布洛赫能带理论是研究固体物理的重要基础。传统的布洛赫能带理论通常认 为体系的哈密顿量 ? 具有厄米性，即满足 ? = 𝐻?，其中隐含的前提假设是系统 完全封闭。然而实际的物理体系会不可避免地与外界环境耦合，体系与环境会互 相交换粒子或能量，因此需要在哈密顿量中引入非厄米项来描述实际体系的演化 过程。例如在折射率为复数的光学体系，存在有限寿命准粒子的相互作用电子体 系，还有开放量子体系中都会出现非厄米的有效哈密顿量。人们在对非厄米体系 的研究中发现，仅靠原有的布洛赫能带理论难以对各种新奇的非厄米现象给出合 理的物理解释。例如，非厄米体系的能量本征态可能全部局域在体系的边界附近 而不是弥散到整个空间，这种现象被称为“非厄米趋肤效应”。有非厄米趋肤效应的 体系还伴有能谱对边界条件的高度敏感性，开放边界条件和周期性边界条件下体 系的能谱会表现出显著的差别。此外，拓扑能带理论的中重要的体边对应原理也 无法直接推广到非厄米体系。为了解释上述现象，人们提出了广义布里渊区的概 念，并由此发展了非布洛赫能带理论。应用非布洛赫能带理论，人们成功预测了 开放边界条件下非厄米体系的能谱，并建立了非厄米体系的体边对应原理。本文将从三个具体的研究工作出发，探讨诸如趋肤效应等非厄米特性对动力 学演化过程的影响。首先，我们研究计算了非厄米体系的格林函数。格林函数是 体系的基本物理量之一，它决定了体系对外界的线性响应，很多动力学演化问题 都涉及格林函数的计算。然而，要计算开放边界条件下非厄米体系的格林函数却 并不容易。目前，人们只能通过数值求逆的方法来计算，但数值计算的难度会随 着体系的尺寸而增加，且不能帮助人们去定性地分析给出预测。因此，我们需要 一个普遍适用的非厄米格林函数的解析公式。最终，我们借助非布洛赫能带理论 成功得出了基于广义布里渊区的格林函数公式。该公式不仅形式简洁，且广泛适 用于各种一维的非厄米体系，该公式还为设计实现信号的单向放大提供了新思路。

Bloch band theory is an important theoretical basis of solid state physics. In standard Bloch band theory, the Hamiltonians are always assumed to be Hermitian, i.e. ? = 𝐻?, with the implicit assumption that the system is closed. However, real systems will inevitably be coupled with the environment and exchange particles or energy, which can result in non-Hermitian Hamiltonians in time evolution. Non-Hermitian effective Hamiltonians can appear in complex-refractive-index photonic systems, interacting electronic systems with finite lifetime quasiparticles and open quantum systems. In recent years, it has been found that the Bloch band theory is not sufficient to explain the novel phe- nomena observed in non-Hermitian systems. For example, all the non-Hermitian eigen- states may be localized near the boundary, rather than extending to the bulk, which is known as the ”non-Hermitian skin effect”. Systems with non-Hermitian skin effect are also accompanied by a high sensitivity of the energy spectrum to boundary conditions, and the spectrum will be significantly different under open/periodic boundary conditions. In addition, the principle of bulk-boundary correspondence, which plays a crucial role in Hermitian topological phases, cannot be directly applied to non-Hermitian systems. The non-Bloch band theory based on the concept of generalized Brillouin zone(GBZ) has been proposed to describe these novel phenomena. Using non-Bloch energy band theory, people have successfully predicted the energy spectrum of non-Hermitian systems under open boundary conditions, and established the bulk-boundary correspondence for non-Hermitian systems.We will introduce three specific works in this paper, to discuss the influence of non- Hermitian properties such as the skin effect on the dynamic evolution process. The first is work is to calculate the Green’s function of the non-Hermitian system. The Green’s function is one of the basic quantities of the system which determines the linear response to the perturbations. The calculation of the Green’s function is involved in the study of many dynamic problems. However, it is not easy to calculate Green’s functions for non- Hermitian systems with open boundary conditions. At present, people can only calculate the Green’s function by numerical methods, but the difficulty of numerical calculation will increase with the size of the system, and it cannot help people to analyze and give predictions. Therefore, we need a universally applicable analytical formula. Finally, we successfully obtained the Green’s function formula with the help of non-Bloch energy band theory and the concept of GBZ. The formula is not only simple, but also widely applicable to various one-dimensional non-Hermitian systems. The formula also provides an efficient guide for designing directional amplifiers.The second work is to simulate the evolution of particles on a one-dimensional bi- partite lossy lattice. In certain parameter intervals, an unexpected high peak of dissipation probability is observed at the edge, which is dudded as ”edge burst”. We found that this phenomenon actually comes from the interplay of ”non-Hermitian skin effect” and ”imaginary gap closing”. Based on this conclusion, we have successfully predicted ”edge burst” in other models. In addition, we also cooperated with the experimental group and successfully observed this phenomenon in the quantum optical platform experiments, which gives a strong verification of our theory.The third work is to explore the evolution of the wave packet starting from the edge of a non-Hermitian system. We found that the Lyapunov exponent changes at differ- ent time scales. We calculated and numerically verified the Lyapunov exponent under different conditions, revealing the influence of non-Hermitian skin effect and boundary conditions on wave packet evolution. In addition, recent studies have shown that some non-Hermitian skin modes could be ”self-healing”. Based on the study of the Lyapunov exponent, we further revised the conditions that the “self-healing” skin state needs to meet.