迁移率边（Mobility Edge）是量子物理中的一个重要概念，最初指的是将绝对连续谱和纯点谱区分开来的临界能量，后来泛指分离不同类型谱的临界能量。由于传统的几乎马蒂厄模型、广义 Harper 模型的谱类型都是纯粹的，不存在迁移率边，因此研究具有丰富谱性质的模型引起了物理学家和数学家的极大兴趣。目前已经发现了三个实验上可行的，且具有精确迁移边的一维拟周期模型。然而，在此之前这方面的理论研究仍停留在数值分析层面。本文严格证明了广义 Aubry-Andre 模型和拟周期镶嵌模型具有分离绝对连续谱和 Anderson 局域的精确迁移率边，而镶嵌 Jacobi 模型具有分离奇异连续谱和 Anderson 局域的精确迁移率边。

The Mobility Edge is a significant concept in quantum physics, originally referring to the critical energy that distinguishes between absolutely continuous spectrum and pure point spectrum. Later, it became a term used to denote the critical energy that separates different types of spectra. Classical examples include the Almost Mathieu model and the generalized Harper model, where the spectra are purely categorized, leading to the absence of mobility edges. This has spurred great interest among physicists and mathematicians in studying models with rich spectral properties. Currently, three experimentally feasible one-dimensional quasiperiodic models with exact mobility edges have been discovered. However, prior to this, theoretical research in this area remained at the numerical analysis level. In this paper, we rigorously prove that the generalized Aubry-Andre model and the
quasiperiodic Mosaic model possess exact mobility edges that separate the absolutely continuous spectrum from Anderson localized states. Furthermore, the Mosaic Jacobi model is also shown to possess exact mobility edges that separate singular continuous spectrum from Anderson localized states.