Energy-critical NLS with quadratic potentials

article Energy-critical NLS with quadratic potentials published yes Rowan Killip author Monica Visan author 056daca0-b8d1-11f0-964f-f91054abf8ca Xiaoyi Zhang author We consider the defocusing H˙ 1-critical nonlinear Schr¨odinger equation in all dimensions (n ≥ 3) with a quadratic potential V (x) = ± 1/2|x|^2. We show global well-posedness for radial initial data obeying ∇u0(x), xu0(x) ∈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. Informa UK Limited2009 eng Communications in Partial Differential Equations 0360-5302 1532-4133 math/061139410.1080/03605300903328109 34121531-1565 yes R. Killip, M. Vişan, and X. Zhang, “Energy-critical NLS with quadratic potentials,” Communications in Partial Differential Equations, vol. 34, no. 12. Informa UK Limited, pp. 1531–1565, 2009. R. Killip, M. Vişan, X. Zhang, Communications in Partial Differential Equations 34 (2009) 1531–1565. Killip, R., Vişan, M., & Zhang, X. (2009). Energy-critical NLS with quadratic potentials. Communications in Partial Differential Equations. Informa UK Limited. <a href="https://doi.org/10.1080/03605300903328109">https://doi.org/10.1080/03605300903328109</a> Killip R, Vişan M, Zhang X. 2009. Energy-critical NLS with quadratic potentials. Communications in Partial Differential Equations. 34(12), 1531–1565. Killip, Rowan, et al. “Energy-Critical NLS with Quadratic Potentials.” Communications in Partial Differential Equations, vol. 34, no. 12, Informa UK Limited, 2009, pp. 1531–65, doi:<a href="https://doi.org/10.1080/03605300903328109">10.1080/03605300903328109</a>. Killip, Rowan, Monica Vişan, and Xiaoyi Zhang. “Energy-Critical NLS with Quadratic Potentials.” Communications in Partial Differential Equations. Informa UK Limited, 2009. <a href="https://doi.org/10.1080/03605300903328109">https://doi.org/10.1080/03605300903328109</a>. Killip R, Vişan M, Zhang X. Energy-critical NLS with quadratic potentials. Communications in Partial Differential Equations. 2009;34(12):1531-1565. doi:<a href="https://doi.org/10.1080/03605300903328109">10.1080/03605300903328109</a> 220582026-06-19T07:57:55Z2026-06-29T10:20:21Z