Spectral analysis of a complex Schrödinger operator in the semiclassical limit

We consider the Dirichlet realization of the operator $-h^2Δ+iV$ in the semi-classical limit $h\to0$, where $V$ is a smooth real potential with no critical points. For a one dimensional setting, we obtain the complete asymptotic expansion, in powers of $h$, of each eigenvalue. In two dimensions we obtain the left margin of the spectrum, under some additional conditions.

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