  Motivated by the Penrose-Onsager criterion for Bose-Einstein condensation we
propose a functional theory for targeting low-lying excitation energies of
bosonic quantum systems through the one-particle picture. For this, we employ
an extension of the Rayleigh-Ritz variational principle to ensemble states with
spectrum $\boldsymbol{w}$ and prove a corresponding generalization of the
Hohenberg-Kohn theorem: The underlying one-particle reduced density matrix
determines all properties of systems of $N$ identical particles in their
$\boldsymbol{w}$-ensemble states. Then, to circumvent the $v$-representability
problem common to functional theories, and to deal with energetic degeneracies,
we resort to the Levy-Lieb constrained search formalism in combination with an
exact convex relaxation. The corresponding bosonic one-body
$\boldsymbol{w}$-ensemble $N$-representability problem is solved
comprehensively. Remarkably, this reveals a complete hierarchy of bosonic
exclusion principle constraints in conceptual analogy to Pauli's exclusion
principle for fermions and recently discovered generalizations thereof.
