In 1983, Klaus studied a class of potentials with bumps and computed the essential spectrum of the associated Schr{o}dinger operator with the help of some localisations at infinity. A key hypothesis is that the distance between two consecutive bumps tends to infinity at infinity. In this article, we introduce a new class of graphs (with patterns) that mimics this situation, in the sense that the distance between two patterns tends to infinity at infinity. These patterns tend, in some way, to asymptotic graphs. They are the localisations at infinity. Our result is that the essential spectrum of the Laplacian acting on our graph is given by the union of the spectra of the Laplacian acting on the asymptotic graphs. We also discuss the question of the stability of the essential spectrum in the appendix.

中文翻译：

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克劳斯稀疏图上离散拉普拉斯算子的本质谱

1983 年，Klaus 研究了一类有颠簸的势能，并在无穷远处的一些定域的帮助下计算了相关的 Schr{o}dinger 算子的基本谱。一个关键假设是两个连续颠簸之间的距离在无穷远处趋于无穷大。在本文中，我们介绍了一类新的图（带有模式）来模拟这种情况，即两个模式之间的距离在无穷远处趋于无穷大。这些模式在某种程度上倾向于渐近图。它们是无穷远的本地化。我们的结果是，作用在我们图上的拉普拉斯算子的本质谱是由作用在渐近图上的拉普拉斯算子的谱的并集给出的。我们还在附录中讨论了基本光谱的稳定性问题。