the bosonic part, we solve the quantum eigenvalue problems for the fermionic part, compute the fermionic density, derive the bosonic wavefunction by solving the bosonic equation and iterate the procedure until convergence is reached (an efficient implementation of this scheme for single BEC was discussed in [@salerno]). In the following we apply this method for BFM in parabolic traps and in optical lattices, concentrating, for simplicity, on the quasi one dimensional case.

Bose-Fermi mixtures in a parabolic trap
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In this section we consider BFM in a trap potential of the form $
 V_{trap}= \frac \varepsilon 2 m   \omega^2 (x-x_0)^2 $ and assume for simplicity $m_b=m_f$, $V_{ext}^b=
V_{ext}^f=V_{trap}$ (the extension to the generic case is straightforward). We normalize Eqs. (\[bose-fermi-eqns\]) by measuring space in harmonic oscillator length units $a_0=\sqrt{\hbar/m \omega}$, time in units of $\hbar/E_0$, with $E_0$ the zero point energy of the oscillator. Introducing the parameter $\chi_{b (bf)}$ as $g_{b (bf)}= (8 \pi \hbar^3
\omega_{\perp} a_{b_0 (bf_0)}/m) \chi_{b (bf)},\;$ where $a_{{b}_0}, a_{{bf}_0}$ are scattering lengths in zero magnetic field (we use $\chi_{b (bf)}$ to change nonlinearities by means of external magnetic fields via Feshbach resonances), and rescaling wavefunctions according to $\psi^b \rightarrow (4 \pi g_{{b}_0}
N_b /m \omega a^2_{\perp})^{1/2} 