Essential spectrum of the discrete Laplacian on a perturbed periodic graph

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• 作者：Itaru Sasaki^a ; ^{isasaki@shinshu-u.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Akito Suzuki^b ; ^{akito@shinshu-u.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Infinite graph ; Essential spectrum ; Perturbation theory ; Discrete Laplacian ; Pendant ; Random graph
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：446
• 期：2
• 页码：1863-1881
• 全文大小：459 K

文摘

We address the Laplacian on a perturbed periodic graph which might not be a periodic graph. We give a criterion for the essential spectrum of the Laplacian on the perturbed graph to include that on the unperturbed graph. This criterion is applicable to a wide class of graphs obtained by a non-compact perturbation such as adding or removing infinitely many vertices and edges. Using this criterion, we demonstrate how to determine the spectra of cone-like graphs, the upper-half plane, and graphs obtained from Z²${\mathbb{Z}}^2$ by randomly adding vertices.