Semilinear Schrödinger Flows on Hyperbolic Spaces: Scattering in H^1

Abstract

We prove global well-posedness and scattering in $H^1$ for the defocusing nonlinear Schrödinger equations \begin{equation*} \begin{cases} &(i\partial_t+Δ_\g)u=u|u|^{2σ}; &u(0)=ϕ, \end{cases} \end{equation*} on the hyperbolic spaces $\H^d$, $d\ge 2$, 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.