  The spectrum of the focusing Zakharov-Shabat operator on the circle is
studied, and its explicit dependence on the presence of a semiclassical
parameter is also considered. Several new results are obtained. In particular:
(i) it is proved that the resolvent set is comprised of two connected
components, (ii) new bounds on the location of the Floquet and Dirichlet
spectra are obtained, some of which depend explicitly on the value of the
semiclassical parameter, (iii) it is proved that the spectrum localizes to a
"cross" in the spectral plane in the semiclassical limit. The results are
illustrated by discussing several examples in which the spectrum is computed
analytically or numerically.
