Exponential Asymptotics in a Singular Limit for n-Level Scattering Systems

International audience ; The singular limit $\epsilon\rightarrow 0$ of the S-matrix associated with the equation $i\epsilon d\psi(t)/dt=H(t)\psi(t)$ is considered, where the analytic generator $H(t)\in M_n(Ç)$ is such that its spectrum is real and nondegenerate for all $t\in{\bf R}$. Sufficient conditions allowing us to compute asymptotic formulas for the exponentially small off-diagonal elements of the S-matrix as $\epsilon\rightarrow 0$ are made explicit and a wide class of generators for which these conditions are verified is defined. These generators are obtained by means of generators whose spectrum exhibits eigenvalue crossings which are perturbed in such a way that these crossings turn into avoided crossings. The exponentially small asymptotic formulas which are derived are shown to be valid up to exponentially small relative error by means of a joint application of the complex Wentzel--Kramers--Brillouin (WKB) method together with superasymptotic renormalization. This paper concludes with the application of these results to the study of quantum adiabatic transitions in the time-dependent Schrödinger equation and of the semiclassical scattering properties of the multichannel stationary Schrödinger equation. The results presented here are a generalization to n-level systems, $n\geq 2$, of results previously known for two-level systems only.