# Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle

## TL;DR

This paper derives Bohr-Sommerfeld type quantization conditions for semiclassical eigenvalues of the non-selfadjoint Zakharov-Shabat operator on the circle using an exact WKB method, considering the presence or absence of real turning points.

## Contribution

It introduces a novel approach to quantization conditions for non-selfadjoint operators on the circle using exact WKB analysis.

## Key findings

- Quantization conditions depend on the action integral around the circle.
- Conditions vary with the presence or absence of real turning points.
- Provides a framework for analyzing semiclassical eigenvalues in non-selfadjoint systems.

## Abstract

Bohr-Sommerfeld type quantization conditions of semiclassical eigenvalues for the non-selfadjoint Zakharov-Shabat operator on the circle are derived using an exact WKB method. The conditions are given in terms of the action associated with the unit circle or the action associated with turning points following the absence or presence of real turning points.