Mathematische Annalen

, 345:133

Semilinear Schrödinger flows on hyperbolic spaces: scattering in H1


    • University of Wisconsin–Madison
  • Gigliola Staffilani
    • Massachusetts Institute of Technology

DOI: 10.1007/s00208-009-0344-6

Cite this article as:
Ionescu, A.D. & Staffilani, G. Math. Ann. (2009) 345: 133. doi:10.1007/s00208-009-0344-6


We prove global well-posedness and scattering in H1 for the defocusing nonlinear Schrödinger equations
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 H1 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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© Springer-Verlag 2009