  Density functional theory constitutes the workhorse of modern electronic
structure calculations due to its favourable computational cost despite the
fact that it usually fails to describe strongly correlated systems. A
particularly promising approach to overcome those difficulties is reduced
density matrix functional theory (RDMFT): It abandons the complexity of the
$N$-particle wave function and at the same time explicitly allows for
fractional occupation numbers. It is the goal of this thesis to initiate and
establish a bosonic RDMFT for both ground state and excited state energy
calculations. Motivated by the Onsager and Penrose criterion which identifies
RDMFT as a particularly suitable approach to describe Bose-Einstein condensates
(BECs), we derive the universal functional for a homogeneous BEC in the
Bogoliubov regime. Remarkably, the gradient of the universal functional is
found to diverge repulsively in the regime of complete condensation. This
introduces the new concept of a BEC force, which provides a universal
explanation for quantum depletion since it is merely based on the geometry of
quantum states. In the second part of the thesis, we propose and work out an
ensemble RDMFT targeting excitations in bosonic quantum systems. This endeavour
further highlights the potential of convex analysis for the development of
functional theories in the future. Indeed by resorting to several concepts from
convex analysis, we succeeded to provide a comprehensive foundation of
$\boldsymbol{w}$-ensemble RDMFT for bosons which is further based on a
generalization of the Ritz variational principle and a constrained search
formalism. In particular, we solve the emerging $N$-representability problem
leading to a generalization of Pauli's famous exclusion principle to bosonic
mixed states.

Quantum Gases;Quantum Physics