We consider the eigenvalues of a Zakharov--Shabat system on the real line with a complex-valued L 1 potential and show that π/2 is the threshold L 1 norm of the potential for the formation of eigenvalues. We obtain best possible lower bounds on the number of eigenvalues for a real potential of one sign. This lower bound is exact for the class of single lobe potentials, that is, positive potentials with a single local maximum that is also global.
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Klaus et al. (2003) studied this question.
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