局部有限图上离散 Schrödinger 算子的亏指数问题
泛函分析
2015-05-18 v4 数学物理
math.MP
谱理论
摘要
复 Hilbert 空间上对称算子的自伴扩张数量由其亏指数刻画。对于局部有限无向简单树，我们证明任何离散 Schrödinger 算子的亏指数要么为零要么为无穷。我们还证明，几乎必然存在一棵树，使得所有离散 Schrödinger 算子本质自伴。此外，我们提供了若干本质自伴性判据。我们也对邻接矩阵的情形给予一定关注，并猜想对于局部有限无向简单图，其亏指数要么为零要么为无穷。除此之外，我们考虑了树和加权图的一些推广。
引用
@article{arxiv.1005.0165,
title = {The problem of deficiency indices for discrete Schr\"odinger operators on locally finite graphs},
author = {Sylvain Golénia and Christoph Schumacher},
journal= {arXiv preprint arXiv:1005.0165},
year = {2015}
}
备注
Typos corrected. References and ToC added. Paper slightly reorganized. Section 3.2, about the diagonalization has been much improved. The older section about the stability of the deficiency indices in now in appendix. To appear in Journal of Mathematical Physics