Semilinear Schr\"odinger Flows on Hyperbolic Spaces: Scattering in H^1

TL;DR

This paper establishes global well-posedness and scattering for the defocusing nonlinear Schrödinger equation on hyperbolic spaces in the energy space, revealing scattering behavior for small nonlinear exponents, unlike the Euclidean case.

Contribution

It proves scattering in H^1 for NLS on hyperbolic spaces for a range of exponents, using noneuclidean Strichartz estimates and Morawetz inequalities, which is novel compared to Euclidean results.

Findings

Scattering occurs in H^1 for small exponents σ on hyperbolic spaces.

Global well-posedness is established for the NLS in H^1.

Scattering in Euclidean spaces is not known or fails for similar exponents.

Abstract

We prove global well-posedness and scattering in $H^1$ for the defocusing nonlinear Schr\"{o}dinger equations \begin{equation*} \begin{cases} &(i\partial_t+\Delta_\g)u=u|u|^{2\sigma}; &u(0)=\phi, \end{cases} \end{equation*} on the hyperbolic spaces $\H^d$, $d\geq 2$, for exponents $\sigma\in(0,2/(d-2))$. The main unexpected conclusion is scattering to linear solutions in the case of small exponents $\sigma$; for comparison, on Euclidean spaces scattering in $H^1$ is not known for any exponent $\sigma\in(1/d,2/d]$ and is known to fail for $\sigma\in(0,1/d]$. Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.