Asymptotic equivalence of identification operators in geometric scattering theory
Year: 2024
Type: article
Abstract: 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}) ).
Source: Documenta Mathematica
DOI: 10.4171/dm/968
Author: Batu Güneysu
Institution: Chemnitz University of Technology
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4404721212