We investigate spectral properties of Kirchhoff Laplacians on radially symmetric antitrees. This class of metric graphs admits a lot of symmetries, which enables us to obtain a decomposition of the corresponding Laplacian into the orthogonal sum of Sturm–Liouville operators. In contrast to the case of radially symmetric trees, the deficiency indices of the Laplacian defined on the minimal domain are at most one and they are equal to one exactly when the corresponding metric antitree has finite total volume. In this case, we provide an explicit description of all self-adjoint extensions including the Friedrichs extension.

Furthermore, using the spectral theory of Krein strings, we perform a thorough spectral analysis of this model. In particular, we obtain discreteness and trace class criteria, a criterion for theKirchhoff Laplacian to be uniformly positive and provide spectral gap estimates. We show that the absolutely continuous spectrumis in a certain sense a rare event, however, we also present several classes of antitrees such that the absolutely continuous spectrum of the corresponding Laplacian is $[O, \infty)$.

中文翻译：

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径向对称反树上的量子图

我们研究了基尔霍夫拉普拉斯算子在径向对称反树上的光谱特性。这类度量图承认很多对称性，这使我们能够将相应的拉普拉斯算子分解为 Sturm-Liouville 算子的正交和。与径向对称树的情况相反，在最小域上定义的拉普拉斯算子的缺陷指数至多为 1，并且当相应的度量反树具有有限的总体积时，它们恰好等于 1。在这种情况下，我们提供了对包括 Friedrichs 扩展在内的所有自伴随扩展的明确描述。

此外，使用 Kerin 弦的谱理论，我们对该模型进行了彻底的谱分析。特别是，我们获得了离散性和迹类标准，这是基尔霍夫拉普拉斯算子一致为正并提供谱间隙估计的标准。我们证明绝对连续谱在某种意义上是一个罕见的事件，然而，我们还提出了几类反树，使得相应拉普拉斯算子的绝对连续谱是 $[O, \infty)$。