IntroductionOur concern in this paper will be with the existence and completeness of wave operators intertwining the negative Laplacian L 0 =-A and the second-order elliptic differential operatorin the three-dimensional Euclidean space R\ where Dj = -id/dXj. In a suitable sense and under appropriate conditions on the coefficient functions a jk (x), bj(x) y and q(x\ L 0 and L may be regarded as selfadjoint operators defined in the Hilbert space L 2 , square integrable functions on R 3 . The wave operators W^ are the strong limits for t-^±oo of exp (itL) exp ( -#L 0 ), and then the scattering operator S is defined as S = WJ W_ (* denotes the adjoint of an operator). The wave operator W ± maps isometrically into the scattering states for L, in other words, the complement of the bound states, or more precisely the subspace of absolute continuity for L, but it does not necessarily map onto this subspace. If it does, W^ is called complete. The completeness of W^ implies that S is unitary, which physicists expect, or sometimes believe, to hold in most problems. We shall prove that W± exist and are complete when L is asymptotically, that is, as \x\ tends to infinity, equal to L 0 . The
Our concern in this paper will be the existence and the isometric property of W±(H 9 H 0 ; J). We shall generalize the result given by Ikebe in [4], where the case P(D)= -A was treated under the assumption that 0 lies in a cylinder. We shall, moreover, discuss the invariance of the wave operators. That is, we shall ask if
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