Regular Bohr-Sommerfeld rules for non-self-adjoint Berezin--Toeplitz operators and complex Lagrangian states

We describe the eigenvalues and eigenvectors of real-analytic, non-self-adjoint Berezin–Toeplitz operators, up to exponentially small error, on complex one-dimensional compact manifolds, under the hypothesis of regularity of the energy levels. These results form a complex version of the Bohr-Sommerfeld quantization conditions; they hold under a hypothesis that the skew-adjoint part is small but can be of principal order with respect to the semiclassical parameter. To this end, we develop a calculus of Fourier Integral Operators and Lagrangian states associated with complex Lagrangians; these tools are of independent interest.