图上广义Harper算子特征值的算术性质
Arithmetic properties of eigenvalues of generalized Harper operators on graphs
摘要：
令$\Qbar$表示复代数数域。一个离散群$G$被称为具有$σ$ -乘子代数特征值性质，如果对于每个矩阵$A$，其元素在复代数数$M_d(\Qbar(G,σ))$上的扭群环中，被认为是$l^2(G)^d$上的算子，$A$的特征值是代数数，其中$σ$是代数乘子。这些算子包括哈珀算子和固态物理中出现的离散磁拉普拉斯算子。我们证明了任何生成的顺从的，自由的或表面的群有这个属性的任何代数乘子$σ$。在特殊情况下，当$σ$是合理的（$σ^n$ =1的一些正整数$n$）这一性质适用于一个更大的类的群体，包含自由群体和顺从群体，并关闭下采取有向工会和扩展与顺从。包括在文件中的其他频谱性质的证明，这样的运营商。
英文摘要：
Let $\Qbar$ denote the field of complex algebraic numbers. A discrete group $G$ is said to have the $σ$-multiplier algebraic eigenvalue property, if for every matrix $A$ with entries in the twisted group ring over the complex algebraic numbers $M_d(\Qbar(G,σ))$, regarded as an operator on $l^2(G)^d$, the eigenvalues of $A$ are algebraic numbers, where $σ$ is an algebraic multiplier. Such operators include the Harper operator and the discrete magnetic Laplacian that occur in solid state physics. We prove that any finitely generated amenable, free or surface group has this property for any algebraic multiplier $σ$. In the special case when $σ$ is rational ($σ^n$=1 for some positive integer $n$) this property holds for a larger class of groups, containing free groups and amenable groups, and closed under taking directed unions and extensions with amenable quotients. Included in the paper are proofs of other spectral properties of such operators.
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