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Going back to considerations of Benjamin (1974), there has been
significant interest in the question of stability for the stationary
periodic solutions of the Korteweg-deVries equation, the so-called
cnoidal waves. In this paper, we exploit the squared-eigenfunction
connection between the linear stability problem and the Lax pair for
the Korteweg-deVries equation to completely determine the spectrum
of the linear stability problem for perturbations that are bounded
on the real line. We find that this spectrum is confined to the
imaginary axis, leading to the conclusion of spectral stability. An
additional argument allows us to conclude the completeness of the
associated eigenfunctions.