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arXiv · 2601.20225

The scattering map for the Schrodinger operator on curved spaces

Andrew Hassell·Qiuye Jia
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Abstract

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.

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Andrew Hassell, Qiuye Jia. 2026-01-28. The scattering map for the Schrodinger operator on curved spaces. https://arxiv.org/abs/2601.20225

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