  Generalizing previous results obtained for the spectrum of the Dirichlet and
Neumann realizations in a bounded domain of a Schr\"odinger operator with a
purely imaginary potential $h^2\Delta+iV$ in the semiclassical limit $h\to 0$
we address the same problem in exterior domains. In particular we obtain the
left margin of the spectrum, and the emptiness of the essential part of the
spectrum under some additional assumptions.
