A Scattering Result of the Radial Cubic Defocusing Schrödinger Equation on the 3d Hyperbolic Space

Abstract

In this paper, we study the defocusing cubic Schrödinger 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.