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

## TL;DR

This paper investigates the conditions under which eigenvalues of a perturbed Zakharov-Shabat operator remain real, using the WKB method and symmetry considerations, relevant for the inverse scattering method in nonlinear Schrödinger equations.

## Contribution

It introduces two quantization conditions for eigenvalues and demonstrates their reality under small non-self-adjoint perturbations with PT-like symmetry.

## Key findings

- Real eigenvalues occur when the potential's square has a simple well.
- Two types of quantization conditions are derived using the exact WKB method.
- Eigenvalues remain real under small PT-symmetric perturbations.

## Abstract

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.