Scattering theory in a pair of spaces
Some elementary notions of two-space scattering theory are discussed. It is shown that for the pair of self-adjoint operators \(H_ 0\), H and identification J the existence of the (strong) wave operator \(W_{\pm}(H,H_ 0;J)\) is equivalent to the existence of weak wave operators \(\tilde W_{\pm}(H,H_ 0;J)\), \(\tilde W_{\pm}(H_ 0,H_ 0;J^*J)\) and to the equality \[ \tilde W_{\pm}(H,H_ 0;J)^*\tilde W_{\pm}(H,H_ 0;J)=\tilde W_{\pm}(H_ 0,H_ 0;J^*J). \] New definition of the completeness of \(W_{\pm}(H,H_ 0;J)\) is proposed. It is related to the asymptotics of exp(iHt) as \(t\to \pm \infty\) and depends on J. Let \(W_{\pm}(H,H_ 0;J)\) exist. Then \(W_{\pm}(H_ 0,H;J^*)\) is shown to exist iff \(W_{\pm}(H,H_ 0;J)\) is complete and \(W_{\pm}(H_ 0,H_ 0;J^*J)\) also exists.
scientific article; zbMATH DE number 3998563