Semiclassical spectral confinement for the sine-Gordon equation

The inverse scattering method for solving the sine-Gordon equation in laboratory coordinates requires the analysis of the Faddeev–Takhtajan eigenvalue problem. This problem is not self-adjoint and the eigenvalues may lie anywhere in the complex plane, so it is of interest to determine conditions on the initial data that restrict where the eigenvalues can be. We establish bounds on the eigenvalues for a broad class of zero-charge initial data that are applicable in the semiclassical or zero-dispersion limit. It is shown that no point off the coordinate axes or turning point curve can be an eigenvalue if the dispersion parameter is sufficiently small.