Complex WKB Method (One-Dimensional Linear Problems the Complex Plane)

The survey is devoted to the complex WKB method which arose as an approach to describing the asymptotic behavior of solutions to one-dimensional ordinary differential equations with semiclassical parameter on the complex plane. Later this method was generalized to the case of difference equations. Related constructions arose when studying exponentially small effects in the problem concerning the adiabatic perturbation of the one-dimensional periodic Schrödinger operator. All these three branches of the method are discussed in the survey from a unified position. The main constructions of the method are described and the proofs are either provided or their ideas are described in detail. Some new finds are published for the first time.