Harper算子的代数几何
The algebraic geometry of Harper operators
摘要：
根据Gieseker，Knörrer和Trubowitz提出的离散化Schr dinger算子的方法，我们从代数几何的角度研究了二维和一维Harper算子的谱理论，作为磁Laplacian算子的离散化模型.我们描述了一个相关的家庭的布洛赫品种的几何形状，并计算其态密度。最后，我们还计算了一些基于态密度的谱函数。讨论了参数为有理数和无理数的情形的区别：对于二维Harper算子，Bloch簇的紧化在有理数情形下是普通簇，在无理数情形下是ind-pro-簇.这引起了，在代数几何水平的布洛赫品种，一个现象类似于霍夫施塔特蝴蝶在谱理论。在二维中，态密度可以用费米曲线上的周期积分表示，其中所得的椭圆积分与参数无关。在一维，几乎马蒂厄算子，与一个类似的参数，我们发现通常的依赖性的谱密度的参数，这引起了著名的霍夫施塔特蝴蝶图片。
英文摘要：
Following an approach developed by Gieseker, Knörrer and Trubowitz for discretized Schrödinger operators, we study the spectral theory of Harper operators in dimension two and one, as a discretized model of magnetic Laplacians, from the point of view of algebraic geometry. We describe the geometry of an associated family of Bloch varieties and compute their density of states. Finally, we also compute some spectral functions based on the density of states. We discuss the difference between the cases with rational or irrational parameters: for the two dimensional Harper operator, the compactification of the Bloch variety is an ordinary variety in the rational case and an ind-pro-variety in the irrational case. This gives rise, at the algebro-geometric level of Bloch varieties, to a phenomenon similar to the Hofstadter butterfly in the spectral theory. In dimension two, the density of states can be expressed in terms of period integrals over Fermi curves, where the resulting elliptic integrals are independent of the parameters. In dimension one, for the almost Mathieu operator, with a similar argument we find the usual dependence of the spectral density on the parameter, which gives rise to the well known Hofstadter butterfly picture.
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