Boson gas in a periodic array of tubes

We report the thermodynamic properties of an ideal boson gas confined in an infinite periodic array of channels modeled by two, mutually perpendicular, Kronig-Penney delta-potentials. The particle's motion is hindered in the $x$-$y$ directions, allowing tunneling of particles through the walls, while no confinement along the $z$ direction is considered. It is shown that there exists a finite Bose-Einstein condensation (BEC) critical temperature $T_{c}$ that decreases monotonically from the 3D ideal boson gas (IBG) value $T_{0}$ as the strength of confinement $P_{0}$ is increased while keeping the channel's cross section, $a_x a_y$ constant. In contrast, $T_{c}$ is a non-monotonic function of the cross-section area for fixed $P_{0} $. In addition to the BEC cusp, the specific heat exhibits a set of maxima and minima. The minimum located at the highest temperature is a clear signal of the confinement effect which occurs when the boson wavelength is twice the cross-section side size. This confinement is amplified when the wall strength is increased until a dimensional crossover from 3D to 1D is produced.