A complex WKB method for adiabatic problems

This work is devoted to a new version of the complex WKB method suited for adiabatic perturbations of one-dimensional periodic Schrodinger operators. Therefore, we introduce an additional parameter, and it is this parameter (and not the variable of the equation) that will become complex. This naturally leads to canonical domains where we construct solutions of the Schrodinger equation with a standard asymptotic behavior. These can be used to compute the asymptotics of the exponentially small coefficients of transfer matrices (e.g., scattering matrices, monodromy matrices, etc.). We give an example of such a computation.