Existence of Wave Operators with Time-Dependent Modifiers for the Schrödinger Equations with Long-Range Potentials on Scattering Manifolds

Abstract

We construct time-dependent wave operators for Schrödinger 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.