  We study the eigenvalues of the self-adjoint Zakharov-Shabat operator
corresponding to the defocusing nonlinear Schrodinger equation in the inverse
scattering method. Real eigenvalues exist when the square of the potential has
a simple well. We derive two types of quantization condition for the
eigenvalues by using the exact WKB method, and show that the eigenvalues stay
real for a sufficiently small non-self-adjoint perturbation when the potential
has some PT-like symmetry.
