Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps

• Published in: Journal de mathématiques pures et appliquées Vol. 98; no. 4; pp. 450 - 478
• Main Author: Chen, Xuwen
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Masson SAS 01.10.2012; Elsevier
• Subjects: Anisotropic switchable quadratic trap; Collapsing estimate; Exact sciences and technology; General mathematics; General, history and biography; Gross–Pitaevskii hierarchy; Logic and foundations; Mathematical analysis; Mathematical logic, foundations, set theory; Mathematics; Metaplectic representation; Partial differential equations; Recursion theory; Sciences and techniques of general use; secondary; Collapsing estimate; Metaplectic representation; Anisotropic switchable quadratic trap; Gross–Pitaevskii hierarchy; primary; Non linear equation; primary 35Q55, 35A02, 81V70; secondary 35A23, 35B45, 81Q05; Gross-Pitaevskii hierarchy; Quadratic field; Anisotropic equation; Schrödinger equation
• ISSN: 0021-7824
• DOI: 10.1016/j.matpur.2012.02.003

We consider the 2d and 3d many body Schrödinger equations in the presence of anisotropic switchable quadratic traps. We extend and improve the collapsing estimates in Klainerman and Machedon (2008) [25] and Kirkpatrick, Schlein and Staffilani (2011) [23]. Together with an anisotropic version of the generalized lens transform in Carles (2011) [3], we derive rigorously the cubic NLS with anisotropic switchable quadratic traps in 2d through a modified Elgart–Erdös–Schlein–Yau procedure. For the 3d case, we establish the uniqueness of the corresponding Gross–Pitaevskii hierarchy without the assumption of factorized initial data. On considère les équations de Schrödinger à plusieurs corps en présence de pièges quadratiques anisotropes et commutables pour les dimensions 2 et 3. On étend et on améliore les estimations dʼécroulement de Klainerman et Machedon (2008) [25] et de Kirkpatrick, Schlein et Staffilani (2011) [23]. En utilisant une version anisotrope de la transformation lenticulaire généralisée de Carles (2011) [3] on déduit rigoureusement, pour la dimension 2, la cubique NSL en présence de pièges quadratiques anisotropes et commutables par la méthode de Elgart–Erdös–Schlein–Yau modifiée. Pour la dimension 3 on établit lʼunicité de la hiérarchie de Gross–Pitaevskii correspondante sans lʼhypothèse dʼune donnée initiale factorisée.