  We consider the 2d and 3d many body Schr\"odinger equations in the presence
of anisotropic switchable quadratic traps. We extend and improve the collapsing
estimates in Klainerman-Machedon [24] and Kirkpatrick-Schlein-Staffilani [23].
Together with an anisotropic version of the generalized lens transform in
Carles [3], we derive rigorously the cubic NLS with anisotropic switchable
quadratic traps in 2d through a modified Elgart-Erd\"os-Schlein-Yau procedure.
For the 3d case, we establish the uniqueness of the corresponding
Gross-Pitaevskii hierarchy without the assumption of factorized initial data.
