We provide a systematic and self-consistent method to calculate the generalized Brillouin zone (GBZ) analytically in one-dimensional non-Hermitian systems, which helps us to understand the non-Hermitian bulk-boundary correspondence. In general, a $n$-band non-Hermitian Hamiltonian is constituted by $n$ distinct sub-GBZs, each of which is a piecewise analytic closed loop. Based on the concept of resultant, we can show that all the analytic properties of the GBZ can be characterized by an algebraic equation, the solution of which in the complex plane is dubbed as auxiliary GBZ (aGBZ). We also provide a systematic method to obtain the GBZ from aGBZ. Two physical applications are also discussed. Our method provides an analytic approach to the spectral problem of open boundary non-Hermitian systems in the thermodynamic limit.

中文翻译：

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非埃尔米特体边界函和辅助广义布里渊区理论

我们提供了一种系统的，自洽的方法来分析一维非Hermitian系统中的广义布里渊区（GBZ），这有助于我们理解非Hermitian体边界。一般来说，$ñ$带非赫米特哈密顿量由 $ñ$不同的子GBZ，每个子GBZ都是分段分析的闭环。基于结果的概念，我们可以证明GBZ的所有解析性质都可以用一个代数方程来表征，该方程在复平面中的解称为辅助GBZ（aGBZ）。我们还提供了一种从aGBZ获取GBZ的系统方法。还讨论了两个物理应用程序。我们的方法为热力学极限中的开放边界非Hermitian系统的光谱问题提供了一种解析方法。