  Trapped degenerate dipolar Bose and Fermi gases of cylindrical symmetry with
the polarization vector along the symmetry axis are only stable for the
strength of dipolar interaction below a critical value. In the case of bosons,
the stability of such a dipolar Bose-Einstein condensate (BEC) is investigated
for different strengths of contact and dipolar interactions using variational
approximation and numerical solution of a mean-field model. In the disk shape,
with the polarization vector perpendicular to the plane of the disk, the atoms
experience an overall dipolar repulsion and this fact should contribute to the
stability. However, a complete numerical solution of the dynamics leads to the
collapse of a strongly disk-shaped dipolar BEC due to the long-range
anisotropic dipolar interaction. In the case of fermions, the stability of a
trapped single-component degenerate dipolar Fermi gas is studied including the
Hartree-Fock exchange and Brueckner-Goldstone correlation energies in the local
density approximation valid for a large number of atoms. Estimates for the
maximum allowed number of polar Bose and Fermi molecules in BEC and degenerate
Fermi gas are given.

Quantum Gases