A Scattering Result of the Radial Cubic Defocusing Schr\"odinger Equation on the 3d Hyperbolic Space

TL;DR

This paper proves that the radial cubic defocusing Schrödinger equation on 3D hyperbolic space is globally well-posed and exhibits scattering behavior for initial data with regularity between 15/16 and 1, extending previous work to hyperbolic geometry.

Contribution

It extends the scattering results for the defocusing cubic Schrödinger equation to three-dimensional hyperbolic space for radial data with Sobolev regularity between 15/16 and 1.

Findings

Global well-posedness for 15/16<s<1

Scattering behavior established in hyperbolic space

Extension of prior Euclidean results to hyperbolic geometry

Abstract

In this paper, we study the defocusing cubic Schr\"{o}dinger equation on three dimensional hyperbolic space $\mathbb{H}^3$ with radial initial data in the Sobolev Space $H^s(0<s<1)$. Our main result is that the initial value problem is globally wellposed and scatters for $\frac{15}{16}<s<1$. This is an extension of the work of Staffilani and Yu to the three dimensional hyperbolic space.