band (shadow region with positive slope) have also been shown. The fermionic band is half filled. Panels (c) and (d): the same as panels (a),(b), but for the inter-site symmetric case with $N_b=1.47$ . Plotted quantities are in normalized units.[]{data-label="fig3"}](fig3c.eps "fig:"){width="4.2cm" height="4.2cm"} ![Panel (a). On-site symmetric densities for the case of $\chi_{b}=1, \chi_{bf}=-1$, $N_f=20$ and normalized number of bosons $N_b=1$ for an optical lattice of strength $\varepsilon=3$ (thick line refers to bosons). To identify the symmetry of the state the OL is reported with an amplitude scaled by a factor of $60$. Panel (b). Effective potentials (only lower part) and energy levels of the Bose (thin line and open circles) and Fermi (thick line and filled circles) mixture corresponding to the densities in panel(a). Lowest bosonic band (shadow region with negative slope) and lowest fermionic band (shadow region with positive slope) have also been shown. The fermionic band is half filled. Panels (c) and (d): the same as panels (a),(b), but for the inter-site symmetric case with $N_b=1.47$ . Plotted quantities are in normalized units.[]{data-label="fig3"}](fig3d.eps "fig:"){width="4.2cm" height="4.2cm"}

Bose-Fermi mixtures in optical lattices
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In this section we consider a trap potential of the form $V_{trap}=V_0 \cos (2 k_L x) $, 