Asymptotic Equivalence of Identification Operators in Geometric Scattering Theory
Batu Güneysu
TL;DR
The paper studies two-Hilbert-space geometric scattering with measures $\mu_1,\mu_2$ tied by a density $\rho$ and natural identifications $J_{1,2}$ and $\tilde{J}_{1,2}$. It proves a general criterion ensuring $W_\pm(H_2,H_1;J_{1,2})=W_\pm(H_2,H_1;\tilde{J}_{1,2})$ by requiring $J_{1,2}$ and $\tilde{J}_{1,2}$ to be asymptotically $H_1$-equivalent, quantified by $\int (1-1/\rho)^2 \phi_{1}(s,x)\,d\mu_1(x)<\infty$ for all $0<s<1$, where $\phi_{1}$ encodes smoothing of $H_1$. The authors show this criterion is automatically satisfied in common geometric settings, including noncompact Riemannian manifolds with quasi-isometric metrics and complete weighted graphs, and they derive explicit verifications via heat-kernel bounds (Li–Yau) and curvature-determined distortion functions. Consequently, the two wave operators coincide, with existence and completeness carried over from $\tilde{J}_{1,2}$ to $J_{1,2}$. The results extend naturally to vector-bundle contexts, illustrating a robust link between different identification maps and stability of the absolutely continuous spectrum under geometric variations.
Abstract
Given two measures $μ_1$ and $μ_2$ on a measurable space $X$ such that $dμ_2=ρ_{1,2} \, dμ_1$ for some bounded measurable function $ρ_{1,2}:X\to (0,\infty)$, there exist two natural identification operators $J_{1,2},\tilde{J}_{1,2}:L^2(X,μ_1)\to L^2(X,μ_2)$, namely the unitary $J_{1,2}ψ:=ψ/\sqrt{ρ_{1,2}}$ and the trivial $\tilde{J}_{1,2}ψ:=ψ$. Given self-adjoint semibounded operators $H_j$ on $L^2(X,μ_j)$, $j=1,2$, 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 $W_\pm(H_2,H_1;J_{1,2}))$.