On the Essential Spectrum of Schrödinger Operators on Trees

Jonathan Breuer; Sergey Denisov; Latif Eliaz

doi:10.1007/s11040-018-9290-4

On the Essential Spectrum of Schrödinger Operators on Trees

Jonathan Breuer^*, Sergey Denisov, Latif Eliaz

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

It is known that the essential spectrum of a Schrödinger operator H on ℓ²(ℕ) is equal to the union of the spectra of right limits of H. The natural generalization of this relation to ℤⁿ 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.

Original language	American English
Article number	33
Journal	Mathematical Physics Analysis and Geometry
Volume	21
Issue number	4
DOIs	https://doi.org/10.1007/s11040-018-9290-4
State	Published - 1 Dec 2018

Bibliographical note

Funding Information:
Supported in part by the Israel Science Foundation (Grant No. 399/16) and in part by the United States-Israel Binational Science Foundation (Grant No. 2014337)

Funding Information:
Supported in part by the Israel Science Foundation (Grant No. 399/16) and in part by the United States-Israel Binational Science Foundation (Grant No. 2014337) Work on section 4was supported by grant RSF-14-21-00025 and research conducted on other sections was supported by grant NSF-DMS-1464479 and by Van Vleck Professorship Research Award

Publisher Copyright:
© 2018, Springer Nature B.V.

Keywords

Essential spectrum
Graph Laplacian
Right limits
Schrödinger operators

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abstract = "It is known that the essential spectrum of a Schr{\"o}dinger operator H on ℓ2(ℕ) is equal to the union of the spectra of right limits of H. The natural generalization of this relation to ℤn 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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author = "Jonathan Breuer and Sergey Denisov and Latif Eliaz",

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AU - Denisov, Sergey

AU - Eliaz, Latif

N1 - Funding Information: Supported in part by the Israel Science Foundation (Grant No. 399/16) and in part by the United States-Israel Binational Science Foundation (Grant No. 2014337) Funding Information: Supported in part by the Israel Science Foundation (Grant No. 399/16) and in part by the United States-Israel Binational Science Foundation (Grant No. 2014337) Work on section 4was supported by grant RSF-14-21-00025 and research conducted on other sections was supported by grant NSF-DMS-1464479 and by Van Vleck Professorship Research Award Publisher Copyright: © 2018, Springer Nature B.V.

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N2 - It is known that the essential spectrum of a Schrödinger operator H on ℓ2(ℕ) is equal to the union of the spectra of right limits of H. The natural generalization of this relation to ℤn 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.

AB - It is known that the essential spectrum of a Schrödinger operator H on ℓ2(ℕ) is equal to the union of the spectra of right limits of H. The natural generalization of this relation to ℤn 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.

KW - Essential spectrum

KW - Graph Laplacian

KW - Right limits

KW - Schrödinger operators

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JO - Mathematical Physics Analysis and Geometry

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M1 - 33

ER -

On the Essential Spectrum of Schrödinger Operators on Trees

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Keywords

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