Adiabatic Asymptotics of the Reflection Coefficient

The properties of the functions f and M are well known (see, e.g., [1]). The function R = −MM , defined for k ∈ R, |R| = 1, is called the reflection coefficient. If v(x) = p( x), where > 0 is a small parameter, then all these objects become functions of , and we may ask about the asymptotic behavior as → 0 of, say, the solution f , via which M , R, and ψ can be expressed. Such asymptotics is said to be quasiclassical. The possibility itself of constructing this asymptotics, as well as its form, depend crucially on the location of the curve k + p(ξ) = E, k ∈ C, ξ ∈ R, on the phase plane (k, ξ), i.e., on the graph of the potential p. The example of