Alain JOYE
Exponential Asymptotics in a Singular Limit for 
$n$-Level Scattering Systems
(99K, Latex)

ABSTRACT.  The singular limit $\eps\ra 0$ of the $S$-matrix associated with
the equation $i\eps d\psi(t)/dt=H(t)\psi(t)$ is considered,
where the analytic generator $H(t)\in M_n(\C)$ is such that its
spectrum is real and non-degenerate for all $t\in\R$. Sufficient
conditions allowing to compute asymptotic formulas for the
exponentially small off-diagonal elements of the $S$-matrix as
$\eps\ra 0$ are explicited 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 to 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 WKB method together with
superasymptotic renormalization. The application of these
results to the study of quantum adiabatic transitions in the
time dependent Schr\"odinger equation and of the semiclassical
scattering properties of the multichannel stationary
Schr\"odinger equation closes this paper. The results presented
here are a generalization to $n$-level systems, $n\geq 2$, of
results previously known for $2$-level systems only.