On the Essential Spectrum of Schrödinger Operators on Trees

被引:0
作者
Jonathan Breuer
Sergey Denisov
Latif Eliaz
机构
[1] The Hebrew University of Jerusalem,Institute of Mathematics
[2] University of Wisconsin-Madison,Department of Mathematics
来源
Mathematical Physics, Analysis and Geometry | 2018年 / 21卷
关键词
Right limits; Schrödinger operators; Graph Laplacian; Essential spectrum; Primary 34L05; Secondary 35J10;
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摘要
It is known that the essential spectrum of a Schrödinger operator H on ℓ2ℕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell ^{2}\left (\mathbb {N}\right )$\end{document} is equal to the union of the spectra of right limits of H. The natural generalization of this relation to ℤn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {Z}^{n}$\end{document} is known to hold as well. In this paper we generalize the notion of right limits to general infinite connected graphs and construct examples of graphs for which the essential spectrum of the Laplacian is strictly bigger than the union of the spectra of its right limits. As these right limits are trees, this result is complemented by the fact that the equality still holds for general bounded operators on regular trees. We prove this and characterize the essential spectrum in the spherically symmetric case.
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