Spectral semi-classical analysis of a complex Schrödinger operator in exterior domains

Generalizing previous results obtained for the spectrum of the Dirichlet and Neumann realizations in a bounded domain of a Schrödinger operator with a purely imaginary potential -h²Δ + iV in the semiclassical limit h → 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.