  In this paper we examine the semiclassical behavior of the scattering data of
a non-self-adjoint Dirac operator with a rapidly oscillating potential that is
complex analytic in some neighborhood of the real line. Some of our results are
rigorous and quite general. On the other hand, complete and concrete
understanding requires the investigation of the WKB geometry of specific
examples. For such detailed computations we use a particular example that has
been investigated numerically more than 20 years ago by Bronski and Miller and
rely heavily on their numerical computations. Mostly employing the exact WKB
method, we provide the complete rigorous uniform semiclassical analysis of the
Bohr-Sommerfeld condition for the location of the eigenvalues across unions of
analytic arcs as well as the associated norming constants. For the reflection
coefficient as well as the eigenvalues near 0 in the spectral plane, we employ
instead an older theory that has been developed in great detail by Olver. Our
analysis is motivated by the need to understand the semiclassical behaviour of
the focusing cubic NLS equation with initial data $A\exp\{iS/\epsilon\}$, in
view of the well-known fact discovered by Zakharov and Shabat that the spectral
analysis of the Dirac operator enables the solution of the NLS equation via
inverse scattering theory.
