The magnetic Laplacian (also called the line bundle Laplacian) on a connected weighted graph is a self-adjoint operator wherein the real-valued adjacency weights are replaced by unit complex-valued weights $$\{\omega _{xy}\}_{xy\in E}$$ , satisfying the condition that $$\omega _{xy}=\overline{\omega _{yx}}$$ for every directed edge xy. When properly interpreted, these complex weights give rise to magnetic fluxes through cycles in the graph. In this paper we establish the spectrum of the magnetic Laplacian, as a set of real numbers with multiplicities, on the Sierpinski gasket graph (SG) where the magnetic fluxes equal $$\alpha $$ through the upright triangles, and $$\beta $$ through the downright triangles. This is achieved upon showing the spectral self-similarity of the magnetic Laplacian via a 3-parameter map $${\mathcal {U}}$$ involving non-rational functions, which takes into account $$\alpha $$ , $$\beta $$ , and the spectral parameter $$\lambda $$ . In doing so we provide a quantitative answer to a question of Bellissard [Renormalization Group Analysis and Quasicrystals (1992)] on the relationship between the dynamical spectrum and the actual magnetic spectrum. Our main theorems lead to two applications. In the case $$\alpha =\beta $$ , we demonstrate the approximation of the magnetic spectrum by the filled Julia set of $${\mathcal {U}}$$ , the Sierpinski gasket counterpart to Hofstadter’s butterfly. Meanwhile, in the case $$\alpha ,\beta \in \{0,\frac{1}{2}\}$$ , we can compute the determinant of the magnetic Laplacian and the corresponding asymptotic complexity.

中文翻译：

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谢尔宾斯基垫片上磁性拉普拉斯算子的频谱抽取：解决霍夫施塔特-谢尔宾斯基蝴蝶

连通加权图上的磁拉普拉斯算子（也称为线丛拉普拉斯算子）是一个自伴随算子，其中实值邻接权重被单位复值权重代替 $$\{\omega _{xy}\}_ {xy\in E}$$ ，对于每条有向边 xy 满足 $$\omega _{xy}=\overline{\omega _{yx}}$$ 的条件。如果正确解释，这些复杂的权重会通过图中的循环产生磁通量。在本文中，我们在谢尔宾斯基垫片图 (SG) 上建立了磁性拉普拉斯算子的频谱，作为一组具有多重性的实数，其中通过直立三角形的磁通量等于 $$\alpha $$，而 $$\beta $$ 通过彻头彻尾的三角形。这是通过涉及非有理函数的 3 参数映射 $${\mathcal {U}}$$ 显示磁性拉普拉斯算子的光谱自相似性来实现的，该映射考虑了 $$\alpha $$ , $$ \beta $$ ，以及光谱参数 $$\lambda $$ 。在这样做时，我们对 Bellissard [重整化群分析和准晶体 (1992)] 关于动力学谱和实际磁谱之间关系的问题提供了定量答案。我们的主要定理导致两个应用。在 $$\alpha =\beta $$ 的情况下，我们通过填充的 Julia 集 $${\mathcal {U}}$$ 证明了磁谱的近似值，这是霍夫施塔特蝴蝶的谢尔宾斯基垫圈对应物。同时，在 $$\alpha ,\beta \in \{0,\frac{1}{2}\}$$ 的情况下，