Analysis and Computation for Ground State Solutions of Bose-Fermi Mixtures at Zero Temperature

Previous numerical studies on the ground state structure of Bose–Fermi mixtures mostly relied on Thomas–Fermi (TF) approximation for the Fermi gas. In this paper, we establish the existence and uniqueness of ground state solutions of Bose–Fermi mixtures at zero temperature for both a coupled Gross–Pitaevskii (GP) equations model and a model with TF approximation for fermions. To prove the uniqueness, the key is to estimate the L∞ bounds of the ground state solution. By implementing an efficient method—gradient flow with discrete normalization with backward Euler finite difference discretization—to compute the coupled GP equations, we report extensive numerical results in one and two dimensions. The numerical experiments show that we can also extract many interesting phenomena without reference to TF approximation for the fermions. Finally, we numerically compare the ground state solutions for the coupled GP equations model and the model with TF approximation for fermions as well as for the model with TF approximations for both bosons and fermions.