  We experimentally demonstrate the trapping of a propagating Bose-Einstein
Condensate in a Bragg cavity produced by an attractive optical lattice with a
smooth envelope. As a consequence of the envelope, the band gaps become
position-dependent and act as mirrors of finite and velocity-dependent
reflectivity. We directly observe both the oscillations of the wave packet
bouncing in the cavity provided by these spatial gaps and the tunneling out for
narrow classes of velocity. Synchronization of different classes of velocity
can be achieved by proper shaping of the envelope. This technique can generate
single or multiple tunnel barriers for matter waves with a tunable transmission
probability, equivalent to a standard barrier of submicron size.
