Scattering theory for hyperbolic equations of order 2m

Abstract

Given self-adjoint operators Hj, on Hilbert spaces ℋ︁j, j = 0,l, and J ∈ ℬ︁ (ℋ︁0, ℋ︁1) (where ℬ︁ (ℋ︁0 ℋ︁1) denotes the set of bounded linear operators from ℋ︁0to ℋ︁1), define the wave operators where P0 is the projection onto the subspace for absolute continuity for H0. We use (i) to study the scattering problem associated with a pair of equations each of the form where L is a positive, self-adjoint operator on a Hilbert space X, m is a positive integer and the αj are distinct positive constants. Methods patterned after those of Kato are used to study two equations (that is for L = L0 and L = Ll) each of the form (ii). We show that they are equivalent to equations of the form where each Ĥk is a self-adjoint operator on an associated Hilbert space ℋ︁k. Now suppose∼he-wave operators W±,(L1 L0) exist and are complete. Then we can find a J ∈ ℬ︁(H1 H0) such that W+(Ĥl, Ĥ0,J) exists. In the case where Lo and L1 have the same domain, ℋ︁1 and ℋ︁0 are equal as vector spaces, and under certain conditions (on Li, i = 0, 1) ℋ︁0 and ℋ︁1 have equivalent norms. Assuming these conditions, let J'∈ ℬ︁(ℋ︁1' ℋ︁0) be the identity map. We show that (with an additional assumption on L0 and L1) W+(Ĥ1Ĥ0,J) exists andisequal to W+(Ĥl,Ĥ0, J).