Asymptotic Equivalence of Identification Operators in Geometric Scattering Theory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929211051081728 |
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| author | Güneysu, Batu |
| author_facet | Güneysu, Batu |
| contents | 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 $W_\pm(H_2,H_1;J_{1,2}))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08540 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic Equivalence of Identification Operators in Geometric Scattering Theory Güneysu, Batu Mathematical Physics Differential Geometry Spectral Theory 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 $W_\pm(H_2,H_1;J_{1,2}))$. |
| title | Asymptotic Equivalence of Identification Operators in Geometric Scattering Theory |
| topic | Mathematical Physics Differential Geometry Spectral Theory |
| url | https://arxiv.org/abs/2401.08540 |