论文标题
离散一维的ErgodicSchrödinger操作员的差距标签
Gap Labelling for Discrete One-Dimensional Ergodic Schrödinger Operators
论文作者
论文摘要
在这项调查中，我们通过Schwartzman同态介绍了和证明差距标记定理的差距标记定理。为了保持相对独立的论文，我们包括有关状态综合密度的背景，1D操作员的振荡定理以及Schwartzman同态的构建。我们用一些示例说明了结果。特别是，我们展示了如何使用Schwartzman形式主义来恢复经典的差距定理，以获得几乎有周期性的潜力。我们还考虑了由有限维托里（Tori）仿射同构产生的子缩影和操作员产生的运营商。在后一种情况下，可以使用间隙标记定理表明与合适转换产生的电势相关的光谱（例如Arnold的Cat Map）是一个间隔。
In this survey, we give an introduction to and proof of the gap labelling theorem for discrete one-dimensional ergodic Schrödinger operators via the Schwartzman homomorphism. To keep the paper relatively self-contained, we include background on the integrated density of states, the oscillation theorem for 1D operators, and the construction of the Schwartzman homomorphism. We illustrate the result with some examples. In particular, we show how to use the Schwartzman formalism to recover the classical gap-labelling theorem for almost-periodic potentials. We also consider operators generated by subshifts and operators generated by affine homeomorphisms of finite-dimensional tori. In the latter case, one can use the gap-labelling theorem to show that the spectrum associated with potentials generated by suitable transformations (such as Arnold's cat map) is an interval.
