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This thesis is devoted to a rigorous study of interacting quantum gases. Themain objects of interest are the closely related concepts of excitationspectrum and quasiparticles. The immediate motivation of this work is topropose a spectral point of view concerning these two concepts. In the firstpart of this thesis we discuss the concepts of excitation spectrum andquasiparticles. We provide an overview of physical motivations and based onthat we propose a spectral and Hamiltonian-based approach towards these terms.Based on that, we formulate definitions and propositions related to theseconcepts. In the second part we recall the Bogoliubov andHartree-Fock-Bogoliubov approximations, which in the physics literature areused to obtain the quasiparticle picture. We show how these two approaches fitinto a universal scheme which allows us to arrive at a quasiparticle picture ina more general setup. This scheme is based on the minimization of Hamiltoniansover the so-called Gaussian states. Its abstract formulation is the content ofBeliaev's Theorem. In the last part we present a rigorous result concerning thejustification of the Bogoliubov approximation. This justification employs theconcept of the mean-field and infinite-volume limit. We show that for a largenumber of particles, a large volume and a sufficiently high density, thelow-lying energy-momentum spectrum of the homogeneous Bose gas is welldescribed by the Bogoliubov approximation. This result, which is formulated inthe form of a theorem, can be seen as the main result of this thesis.

Excitation spectrum and quasiparticles in quantum gases. A rigorous approach