A toroidal compactification of the two dimensional Bloch-manifold
A toroidal compactification of the two dimensional Bloch-manifold
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二维布洛赫流形的环形紧化
DOI：
10.3929/ethz-a-000579584
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发表时间：
1988
期刊：
影响因子：
--
通讯作者：
D. Bättig
中科院分区：
文献类型：
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作者：
D. Bättig
我们研究二维拉普拉斯差分算子的谱问题，其势为 V，在晶格上呈周期性。与此问题相关的是非紧表面，即所谓的布洛赫流形。在本论文中，我们证明，由于八个一维谱问题，通过环形嵌入的方法，可以通过添加八条曲线来压缩表面，即 1.) 具有一定数量的普通双点的四条有理曲线，仅取决于晶格的大小。这些曲线与电势无关。 2.) 四个超椭圆曲线，每个都由以下谱问题引起：具有势的一维拉普拉斯差分算子，来自对晶格两个方向之一上的势 V 进行平均。证明以两种不同的方式完成。首先通过在环面嵌入上构建无限向量束。其次，用组合方法计算布洛赫流形定义多项式的极点。
We study the spectral problem for the two-dimensional laplace-difference Operator with a potential V, periodic on a lattice. Associated to this problem is a non-compact surface, the so called Bloch-manifold. In this thesis we show, that by methods of toroidal embedding one can compactify the surface by adding eight curves, due to eight one-dimensional spectral problems , namely 1.) four rational curves with a certain number of ordinary double points, depending only on the size of the lattice. These curves are independent of the potential. 2.) four hyperelliptic curves, each due to the following spectral problem : the one-dimensional laplace difference Operator with potential, comeing from averaging the potential V over one of the two directions of the lattice. The proof is done in two different ways . First by construction of an infinite vectorbundle on the torus embedding. Second by Computing the poles of the defining polynomial of the Bloch-manifold with combinatorial methods.