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Energy-critical NLS with potentials of quadratic growth

The first author is supported by NSF grants DMS-0838680, DMS-1265868, DMS-0901166, DMS-1161396.

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  • We consider the global wellposedness problem for the nonlinear Schrödinger equation

    $i\partial_t u = [-\tfrac{1}{2} Δ + V(x)] u ± |u|^{4/(d-2)} u, \ u(0)∈ Σ(\mathbf{R}^d),$

    where $Σ$ is the weighted Sobolev space $\dot{H}^1 \cap |x|^{-1} L^2$. The case $V(x) = \tfrac{1}{2}|x|^2$ 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.

    Mathematics Subject Classification: Primary: 35Q55; Secondary: 53C35.


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