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In this article, we give a description of the spectrum of a class of non-selfadjoint perturbations of selfadjoint   $\hbar$  -pseudo-differential operators in dimension one and we show that it is given by Bohr–Sommerfeld quantization conditions. To achieve this, we make use of previous work by Michael Hitrik, Anders Melin, and Johannes Sjöstrand. We also give an application of our result in the case of   $\mathcal{P} \mathcal{T}$  -symmetric pseudo-differential operators.

Bohr–Sommerfeld Quantization Conditions for Non-Selfadjoint Perturbations of Selfadjoint Operators in Dimension One