We consider the Faddeev-Takhtajan eigenvalue problem, for which the
Sine-Gordon equation is the isospectral flow. Based on some intuition
provided by some results of Klaus and Shaw on the Zakharov-Shabat
eigenvalue problem, as well as some special exactly solvable
potentials constructed by Miller and Buckingham, we are able to prove
under certain monotonicity assumptions the point spectrum of the
Faddeev-Takhtajan eigenvalue problem is simple and lies on the unit circle.
This result (as well as that of Klaus-Shaw) can be considered a
generalization of the Krein stability theory for symplectic matrices.
We will develop these connections throughout the talk.
This is joint work with Mat Johnson.
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