Bosons among Kronig-Penney layers

We describe the statistical behavior of an ideal boson gas among periodic plane layers which are simulated by an external Kronig-Penney potential in the $z$-direction while bosons are allowed to be free in the other two directions. The critical temperature goes from the 3D ideal boson gas critical temperature to that of a quasi-2D boson gas inside a slab of wide $a$, as $P = mV_0 ab/\hbar^2$ goes from zero to infinity. When we turn on a small repulsive (attractive) interaction between bosons, the critical temperature increases (decreases) compared to the ideal case with the same $P$ and $a$. We also calculate and discuss other thermodynamic properties such as the specific heat.