  We analyze the divergent zero-point energy of a dilute and ultracold gas of
atoms in D spatial dimensions. For bosonic atoms we explicitly show how to
regularize this divergent contribution, which appears in the Gaussian
fluctuations of the functional integration, by using three different
regularization approaches: dimensional regularization, momentum-cutoff
regularization and convergence-factor regularization. In the case of the ideal
Bose gas the divergent zero-point fluctuations are completely removed, while in
the case of the interacting Bose gas these zero-point fluctuations give rise to
a finite correction to the equation of state. The final convergent equation of
state is independent of the regularization procedure but depends on the
dimensionality of the system and the two-dimensional case is highly nontrivial.
We also discuss very recent theoretical results on the divergent zero-point
energy of the D-dimensional superfluid Fermi gas in the BCS-BEC crossover. In
this case the zero-point energy is due to both fermionic single-particle
excitations and bosonic collective excitations, and its regularization gives
remarkable analytical results in the BEC regime of composite bosons. We compare
the beyond-mean-field equations of state of both bosons and fermions with
relevant experimental data on dilute and ultracold atoms quantitatively
confirming the contribution of zero-point-energy quantum fluctuations to the
thermodynamics of ultracold atoms at very low temperatures.
