Scattering theory for hyperbolic equations of order 2m

摘要

AbstractGiven self‐adjoint operatorsHj, on Hilbert spaces ℋ︁j,j= 0,l, andJ∈ ℬ︁ (ℋ︁0, ℋ︁1) (where ℬ︁ (ℋ︁0ℋ︁1)denotes the set of bounded linear operators from ℋ︁0to ℋ︁1), define the wave operatorswhereP0is 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 formwhereLis a positive, self‐adjoint operator on a Hilbert spaceX,mis a positive integer and the αjare distinct positive constants. Methods patterned after those of Kato are used to study two equations (that is forL=L0andL=Ll) each of the form (ii). We show that they are equivalent to equations of the formwhere each Ĥkis a self‐adjoint operator on an associated Hilbert space ℋ︁k. Now suppose∼he‐wave operatorsW±,(L1L0) exist and are complete. Then we can find aJ∈ ℬ︁(H1H0) such thatW+(Ĥl, Ĥ0,J) exists. In the case whereLoandL1have the same domain, ℋ︁1and ℋ︁0are equal as vector spaces, and under certain conditions (on Li,i= 0, 1) ℋ︁0and ℋ︁1have equivalent norms. Assuming these conditions, letJ'∈ ℬ︁(ℋ︁1' ℋ︁0) be the identity map. We show t