On the Eigenvalues of Zakharov--Shabat Systems

Abstract

We consider the eigenvalues of a Zakharov--Shabat system on the real line with a complex-valued L1 potential and show that $\pi/2$ is the threshold L1 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.