The spectrum of a Schr\"odinger operator in a wire-like domain with a purely imaginary degenerate potential in the semiclassical limit

Consider a two-dimensional domain shaped like a wire, not necessarily of uniform cross section. Let $V$ denote an electric potential driven by a voltage drop between the conducting surfaces of the wire. We consider the operator ${\mathcal{A}}_h=-h^2\mathrm{\Delta }+iV$ in the semi-classical limit $h\to 0$. We obtain both the asymptotic behaviour of 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 potentials for which the set where the current (or $\nabla V$) is normal to the boundary is discrete, in contrast with the present case where $V$ is constant along the conducting surfaces.