Nonlinear Schr\"odinger equation on real hyperbolic spaces

TL;DR

This paper establishes sharp dispersive and Strichartz estimates for the nonlinear Schr"odinger equation on real hyperbolic spaces, leading to new global well-posedness and scattering results for a wide range of nonlinearities without radial or gauge invariance assumptions.

Contribution

It provides the first sharp dispersive estimates on hyperbolic spaces and demonstrates global well-posedness and scattering for NLS with broad nonlinearities, surpassing Euclidean limitations.

Findings

Sharp dispersive and Strichartz estimates obtained

Global well-posedness for small data in L^2 and H^1

Scattering results for all subcritical powers without radial or gauge assumptions

Abstract

We consider the Schr\"odinger equation with no radial assumption on real hyperbolic spaces. We obtain sharp dispersive and Strichartz estimates for a large family of admissible pairs. As a first consequence, we get strong well-posedness results for NLS. Specifically, for small intial data, we prove $L^2$ and $H^1$ global well-posedness for any subcritical nonlinearity (in contrast with the Euclidean case) and with no gauge invariance assumption on the nonlinearity $F$. On the other hand, if $F$ is gauge invariant, $L^2$ charge is conserved and hence, as in the Euclidean case, it is possible to extend local $L^2$ solutions to global ones. The corresponding argument in $H^1$ requires the conservation of energy, which holds under the stronger condition that $F$ is defocusing. Recall that global well-posedness in the gauge invariant case was already proved by Banica, Carles & Staffilani, for small radial $L^2$ data and for large radial $H^1$ data. The second important application of our global Strichartz estimates is "scattering" for NLS both in $L^2$ and in $H^1$, with no radial or gauge invariance assumption. Notice that, in the Euclidean case, this is only possible for the critical power $\gamma=1+\frac4n$ and can be false for subcritical powers while, on hyperbolic spaces, global existence and scattering of small $L^2$ solutions holds for all powers $1<\gamma\le1+\frac4n$. If we restrict to defocusing nonlinearities $F$, we can extend the $H^1$ scattering results of Banica, Carles & Staffilani to the nonradial case. Also there is no distinction anymore between short range and long range nonlinearity : the geometry of hyperbolic spaces makes every power-like nonlinearity short range.