通过几何结构生成光谱间隙
Generating spectral gaps by geometry
摘要：
受周期势薛定谔算子分析的启发，我们考虑以下抽象情形：设$Δ_X$是非紧黎曼覆盖流形$X$上的拉普拉斯算子，其中离散等距群$Γ$作用于其上，使得商$X/Γ$是紧流形.本文证明了算子$Δ_X$与一类合适的具有非交换覆盖变换群$Γ$的流形$X$相关联时，存在有限个谱间隙.这个结果是基于非阿贝尔Floquet理论以及最小最大原理。类型I的群指定一类满足主定理假设的例子。
英文摘要：
Motivated by the analysis of Schrödinger operators with periodic potentials we consider the following abstract situation: Let $Δ_X$ be the Laplacian on a non-compact Riemannian covering manifold $X$ with a discrete isometric group $Γ$ acting on it such that the quotient $X/Γ$ is a compact manifold. We prove the existence of a finite number of spectral gaps for the operator $Δ_X$ associated with a suitable class of manifolds $X$ with non-abelian covering transformation groups $Γ$. This result is based on the non-abelian Floquet theory as well as the Min-Max-principle. Groups of type I specify a class of examples satisfying the assumptions of the main theorem.
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