  The increasing interest in the Mueller density-matrix-functional theory has
led us to a systematic mathematical investigation of its properties. This
functional is similar to the Hartree-Fock functional, but with a modified
exchange term in which the square of the density matrix \gamma(X, X') is
replaced by the square of \gamma^{1/2}(X, X'). After an extensive introductory
discussion of density-matrix-functional theory we show, among other things,
that this functional is convex (unlike the HF functional) and that energy
minimizing \gamma's have unique densities \rho(x), which is a physically
desirable property often absent in HF theory. We show that minimizers exist if
N \leq Z, and derive various properties of the minimal energy and the
corresponding minimizers. We also give a precise statement about the equation
for the orbitals of \gamma, which is more complex than for HF theory. We state
some open mathematical questions about the theory together with conjectured
solutions.
