Methods of Harmonic Analysis Applied to Bose-Einstein Condensation

This dissertation studies questions whose origin is in quantum statistical mechanics and is concerned about the evolution of large numbers of quantum spinless, interacting particles. More specifically, we study the analysis of the $N$-particle linear Schr\"{o}dinger equation as $N\rightarrow \infty $ and discuss rigorously how the $1$-body nonlinear Schr\"{o}dinger equation comes from this limit process. Such problems arise in Bose-Einstein condensation. In the first part of this dissertation, we consider the 2d and 3d many body Schr\"{o}dinger equations in the presence of anisotropic switchable quadratic traps. We extend and improve the collapsing estimates in Klainerman-Machedon [29] and Kirkpatrick-Schlein-Staffilani [27]. Combining with an anisotropic version of the generalized lens transform as in Carles [3], we offer a rigorous derivation of the cubic NLS with anisotropic switchable quadratic traps in 2d through an appropriately modified procedure in Elgart-Erd\"{o}s-Schlein-Yau [12, 13, 14, 15, 16, 17, 18] which is based on a kinetic hierarchy. For the 3d case, we establish the uniqueness of the corresponding Gross-Pitaevskii hierarchy without the assumption of factorized initial data. In the second part of this thesis, we consider the Hamiltonian evolution of $N$ weakly interacting Bosons. Assuming triple collisions, its mean field approximation is given by a quintic Hartree equation. We construct a second-order correction to the mean field approximation using a kernel $k(t,x,y)\ $ describing pair creation and derive an evolution equation for $k$. We show the global existence for the resulting evolution equation for the correction and establish an apriori estimate comparing the approximation to the exact Hamiltonian evolution. Our error estimate is global and uniform in time. Comparing with the work of Rodnianski and Schlein [35], and Grillakis, Machedon and Margetis [21,22], where the error estimate grows in time, our approximation tracks the exact dynamics for all time with an error of the order of $O\left( 1/\sqrt{N}\right) .