  One of the difficulties associated with the scattering problems arising in
connection with integrable systems is that they are frequently
non-self-adjoint, making it difficult to determine where the spectrum lies. In
this paper, we consider the problem of locating and counting the discrete
eigenvalues associated with the scattering problem for which the sine-Gordon
equation is the isospectral flow. In particular, suppose that we take an
initially stationary pulse for the sine-Gordon equation, with a profile that
has either one extremum point of height less than pi and topological charge 0,
or is monotone with topological charge +-1. Then we show that the point
spectrum lies on the unit circle and is simple. Furthermore, we give a count of
the number of eigenvalues. This result is an analog of that of Klaus and Shaw
for the Zakharov-Shabat scattering problem. We also relate our results, as well
as those of Klaus and Shaw, to the Krein stability theory for symplectic
matrices. In particular we show that the scattering problem associated to the
sine-Gordon equation has a symplectic structure, and under the above conditions
the point eigenvalues have a definite Krein signature, and are thus simple and
lie on the unit circle.
