Rigorous Density Functional Theory for Inhomogeneous Bose-Condensed Fluids

The density functional theory originally developed by Hohenberg, Kohn and Sham provides a rigorous conceptual framework for dealing with inhomogeneous interacting Fermi systems. We extend this approach to deal with inhomogeneous interacting Bose-condensed systems, limiting this presentation to setting up the formalism to deal with ground state $(T=0)$ properties. The key new feature is that one must deal with energy functionals of both the local density $n({\bf r})$ and the local complex macroscopic wavefunction $\Phi ({\bf r})$ associated with the Bose broken-symmetry (the local condensate density is $n_{c}({\bf r}) = \vert \Phi ({\bf r}) \vert ^{2}$). Implementing the Kohn-Sham scheme, we reduce the problem to a gas of weakly-interacting Bosons moving in self-consistent diagonal and off-diagonal one-body potentials. Our formalism should provide the basis for studies of the surface properties of liquid $^4$He as well as the properties of Bose-condensed atomic gases trapped in external potentials.