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

For an ideal Bose-gas within a multi-slabs periodic structure, we report a dimensional 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 potential perpendicular to the slabs of width $a$ and separated by a distance $b$. The ability of the particles to jump between adjacent slabs is determined by the hight $V_0$ and width $b$ of the potential barrier. Contrary to what happens in the boson gas inside a zero-width multilayers case, where the critical temperature diminishes and goes up again as a function of the wall separation, here the $T_c$ decreases continuously as the potential barrier height and the cell size $a+b$ increase. We plot the surface $T_c = 10^{-6}$ showing two prominent regions in the parameters space, which suggest a phase transition BEC-NOBEC at $T \neq 0$. %The position of the phase transition surface is almost independent of the ratio $r=b/a$ while the cell size $a+b$ is almost proportional to the square root of the height of the potential barriers. The specific heat shows a crossover from 3D to 2D when the height of the potential or the barrier width increase, in addition to the well known peak related to the Bose-Einstein condensation.