{"date_updated":"2026-06-29T10:20:21Z","volume":34,"issue":"12","type":"journal_article","oa":1,"month":"11","oa_version":"Preprint","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication":"Communications in Partial Differential Equations","author":[{"last_name":"Killip","full_name":"Killip, Rowan","first_name":"Rowan"},{"last_name":"Visan","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","first_name":"Monica","full_name":"Visan, Monica"},{"full_name":"Zhang, Xiaoyi","first_name":"Xiaoyi","last_name":"Zhang"}],"article_type":"original","_id":"22058","scopus_import":"1","arxiv":1,"OA_place":"repository","doi":"10.1080/03605300903328109","quality_controlled":"1","mathsc":["35Q55","35B33"],"publication_status":"published","date_created":"2026-06-19T07:57:55Z","intvolume":"        34","article_processing_charge":"No","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.math/0611394","open_access":"1"}],"publisher":"Informa UK Limited","year":"2009","citation":{"chicago":"Killip, Rowan, Monica Vişan, and Xiaoyi Zhang. “Energy-Critical NLS with Quadratic Potentials.” <i>Communications in Partial Differential Equations</i>. Informa UK Limited, 2009. <a href=\"https://doi.org/10.1080/03605300903328109\">https://doi.org/10.1080/03605300903328109</a>.","ama":"Killip R, Vişan M, Zhang X. Energy-critical NLS with quadratic potentials. <i>Communications in Partial Differential Equations</i>. 2009;34(12):1531-1565. doi:<a href=\"https://doi.org/10.1080/03605300903328109\">10.1080/03605300903328109</a>","ista":"Killip R, Vişan M, Zhang X. 2009. Energy-critical NLS with quadratic potentials. Communications in Partial Differential Equations. 34(12), 1531–1565.","mla":"Killip, Rowan, et al. “Energy-Critical NLS with Quadratic Potentials.” <i>Communications in Partial Differential Equations</i>, vol. 34, no. 12, Informa UK Limited, 2009, pp. 1531–65, doi:<a href=\"https://doi.org/10.1080/03605300903328109\">10.1080/03605300903328109</a>.","apa":"Killip, R., Vişan, M., &#38; Zhang, X. (2009). Energy-critical NLS with quadratic potentials. <i>Communications in Partial Differential Equations</i>. Informa UK Limited. <a href=\"https://doi.org/10.1080/03605300903328109\">https://doi.org/10.1080/03605300903328109</a>","short":"R. Killip, M. Vişan, X. Zhang, Communications in Partial Differential Equations 34 (2009) 1531–1565.","ieee":"R. Killip, M. Vişan, and X. Zhang, “Energy-critical NLS with quadratic potentials,” <i>Communications in Partial Differential Equations</i>, vol. 34, no. 12. Informa UK Limited, pp. 1531–1565, 2009."},"day":"01","external_id":{"arxiv":["math/0611394"]},"title":"Energy-critical NLS with quadratic potentials","OA_type":"green","page":"1531-1565","date_published":"2009-11-01T00:00:00Z","publication_identifier":{"issn":["0360-5302","1532-4133"]},"language":[{"iso":"eng"}],"das_tickbox":"1","extern":"1","abstract":[{"lang":"eng","text":"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\r\ncouple of simplifications discovered while treating the problem with quadratic potential."}]}