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

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344 KATO rather than by E and H ? These two modes of identification are clearly different, for E1 f &(x) implies that the dielectric constant and magnetic permeability are different for the two fields. Of course there are many other different ways of identification which could claim equal right to the above ones. It would be futile to try to decide which identification is the “cor- rect” one; it is not a mathematical problem, perhaps not even a physical one. Thus one must admit that in general there can be many scattering theories for a given pair U, , U, of groups, according to different choices of the identification operator J. Fortunately, however, there are practically not too many different scattering theories for the given pair U, , U, , for the difference in J is often irrelevant asymptotically, and it is exactly the asymptotic behavior of the systems that scattering theory is concerned with. Mathematically, two identification operators J and 9 are (asymptotically) equivalent if s-lim (J - /) Ul(t) PI = 0. In such a case the wave operators obtained by using J and J are the same, as is easily seen from (1.2). It can be shown in many cases that all reasonable identifications are equivalent so that we have essentially a unique scattering theory. In particular this is the case with the Maxwell equations mentioned above. Intuitively this is due to the fact that the waves eventually go to infinity, where E, and E,(x) are assumed to be equal asymptotically, and thus the identifications by E and H and by D and B are equivalent. It might appear, after all, that we have arrived at a rather trivial conclusion. But there is at least one positive result of these considera- tions. Namely, among all equivalent identifications one can choose a particular J which is mathematically most convenient. For example, it happens frequently that there is a unitary J on 6, to & among equivalent identifications. In this case (1.2) gives J-‘W* = y&&y cl&(- t) l&(t) Pl , (1.3) where us(t) = J-‘U,( t) J is a unitary group that acts in the same space 5, as Ul(t) does. Thus we have reduced the problem to that of wave operators of Schradinger type, to which existing results may be applicable. This also explains why the wave operators are partially isometric in many cases. Even if the reduction (1.3) to the case of SchrSdinger type is available, it does not necessarily follow that the scattering for the wave equations can be handled by the known results. The difficulty is that the deviation of 0, from U, is often larger than such a deviation
SCATTERING THEORY WITH TWO HILBERT SPACES 345 encountered in the ordinary Schrodinger equations, chiefly because one or both of the generators of these groups are usually first-order differential operators with variable coefficients. But this is no more than a technical difficulty, which could exist in Schrodinger equations as well if one considered more general types of equations. 2. INTERTWINING OPERATORS Let .sj, , j = 1, 2, be two separable Hilbert spaces. We use the same symbols 11 11 and ( , ) for the norms and inner products in Z& and g2 , but there is no possibility of confusion. For each j we consider a continuous unitary group U, = (Uj(t)), - CO < t < co. We denote by Hj the selfadjoint generator of Ui so that Uj(t) = e-iM*. The spectral family associated with Hi is denoted by {E,(X)). DEFINITION 2.1. T E I?(&, s2) is called an intertwining operator for the pair U, , U, (or for the pair HI , Hz) if the following equivalent conditions are satisfied: H,TI TH, , (2-l) b(t) T = TU,(t), -co
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JOURNAL OF FUNCTIONAL ANALYSIS 1, 2
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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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PLANCHERRL FORMULA OF NILPOTENT GRO
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JOURNAL OF FUNCTIONAL ANALYSIS 1, 2
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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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SIZES OF COMPACT SUBSETS OF HILBERT
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