# Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential

## TL;DR

This paper rigorously analyzes the semiclassical scattering data of a non-self-adjoint Dirac operator with analytic potential using the exact WKB method, with implications for the focusing cubic NLS equation.

## Contribution

It provides a complete uniform semiclassical analysis of the reflection coefficient and eigenvalue conditions for the Dirac operator with analytic potential.

## Key findings

- Uniform semiclassical analysis of reflection coefficient
- Derivation of Bohr-Sommerfeld eigenvalue condition
- Implications for inverse scattering in NLS equation

## Abstract

In this paper we examine the semiclassical behaviour of the scattering data of a non-self-adjoint Dirac operator with analytic potential decaying at infinity. In particular, employing the exact WKB method, we provide the complete rigorous uniform semiclassical analysis of the reflection coefficient and the Bohr-Sommerfeld condition for the location of the eigenvalues. Our analysis has some interesting consequences concerning the focusing cubic NLS equation, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator is the basis of the solution of the NLS equation via inverse scattering theory.