Wave and Scattering Operators for Second-Order Elliptic Operators in $R^3$

Abstract

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

Existence and Completeness of the Wave Operators for Higher Order Differential Operators

“…Trace class method is one of the important methods applicable to the theory of differential operators. For example, the second-order elliptic differential operator in 3 was considered by Ikebe and Tayoshi (1968), also by using a trace condition. For some other results connected to the trace condition (see , 1963, 1969, Birman and Entina, 1964.…”