  We consider the Bose gas on a $d$-dimensional anisotropic lattice employing
the imperfect (mean-field) gas as a prototype example. We study the dimensional
crossover arising as a result of varying the dispersion relation at finite
temperature $T$. We analyze in particular situations where one of the relevant
effective dimensionalities is located at or below the lower critical dimension,
so that the Bose-Einstein condensate becomes expelled from the system by
anisotropically modifying the lattice parameters controlling the kinetic term
in the Hamiltonian. We clarify the mechanism governing this phenomenon.
Subsequently we study the thermodynamic Casimir effect occurring in this
system. We compute the exact profile of the scaling function for the Casimir
energy. As an effect of strongly anisotropic scale invariance, the Casimir
force below or at the critical temperature $T_c$ may be repulsive even for
periodic boundary conditions. The corresponding Casimir amplitude is universal
only in a restricted sense, and the power law governing the decay of the
Casimir interaction becomes modified. We also demonstrate that, under certain
circumstances, the scaling function is constant for suffciently large values of
the scaling variable, and in consequence is not an analytical function. At $T >
T_c$ the Casimir-like interactions reflect the structure of the correlation
function, and, for certain orientations of the confining walls, show
exponentially damped oscillatory behavior so that the corresponding force is
attractive or repulsive depending on the distance.
