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

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JOURNAL OF FUNCTIONAL ANALYSIS 1, 342-369 (1967) Scattering Theory with Two Hilbert Spaces TOSIO KATO* Department of Mathematics, University of California, Berkeley, California 94720 Communicated by Ralph Phillips Received May 9, 1967 1. INTRODUCTION The purpose of this paper is to give a general scattering theory which can be applied, among others, to the scattering for wave equations. In previous publications (see e.g. [I], Chapter X) we have developed scattering theory applicable to Schrcdinger equations. In this theory one is concerned with two unitary groups Uj(t) = e-itHj, - CO < t < co, i = 1,2 in a Hilbert space 5 and tries to construct the wave operators where Pi denotes the projection of 5 on the subspace of absolute continuity for Hi. If W+ exists, it is a partial isometry in 8 with initial projection Pi and final projection < Pz . W, is said to be complete if the final projection is equal to Pz . Similar results hold for W- . If both W, exist and are complete, the scattering operator S = W$W- is unitary on P&. Sufficient conditions for the existence and completeness of the wave operators have been studied extensively (see, e.g., [II-131). Scattering theories of different types have appeared more recently. These are concerned with wave equations, in the ordinary as well as generalized sense. Here it has been observed that one has to do with unitary groups Uj(t) acting in d@erent HiZbert spaces $$ , j = 1,2. It is true that the two spaces are often the same vector space L! equipped with different inner products, so that the wave operators can still * Part of this work was done while the author held a Miller Professorship. It was partly supported by Air Force Office of Scientific Research, grant 553-64. 342
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,
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PLANCHEREL FORMULA OF NILPOTENT GRO
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and PLANCHEREL FORMULA OF NILPOTENT
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PLANCHERRL FORMULA OF NILPOTRNT GRO
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VECTEURS SePARATEURs DES ALGhBRES D
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VECTEURS SfiPARATEURS DES ALGkBRES
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VECTEURS SltPARATEURS DES ALGhBRBS
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VECTEURS SiPARATEURS DES ALGBBRES D
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