The scattering map for the Schrodinger operator on curved spaces

TL;DR

The paper analyzes the scattering of Schrödinger equations on curved backgrounds with spacetime-compact perturbations, introducing a novel 1-cusp framework to encode asymptotic data. It establishes that the scattering map $S$ is an elliptic 1c-1c Fourier integral operator of order zero, with a canonical relation given by the classical scattering map, and proves a corresponding Egorov-type theorem. By developing parabolic (ps) and 1-cusp calculi, including Poisson operators and forward/backward sojourn relations, the work provides a robust microlocal-analytic description of the propagation of asymptotic data and the interaction with metric perturbations. The results yield precise wavefront set transformations, boundedness on 1c-Sobolev spaces, and noncompactness criteria that tie the operator's behavior to the geometry of the classical flow, offering a framework for inverse problems and stability analyses in curved-space Schrödinger scattering. The combination of 1c-ps and 1c-1c Fourier integral calculus generalizes known scattering results (e.g., Melrose–Zworski, Vasy) to a setting tailored to Schrödinger dynamics, with potential applications to inverse problems on nontrivial geometries.

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.

Figures (1)

• Figure 1: On the left: the radially compactified spacetime $\overline{\mathbb{R}^{n+1}_{z,t}}$. The function $x_{\mathrm{ps}} = (1+t^2+|z|^2)^{-1/2}$ defines (i.e. vanishes simply at) spacetime infinity. The radial sets live in phase space $\overline{{}^{\mathrm{ps}} T^* \mathbb{R}^{n+1}}$ over the boundary of $\overline{\mathbb{R}^{n+1}_{z,t}}$; $\mathcal{R}_+$ lies over the top, or 'northern' hemisphere, while the radial set $\mathcal{R}_-$ lies over the bottom or 'southern' hemisphere. The intersection of the northern and southern hemispheres is referred to as the equator. On the right: Projection to $\overline{\mathbb{R}^{n+1}}$ of bicharacteristics starting from the boundary of $\mathcal{R}_-$, which lies over the equator. Each bicharacteristic lies in a single level set of $t$.

Theorems & Definitions (120)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Corollary 1.4
• Corollary 1.5
• Corollary 1.6
• Remark 1.7
• Definition 2.1
• Remark 2.2
• Definition 2.3