Nonlinear Schrödinger equation on real hyperbolic spaces

Abstract

We consider the Schrödinger 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 $\gamma =1+\frac{4}{n}$ and can be false for subcritical powers while, on hyperbolic spaces, global existence and scattering of small $L^2$ solutions holds for all powers $1<\gamma \le 1+\frac{4}{n}$. 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.

Version 1 : 18 January 2008. Version 2 : 29 February 2008

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