# Bohr--Sommerfeld rules for systems

## TL;DR

This paper develops a comprehensive Bohr--Sommerfeld quantization rule for semiclassical 2x2 systems with eigenvalue crossings, providing explicit formulas with geometric phase corrections relevant to Dirac operators.

## Contribution

It introduces a complete formulation of Bohr--Sommerfeld rules for 2x2 systems with crossings, extending previous scalar and scalar-like semiclassical quantization methods.

## Key findings

- Derived explicit formulas for quantization conditions.
- Identified geometric phase corrections and their quantization.
- Clarified conditions for phase quantization in 2x2 systems.

## Abstract

We present a complete, self-contained formulation of the Bohr--Sommerfeld quantization rule for a semiclassical self-adjoint $2 \times 2$ system on the real line, arising from a simple closed curve in phase space. We focus on the case where the principal symbol exhibits eigenvalue crossings within the domain enclosed by the curve -- a situation commonly encountered in Dirac-type operators. Building on earlier work on scalar Bohr--Sommerfeld rules and semiclassical treatments of the Harper operator near rational flux quanta, we derive concise expressions for general self-adjoint $2 \times 2$ systems. The resulting formulas give explicit geometric phase corrections and clarify when these phases take quantized values.