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

We consider the operator ${\mathcal{A}}_h=-\mathrm{\Delta }+iV$ in the semi-classical $h\to 0$, 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.