Rigorous Methods for Bohr-Sommerfeld Quantization Rules

TL;DR

This paper rigorously proves Bohr-Sommerfeld quantization rules for specific quantum systems, providing a unified approach that applies to both the Zakharov-Shabat system and the Schrödinger equation with turning points.

Contribution

It introduces a unified method using comparison equations for $2\times 2$ systems to establish quantization rules in different quantum models.

Findings

Proves Bohr-Sommerfeld rules for Zakharov-Shabat and Schrödinger systems.

Provides uniform results across the potential well range.

Uses comparison equations with Weber model for generality.

Abstract

In this work, we prove Bohr-Sommerfeld quantization rules for the self-adjoint Zakharov-Shabat system and the Schr\"odinger equation in the presence of two simple turning points bounding a classically allowed region. In particular, we use the method of comparison equations for $2\times 2$ traceless first-order systems to provide a unified perspective that yields similar proofs in each setting. The use of a Weber model system gives results that are uniform in the eigenvalue parameter over the whole range from the bottom of the potential well up to finite values.