two or more fermionic levels inside. To this regard we consider solitonic BEC states as trapping (expulsive) dot (antidot) potentials and show that the fermionic bound states formed inside are at the bottom of the Fermi sea and therefore well protected by thermal decoherence.

From the other side, we investigate the spectral properties of a BFM both in parabolic and in periodic potentials and demonstrate the existence of BFM gap-solitons in optical lattices (OL).

To this regard, we derive equations of motion for the mixture adopting a mean field approximation for the bosons and assuming non interacting (spin polarized) fermions in true quantum regime. These equations are solved by means of a self-consistent procedure and the spectral properties of BFM, both in parabolic traps and in optical lattices, derived. In particular, we show that fermionic bound state levels at the bottom of the Fermi sea can be trapped inside gap-solitons for both repulsive and attractive boson-boson interactions. We also show that, as the boson-fermion interaction is varied, the trapped levels undergo avoid crossings or level crossings according to the symmetry of the corresponding wavefunctions with respect to the OL. Analytical expressions for energy levels and for the number of fermionic 