Quantization Conditions on Riemannian Surfaces and the Semiclassical Spectrum of the Schrödinger Operator with Complex Potential

We describe the asymptotics of the spectrum of the operator
$$ \widehat H\biggl(x,-\imath h\frac{\partial}{\partial x}\biggr)=-h^2\frac{\partial^2}{\partial x^2}+\imath(\cos x+\cos2x) $$
as $h\to0$ and show that the spectrum concentrates near some graph on the complex plane. We obtain equations for the eigenvalues, which are conditions on the periods of a holomorphic form on the corresponding Riemannian surface.