Asymptotic equivalence of identification operators in geometric scattering theory

Given two measures \mu_{1} and \mu_{2} on a measurable space X such that d\mu_{2}=\rho_{1,2}\,d\mu_{1} for some bounded measurable function \rho_{1,2}:X\rightarrow (0,\infty) , there exist two natural identification operators J_{1,2},\tilde{J}_{1,2}:L^{2}(X,\mu_{1})\rightarrow L^{2}(X,\mu_{2}) , namely the unitary J_{1,2}\psi:=\psi/\sqrt{\rho_{1,2}} and the trivial \tilde{J}_{1,2}\psi:=\psi . Given self-adjoint semibounded operators H_{j} on L^{2}(X,\mu_{j}) , j=1,2 , we prove a natural criterion in a topologic setting for the equality of the two-Hilbert-space wave operators W_{\pm}(H_{2},H_{1};J_{1,2}) and W_{\pm}(H_{2},H_{1};\tilde{J}_{1,2}) , by showing that J_{1,2}-\tilde{J}_{1,2} are asymptotically H_{1} -equivalent in the sense of Kato. It turns out that this criterion is automatically satisfied in typical situations on noncompact Riemannian manifolds and weighted infinite graphs in which one has the existence of completeness W_{\pm}(H_{2},H_{1};\tilde{J}_{1,2}) (and thus a-posteriori of W_{\pm}(H_{2},H_{1};J_{1,2}) ).