Energy-critical NLS with quadratic potentials

Rowan Killip; Monica Visan; Xiaoyi Zhang

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Energy-critical NLS with quadratic potentials

Rowan Killip·Monica Visan·Xiaoyi Zhang

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Abstract

We consider the defocusing $\dot H^1$-critical nonlinear Schrödinger equation in all dimensions ($n\geq 3$) with a quadratic potential $V(x)=\pm \tfrac12 |x|^2$. We show global well-posedness for radial initial data obeying $\nabla u_0(x), xu_0(x) \in L^2$. In view of the potential $V$, this is the natural energy space. In the repulsive case, we also prove scattering. We follow the approach pioneered by Bourgain and Tao in the case of no potential; indeed, we include a proof of their results that incorporates a couple of simplifications discovered while treating the problem with quadratic potential.

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Rowan Killip, Monica Visan, Xiaoyi Zhang. 2010-10-20. Energy-critical NLS with quadratic potentials. https://arxiv.org/abs/math/0611394

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