The scattering map for the Schrodinger operator on curved spaces
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918310598148096 |
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| author | Hassell, Andrew Jia, Qiuye |
| author_facet | Hassell, Andrew Jia, Qiuye |
| contents | Let $P$ be a Schrödinger operator $D_t+Δ_g$ with metric and potential perturbation that are compactly supported in spacetime $\mathbb{R}^{n+1}$.
Here $D_t = -i \partial_t$ and $Δ_g$ is the positive Laplacian.
We consider the scattering map $S$ defined previously by the first author with Gell-Redman and Gomes arXiv:2201.03140, which relates the asymptotic data, as $t \to \pm \infty$, of global solutions $u$ to $Pu = 0$. We show that $S$ is a `1-cusp' Fourier integral operator, where `1-cusp' refers to a pseudodifferential calculus introduced by Vasy and Zachos arXiv:2204.11706 in the completely different setting of inverse problems on asymptotically conic manifolds. Our viewpoint is that 1-cusp geometry is the natural setting for studying the asymptotic data of solutions to Schrödinger's equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20225 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The scattering map for the Schrodinger operator on curved spaces Hassell, Andrew Jia, Qiuye Analysis of PDEs 35S30, 58J40, 35Q41 Let $P$ be a Schrödinger operator $D_t+Δ_g$ with metric and potential perturbation that are compactly supported in spacetime $\mathbb{R}^{n+1}$. Here $D_t = -i \partial_t$ and $Δ_g$ is the positive Laplacian. We consider the scattering map $S$ defined previously by the first author with Gell-Redman and Gomes arXiv:2201.03140, which relates the asymptotic data, as $t \to \pm \infty$, of global solutions $u$ to $Pu = 0$. We show that $S$ is a `1-cusp' Fourier integral operator, where `1-cusp' refers to a pseudodifferential calculus introduced by Vasy and Zachos arXiv:2204.11706 in the completely different setting of inverse problems on asymptotically conic manifolds. Our viewpoint is that 1-cusp geometry is the natural setting for studying the asymptotic data of solutions to Schrödinger's equation. |
| title | The scattering map for the Schrodinger operator on curved spaces |
| topic | Analysis of PDEs 35S30, 58J40, 35Q41 |
| url | https://arxiv.org/abs/2601.20225 |