Kostenko Aleksey, Mugnolo Delio, Nicolussi Noema
Faculty of Mathematics and Physics University of Ljubljana Ljubljana Slovenia.
Institute for Analysis and Scientific Computing Vienna University of Technology Vienna Austria.
J Lond Math Soc. 2022 Mar;105(2):1262-1313. doi: 10.1112/jlms.12539. Epub 2022 Feb 21.
We investigate the relationship between one of the classical notions of boundaries for infinite graphs, , and self-adjoint extensions of the minimal Kirchhoff Laplacian on a metric graph. We introduce the notion of for ends of a metric graph and show that finite volume graph ends is the proper notion of a boundary for Markovian extensions of the Kirchhoff Laplacian. In contrast to manifolds and weighted graphs, this provides a transparent geometric characterization of the uniqueness of Markovian extensions, as well as of the self-adjointness of the Gaffney Laplacian - the underlying metric graph does not have finite volume ends. If, however, finitely many finite volume ends occur (as is the case of edge graphs of normal, locally finite tessellations or Cayley graphs of amenable finitely generated groups), we provide a complete description of Markovian extensions upon introducing a suitable notion of traces of functions and normal derivatives on the set of graph ends.
我们研究了无限图边界的经典概念之一—— ，与度量图上最小基尔霍夫拉普拉斯算子的自伴扩张之间的关系。我们引入了度量图端点的 概念，并表明有限体积图端点是基尔霍夫拉普拉斯算子马尔可夫扩张的恰当边界概念。与流形和加权图不同，这为马尔可夫扩张的唯一性以及加夫尼拉普拉斯算子的自伴性提供了一个清晰的几何特征——基础度量图没有有限体积端点。然而，如果出现有限多个有限体积端点（如正规局部有限镶嵌的边图或顺从有限生成群的凯莱图的情况），我们在引入图端点集上函数和法向导数迹的合适概念后，给出了马尔可夫扩张的完整描述。
