We look at periodic Jacobi matrices on trees. We provide upper and lower bounds on the gap of such operators analogous to the well-known gap in the spectrum of the Laplacian on the upper half-plane with a hyperbolic metric. We make some conjectures about antibound states and make an interesting observation for the so-called rg-model where the underlying graph has r red and g green vertices and where any two vertices of different colors are connected by a single edge.

中文翻译：

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关于树上的周期性Jacobi矩阵的说明

我们看一下树上的周期性Jacobi矩阵。我们提供了此类算子的间隙的上限和下限，类似于使用双曲度量的上半平面上拉普拉斯谱中的众所周知的间隙。我们对反绑定状态做出一些猜想，并对所谓的rg -model进行有趣的观察，其中基础图形具有r红色和g绿色顶点，并且任何颜色的任意两个顶点通过一条边连接。