Abstract
Given self‐adjoint operators H~j~, on Hilbert spaces ℋ︁~j~, j = 0,l, and J ∈ ℬ︁ (ℋ︁~0~, ℋ︁~1~) (where ℬ︁ (ℋ︁~0~ ℋ︁~1~) denotes the set of bounded linear operators from ℋ︁~0~to ℋ︁~1~), define the wave operators magnified image where P~0~ is the projection onto the subspace for absolute continuity for H~0~. We use (i) to study the scattering problem associated with a pair of equations each of the form magnified image where L is a positive, self‐adjoint operator on a Hilbert space X, m is a positive integer and the α~j~ are distinct positive constants. Methods patterned after those of Kato are used to study two equations (that is for L = L~0~ and L = L~l~) each of the form (ii). We show that they are equivalent to equations of the form magnified image where each Ĥ~k~ is a self‐adjoint operator on an associated Hilbert space ℋ︁~k~. Now suppose∼he‐wave operators W~±~,(L~1~ L~0~) exist and are complete. Then we can find a J ∈ ℬ︁(H~1~ H~0~) such that W~+~(Ĥ~l~, Ĥ~0~,J) exists. In the case where L~o~ and L~1~ have the same domain, ℋ︁~1~ and ℋ︁~0~ are equal as vector spaces, and under certain conditions (on L~i~, i = 0, 1) ℋ︁~0~ and ℋ︁~1~ have equivalent norms. Assuming these conditions, let J'∈ ℬ︁(ℋ︁~1~' ℋ︁~0~) be the identity map. We show that (with an additional assumption on L~0~ and L~1~) W~+~(Ĥ~1~Ĥ~0~,J) exists andisequal to W~+~(Ĥ~l~,Ĥ~0~, J).