标题： Bohr--Sommerfeld规则用于系统 显示英文标题

标题： Bohr--Sommerfeld rules for systems

摘要： 我们提出了一种完整的、自包含的玻尔-索末菲量子化规则，适用于实线上由相空间中的简单闭合曲线产生的半经典自伴$2 \times 2$系统。我们关注主符号在曲线所围域内表现出本征值交叉的情况——这是狄拉克型算子中常见的情形。在早期关于标量玻尔-索末菲规则以及哈珀算子在有理通量量子附近的半经典处理工作的基础上，我们识别了量子化条件的额外贡献，并推导出一般自伴$2 \times 2$系统的简洁表达式。所得公式给出了显式的几何相位修正，并阐明了这些相位何时取量子化值。 显示英文摘要

摘要： 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 identify additional contributions to the quantization condition, and 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.