  By introducing a phase field and solving the eigen-functional equation of
particles, we obtain the exact expressions of the ground state energy as a
functional of the particle density for interacting electron/boson systems, and
a two-dimensional electron gas under an external magnetic field, respectively.
With the eigen-functionals of the particles, we can construct the ground state
wave-function of the systems. Moreover, with the expressions of the ground
state energy, we can exactly determine the ground state energy and the ground
state particle density of the systems by taking $% \delta E_g[\rho ]/\delta
\rho (x)=0$.
