Exponential estimates on the one-dimensional Schroedinger equation with bounded analytic potential

Using a Nekhoroshev-like perturbation technique, we investigate the solutions of the one-dimensional stationary Schroedinger equation, with bounded analytic potential. For sufficiently high energy E, we construct (in principle, up to any order in 1/√E) approximate solutions, which resemble free waves, and are very close to the true solutions over very large distances, growing exponentially with √E. For potentials which decay sufficiently rapidly at infinity, we find that the scattering matrix differs from a trivial one by a quantity exponentially small in √E (in particular, the reflection coefficient is exponentially small in √E). Special attention is also devoted to the case of quasi-periodic potentials. These results unify, extend and make quantitative previous results by Fedoryuk, Neishtadt and Delyon and Foulon. Some numerical and analytic tests show that our perturbative construction is, at least in the most relevant point, almost optimal