Abstract. We consider the energy critical nonlinear Schrödinger equation in dimensions d ≥ 3 with a harmonic oscillator potential V (x) = 1 2 |x| 2 . When the nonlinearity is defocusing, we prove global wellposedness for all initial data in the energy space Σ, consisting of all functions u 0 such that both ∇u 0 and xu 0 belong to L 2 . This result extends a theorem of , which treats the radial case. For the focusing problem, we obtain global wellposedness for all data satisfying an analogue of the usual size restriction in terms of the ground state W . The proof uses the concentration compactness variant of the induction on energy paradigm. In particular, we develop a linear profile decomposition adapted to the propagator exp[it(|x| 2 )] for bounded sequences in Σ.

Consider the global wellposedness problem for nonlinear Schrödinger equation2 |x| 2 was recently treated by the author. This note generalizes the results to a class of "approximately quadratic" potentials.We closely follow the previous concentration compactness arguments for the harmonic oscillator. A key technical difference is that in the absence of a concrete formula for the linear propagator, we apply more general tools from microlocal analysis, including a Fourier integral parametrix of Fujiwara.

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Consider the defocusing quintic nonlinear Schrödinger equation on R 3 with initial data in the energy space. This problem is "energy-critical" in view of a certain scale-invariance, which is a main source of difficulty in the analysis of this equation. It is a nontrivial fact that all finiteenergy solutions scatter to linear solutions. We show that this remains true under small compact deformations of the Euclidean metric. Our main new ingredient is a long-time microlocal weak dispersive estimate that accounts for the refocusing of geodesics.

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