  This thesis is devoted to a rigorous study of interacting quantum gases. The
main objects of interest are the closely related concepts of excitation
spectrum and quasiparticles. The immediate motivation of this work is to
propose a spectral point of view concerning these two concepts. In the first
part of this thesis we discuss the concepts of excitation spectrum and
quasiparticles. We provide an overview of physical motivations and based on
that we propose a spectral and Hamiltonian-based approach towards these terms.
Based on that, we formulate definitions and propositions related to these
concepts. In the second part we recall the Bogoliubov and
Hartree-Fock-Bogoliubov approximations, which in the physics literature are
used to obtain the quasiparticle picture. We show how these two approaches fit
into a universal scheme which allows us to arrive at a quasiparticle picture in
a more general setup. This scheme is based on the minimization of Hamiltonians
over the so-called Gaussian states. Its abstract formulation is the content of
Beliaev's Theorem. In the last part we present a rigorous result concerning the
justification of the Bogoliubov approximation. This justification employs the
concept of the mean-field and infinite-volume limit. We show that for a large
number of particles, a large volume and a sufficiently high density, the
low-lying energy-momentum spectrum of the homogeneous Bose gas is well
described by the Bogoliubov approximation. This result, which is formulated in
the form of a theorem, can be seen as the main result of this thesis.
