Geodesic
Asymptotic equivalence of identification operators in geometric scattering theory
Documenta mathematica, Tome 29 (2024) no. 6, pp. 1367-1379

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Given two measures μ1​ and μ2​ on a measurable space X such that dμ2​=ρ1,2​dμ1​ for some bounded measurable function ρ1,2​:X→(0,∞), there exist two natural identification operators J1,2​,J~1,2​:L2(X,μ1​)→L2(X,μ2​), namely the unitary J1,2​ψ:=ψ/ρ1,2​​ and the trivial J~1,2​ψ:=ψ. Given self-adjoint semibounded operators Hj​ on L2(X,μj​), j=1,2, we prove a natural criterion in a topologic setting for the equality of the two-Hilbert-space wave operators W±​(H2​,H1​;J1,2​) and W±​(H2​,H1​;J~1,2​), by showing that J1,2​−J~1,2​ are asymptotically H1​-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±​(H2​,H1​;J~1,2​) (and thus a-posteriori of W±​(H2​,H1​;J1,2​)).
DOI : 10.4171/dm/968
Classification : 35P25, 58J05
Mots-clés : scattering theory, wave operators, Riemannian manifolds, weighted graphs 
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     author = {Batu G\"uneysu},
     title = {Asymptotic equivalence of identification operators in geometric scattering theory},
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     year = {2024},
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Batu Güneysu. Asymptotic equivalence of identification operators in geometric scattering theory. Documenta mathematica, Tome 29 (2024) no. 6, pp. 1367-1379. doi: 10.4171/dm/968

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