The topic of this thesis is the mathematical analysis of physically motivated modelsfor a trapped dilute Bose gas with repulsive pairwise atomic interactions at zero temper- ature. Our goal is to develop the spectral theory for excited many-body quantum states of these systems by accounting for the scattering of atoms in pairs from the macroscopic state (condensate). This general methodology, known as pair-excitation, was introduced in the physics literature in the 1960s – the work of this thesis provides the first compre- hensive mathematical treatment of many aspects of pair-excitation. This includes, e.g., the spectral theory for pair-transformed approximate Hamiltonians, a general existence theory for the pair-excitation kernel, and the connection between the pair-excitation for- malism to quasiparticle excitations in the Bose gas.

We formulate the method of pair-excitation for several historical models of the Bosegas from the physics literature. In particular, we focus on the seminal works of Wu, Fetter, Griffin, and Lee, Huang, and Yang. Each of these models introduce unique features to the mathematical analysis, but the general strategy remains the same: transform the approximate Hamiltonian using a suitably-defined pair-excitation operator. This operator is not determined a priori, but is chosen as part of the problem in order to simplify the expression of excited states of the transformed system.

The study begins with models for the Bose gas in the non-translation-invariant set-ting, where the particles are spatially-confined in an external trapping potential. In this setting, formulating the pair-excitation method entails solving a nonlinear integro-partial- differential equation for the pair-excitation kernel. We provide a general existence theory for this kernel via a variational approach. The kernel which we find allows us to connect the pair-excitation method to the more widely-studied unitary transformation of quadratic Hamiltonians via Bogoliubov rotation. The theory for the kernel also allows us to write a simple formula for excited many-body states, which can be adapted to the various models which we consider in this work.

We then study the problem for the pair-excited transformed approximate Hamilto-nian for Bosons in a periodic box. In this setting, the description of the effective Hamil- tonian in the momentum basis is particularly simple. However, the lack of particle con- servation means that the pair-excitation transform is unbounded in operator norm, and spectral methods developed in earlier chapters are enriched with new tools.