  We consider a Bose-Einstein condensate (BEC), which is characterized by
long-range and anisotropic dipole-dipole interactions and vanishing s-wave
scattering length, in a double-well potential. The properties of this system
are investigated as functions of the height of the barrier that splits the
harmonic trap into two halves, the number of particles (or dipole-dipole
strength) and the aspect ratio $\lambda$, which is defined as the ratio between
the axial and longitudinal trapping frequencies $\omega_z$ and $\omega_{\rho}$.
The phase diagram is determined by analyzing the stationary mean-field
solutions. Three distinct regions are found: a region where the energetically
lowest lying stationary solution is symmetric, a region where the energetically
lowest lying stationary solution is located asymmetrically in one of the wells,
and a region where the system is mechanically unstable. For sufficiently large
aspect ratio $\lambda$ and sufficiently high barrier height, the system
consists of two connected quasi-two-dimensional sheets with density profiles
whose maxima are located either at $\rho=0$ or away from $\rho=0$. The
stability of the stationary solutions is investigated by analyzing the
Bogoliubov de Gennes excitation spectrum and the dynamical response to small
perturbations. These studies reveal unique oscillation frequencies and distinct
collapse mechanisms. The results derived within the mean-field framework are
complemented by an analysis based on a two-mode model.
