Quantized Riemann surfaces and semiclassical spectral series for a non-self-adjoint Schrödinger operator with periodic coefficients

We consider a non-self-adjoint Schrödinger operator describing the motion of a particle in a one-dimensional space with an analytic potential $iV(x)$ that is periodic with a real period $T$ and is purely imaginary on the real axis. We study the spectrum of this operator in the semiclassical limit and show that the points of its spectrum asymptotically belong to the so-called spectral graph. We construct the spectral graph and evaluate the asymptotic form of the spectrum. A Riemann surface of the particle energy-conservation equation can be constructed in the phase space. We show that both the spectral graph and the asymptotic form of the spectrum can be evaluated in terms of integrals of the $p\,dx$ form (where $x\in\mathbb C/T\mathbb Z$ and $p\in\mathbb C$ are the particle coordinate and momentum) taken along basis cycles on this Riemann surface. We use the technique of Stokes lines to construct the asymptotic form of the spectrum.