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Semilinear Schrödinger flows on hyperbolic spaces: scattering in H 1

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Abstract

We prove global well-posedness and scattering in H 1 for the defocusing nonlinear Schrödinger equations

$$\left\{\begin{array}{ll}(i\partial_t+\Delta_g)u=u|u|^{2\sigma};\\u(0)=\phi,\end{array}\right.$$

on the hyperbolic spaces \({\mathbb{H}^d}\), d ≥ 2, for exponents \({\sigma \in (0, 2/(d-2))}\). The main unexpected conclusion is scattering to linear solutions in the case of small exponents σ; 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.

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Correspondence to Alexandru D. Ionescu.

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A. D. Ionescu was supported in part by a Packard Fellowship. G. Staffilani was supported in part by NSF Grant DMS 0602678

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Ionescu, A.D., Staffilani, G. Semilinear Schrödinger flows on hyperbolic spaces: scattering in H 1 . Math. Ann. 345, 133–158 (2009). https://doi.org/10.1007/s00208-009-0344-6

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