  We consider the operator ${\mathcal A}_h=-\Delta+iV$ in the semi-classical
$h\rightarrow 0$, where $V$ is a smooth real potential with no critical points.
We obtain both the left margin of the spectrum, as well as resolvent estimates
on the left side of this margin. We extend here previous results obtained for
the Dirichlet realization of ${\mathcal A}_h$ by removing significant
limitations that were formerly imposed on $V$. In addition, we apply our
techniques to the more general Robin boundary condition and to a transmission
problem which is of significant interest in physical applications.
