Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov-Shabat operator

Abstract

We study the eigenvalues of the self-adjoint Zakharov-Shabat operator corresponding to the defocusing nonlinear Schrödinger 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 conditions for the eigenvalues by using the exact WKB method and show, moreover, that the eigenvalues stay real for a sufficiently small non-self-adjoint perturbation when the potential has some PT-like symmetry.