  In this paper, the ground state Wigner function of a many-body system is
explored theoretically and numerically. First, an eigenvalue problem for Wigner
function is derived based on the energy operator of the system. The validity of
finding the ground state through solving this eigenvalue problem is obtained by
building a correspondence between its solution and the solution of stationary
Schr\"odinger equation. Then, a numerical method is designed for solving
proposed eigenvalue problem in one dimensional case, which can be briefly
described by i) a simplified model is derived based on a quantum hydrodynamic
model [Z. Cai et al, J. Math. Chem., 2013] to reduce the dimension of the
problem, ii) an imaginary time propagation method is designed for solving the
model, and numerical techniques such as solution reconstruction are proposed
for the feasibility of the method. Results of several numerical experiments
verify our method, in which the potential application of the method for large
scale system is demonstrated by examples with density functional theory.
