On scattering for NLS: from Euclidean to hyperbolic space

TL;DR

This paper establishes asymptotic completeness for the defocusing nonlinear Schrödinger equation on hyperbolic space, extending understanding of scattering phenomena from Euclidean to curved geometries.

Contribution

It proves asymptotic completeness in hyperbolic space for radial solutions in dimensions ≥4, and explores how nonlinearities evolve between Euclidean and hyperbolic geometries.

Findings

Asymptotic completeness proven for hyperbolic space

Short range nonlinearities increase as space approaches hyperbolic geometry

Large time behavior of free solutions characterized

Abstract

We prove asymptotic completeness in the energy space for the nonlinear Schrodinger equation posed on hyperbolic space in the radial case, in space dimension at least 4, and for any energy-subcritical, defocusing, power nonlinearity. The proof is based on simple Morawetz estimates and weighted Strichartz estimates. We investigate the same question on spaces which kind of interpolate between Euclidean space and hyperbolic space, showing that the family of short range nonlinearities becomes larger and larger as the space approaches the hyperbolic space. Finally, we describe the large time behavior of radial solutions to the free dynamics.