Complex WKB method for 3-level scattering systems

In this note the S‐matrix naturally associated with a singularly perturbed three‐dimensional system of linear differential equations without turning point on the real axis is considered. It is shown that for a fairly large class of examples, the Complex WKB method gives results far better than what is proven under generic circumstances. In particular, we show how to compute asymptotically all exponentially small off‐diagonal elements of the corresponding S‐matrix.