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 $\rho_{1,2}:X\to (0,\infty)$, there exist two natural identification operators $J_{1,2},\tilde{J}_{1,2}:L^2(X,\mu_1)\to L^2(X,\mu_2)$, 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 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 $W_\pm(H_2,H_1;J_{1,2}))$.