The scattering map for the Schrodinger operator on curved spaces

TL;DR

This paper demonstrates that the scattering map for Schrödinger operators on curved spaces can be characterized as a 1-cusp Fourier integral operator, revealing a natural geometric framework for analyzing asymptotic solution data.

Contribution

It introduces the 1-cusp Fourier integral operator framework to describe the scattering map for Schrödinger operators on curved spaces, connecting it with Vasy and Zachos' calculus.

Findings

The scattering map is a 1-cusp Fourier integral operator.

1-cusp geometry is the natural setting for Schrödinger asymptotics.

The approach links inverse problems on asymptotically conic manifolds with Schrödinger scattering.

Abstract

Let $P$ be a Schr\"odinger operator $D_t+\Delta_g$ with metric and potential perturbation that are compactly supported in spacetime $\mathbb{R}^{n+1}$. Here $D_t = -i \partial_t$ and $\Delta_g$ 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 Schr\"odinger's equation.