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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 estimateson the left side of this margin. We extend here previous results obtained forthe Dirichlet realization of ${\mathcal A}_h$ by removing significantlimitations that were formerly imposed on $V$. In addition, we apply ourtechniques to the more general Robin boundary condition and to a transmissionproblem which is of significant interest in physical applications.

On a Schr\"odinger operator with a purely imaginary potential in the semiclassical limit