
groups Ui(t) acting in the Hilbert spaces Bj , which are the [L2(R3)lS
with certain positive-definite density matrices Z$(x), j = 1, 2, where
E,(x) = El is constant. It is assumed that E,(x) is uniformly bounded
from above and below so that 5, and $s are the same vector space B
with different metrics. The element u = U(X) of 2 is the aggregate
of the electric field E and the magnetic field H. In [S] the identification
is made by the identity of u as an element of 2.
This identification means that the states of the unperturbed and
perturbed fields are identified if they have the same E and H throughout
113. But why should one not identify the two states by D and B
(the electric displacement and magnetic induction), for example,