The scattering map for the Schrodinger operator on curved spaces

Abstract

Let $P$ be a Schrödinger operator $D_t+\hat{I}{”}_g$ with metric and potential perturbation that are compactly supported in spacetime ${\mathbb{R}}^{n+1}$. Here $D_t=-i{\partial }_t$ and $\hat{I}{”}_g$ 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.