Spectral Analysis of a Complex Schrödinger Operator in the Semiclassical Limit

Abstract

We consider the Dirichlet realization of the operator $-h^2\Delta+iV$ in the semiclassical 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 assumptions.