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For an ideal Bose-gas within a multi-slabs periodic structure, we report adimensional crossover and discuss whether a BEC transition at $T_c \neq 0$disappears or not.  The multi-slabs structure is generated via a Kronig-Penney potentialperpendicular to the slabs of width $a$ and separated by a distance $b$. Theability of the particles to jump between adjacent slabs is determined by thehight $V_0$ and width $b$ of the potential barrier. Contrary to what happens inthe boson gas inside a zero-width multilayers case, where the criticaltemperature diminishes and goes up again as a function of the wall separation,here the $T_c$ decreases continuously as the potential barrier height and thecell size $a+b$ increase. We plot the surface $T_c = 10^{-6}$ showing twoprominent regions in the parameters space, which suggest a phase transitionBEC-NOBEC at $T \neq 0$. %The position of the phase transition surface isalmost independent of the ratio $r=b/a$ while the cell size $a+b$ is almostproportional to the square root of the height of the potential barriers. Thespecific heat shows a crossover from 3D to 2D when the height of the potentialor the barrier width increase, in addition to the well known peak related tothe Bose-Einstein condensation.

BEC and dimensional crossover in a boson gas within multi-slabs