局部有限图上离散薛定谔算子的亏指数问题
The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs
摘要：
复希尔伯特空间上的对称算子的自伴扩张的个数可用其亏指数来刻画。给定一棵局部有限无向简单树，证明了任意离散薛定谔算子的亏指数要么为零，要么为无穷大.我们还证明了几乎肯定存在一棵树使得所有离散薛定谔算子本质上是自伴的。此外，我们还给出了本质自伴性的几个判据。我们还讨论了邻接矩阵的一些重要性质，并猜想，给定一个局部有限的无向单图，它的亏指数要么为零，要么为无穷大。除此之外，我们还考虑了树和赋权图的一些推广。
英文摘要：
The number of self-adjoint extensions of a symmetric operator acting on a complex Hilbert space is characterized by its deficiency indices. Given a locally finite unoriented simple tree, we prove that the deficiency indices of any discrete Schrödinger operator are either null or infinite. We also prove that almost surely, there is a tree such that all discrete Schrödinger operators are essentially self-adjoint. Furthermore, we provide several criteria of essential self-adjointness. We also adress some importance to the case of the adjacency matrix and conjecture that, given a locally finite unoriented simple graph, its the deficiency indices are either null or infinite. Besides that, we consider some generalizations of trees and weighted graphs.
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