Dynamical and Spectral Properties of Bose Gases with Singular Interactions

Bose-Einstein condensates have been experimentally realized in cold Bose gases for the first time in 1995. This discovery has renewed interest in the theory of interacting Bose gases; however, the mathematical understanding of Bose-Einstein condensation is still incomplete. In this thesis we derive e ective theories for the dynamical and spectral properties of Bose gases. Our theoretical description is based on scaling regimes where the number of particles N becomes large and the two-body interaction potential scales as N3 1V(N ), for a xed (01]. The case = 1 is known as Gross-Pitaevskii regime. Our rst result concerns the time evolution of Bose-Einstein condensates after the trapping potential is removed. In previous works, it was shown that the many-body dynamics can be approximated by a one-body nonlinear Schrodinger equation at the level of reduced density matrices. We provide a more precise approximation of the time evolution: in a second-quantized setting, we construct a unitary e ective dynamics, which, in the limit of large N, approximates the fully evolved state in Fock space norm. Our result comes with an explicit bound on the rate of convergence. It is valid for < 1. Our second result addresses the properties of the ground state wave function. It has been shown before that the ground state wave function exhibits condensation in the limit N , i.e. the one-particle density matrix associated to the ground state approaches a projection. We prove the optimal bound for the convergence rate, for a translation invariant system. To do this, we estimate uniformly in N the number of excited particles. This result is valid for = 1. Our last result concerns the ground state energy and the excitation spectrum of the translation invariant system. We prove the validity of Bogoliubov theory; this allows us to compute the ground state energy and the low-lying excitation spectrum up to an error that vanishes for large N. This result holds for < 1. This thesis is based on the articles [11], [9] and [10].