The Essential Spectrum of the Discrete Laplacian on Klaus-sparse Graphs
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The Essential Spectrum of the Discrete Laplacian on Klaus-sparse Graphs ´ 1 Sylvain Golenia
· Franc¸oise Truc2
Received: 10 April 2020 / Accepted: 28 October 2020 / © Springer Nature B.V. 2020
Abstract In 1983, Klaus studied a class of potentials with bumps and computed the essential spectrum of the associated Schr¨odinger 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. Keywords Essential spectrum · Locally finite graphs · Self-adjointness · Spectral graph theory
1 Introduction The computation of the essential spectrum of an operator is a standard question in spectral theory. For a large family of Schr¨odinger operators, it is well-known that the essential spectrum is characterised by the behaviour at infinity of the potential. In
Sylvain Gol´enia
[email protected] Franc¸oise Truc [email protected] 1
University of Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400, Talence, France
2
Institut Fourier, UMR 5582 du CNRS Universit´e de Grenoble I, BP 74, 38402, Saint-Martin d’H`eres, France
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Math Phys Anal Geom
(2020) 23:45
1983, Klaus introduces in his article [18] a type of potential with bumps with a crucial feature, that is, the distance between two such bumps tends to infinity. He computes the essential spectrum of this 1d (continuous) Schr¨odinger operator in terms of the union of the spectrum of some simpler operators. The common patterns, that define the localisations at infinity, are given by the behaviour of the potential. In [21], this notion is illustrated by the use of right-limits, see also [6] for this concept and references therein. The example of Klaus was generalised to higher dimensions and encoded in some C ∗ -algebraic context in [13, 14], see also [22, 23]. We refer to [10] for more general results and historical references. We mention also [24, 25] for recent developments in a sparse context. In the context of graphs, the computation of essential spectra is done in many places e.g., [19, 26, 28]. In [5, 9] they extend the notion of right-limit and introduce R-limits in the context of discrete graphs. We refer to [11, 15] for a C ∗ -algebra approach. Our motivation in this paper is to analyse a graph analog of the example of Klaus where the “bumps” are no more due to a potential but to patterns coming from the structure of the graph. We call this family of graphs Klaus-sparse graphs. The approaches of
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