Existence of wave operators with time-dependent modifiers for the Sch\"odinger equations with long-range potentials on scattering manifolds

TL;DR

This paper develops time-dependent wave operators for Schrödinger equations with long-range potentials on scattering manifolds, advancing the mathematical understanding of quantum scattering in complex geometric settings.

Contribution

It introduces a novel construction of wave operators using a two space scattering theory framework on asymptotically conic manifolds, including solutions to the Hamilton-Jacobi equation.

Findings

Existence of modified wave operators for long-range potentials.

Construction of exact solutions to the Hamilton-Jacobi equation.

Application of two space scattering theory formalism.

Abstract

We construct time-dependent wave operators for Schr\"{o}dinger equations with long-range potentials on a manifold $M$ with asymptotically conic structure. We use the two space scattering theory formalism, and a reference operator on a space of the form $\mathbb{R} \times \partial M$, where $\partial M$ is the boundary of $M$ at infinity. We construct exact solutions to the Hamilton-Jacobi equation on the reference system $\mathbb{R} \times \partial M$, and prove the existence of the modified wave operators.