On scattering for NLS: from Euclidean to hyperbolic space

Details On scattering for NLS: from Euclidean to hyperbolic space On scattering for NLS: from Euclidean to hyperbolic space

2008-01-15

arxiv.org/abs/0801.2227v2

Discrete and Continuous Dynamical Systems: Series A 24, 4 (2009) 1113-1127

Mathematics

Analysis of PDEs

13 pages. References updated; see Remark 1.1

Scientific paper

10.3934/dcds.2009.24.1113

We prove asymptotic completeness in the energy space for the nonlinear Schrodinger equation posed on hyperbolic space in the radial case, in space dimension at least 4, and for any energy-subcritical, defocusing, power nonlinearity. The proof is based on simple Morawetz estimates and weighted Strichartz estimates. We investigate the same question on spaces which kind of interpolate between Euclidean space and hyperbolic space, showing that the family of short range nonlinearities becomes larger and larger as the space approaches the hyperbolic space. Finally, we describe the large time behavior of radial solutions to the free dynamics.

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