## Abstract

It is known that the essential spectrum of a Schrödinger operator H on ℓ^{2}(ℕ) is equal to the union of the spectra of right limits of H. The natural generalization of this relation to ℤ^{n} 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 |
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Article number | 33 |

Journal | Mathematical Physics Analysis and Geometry |

Volume | 21 |

Issue number | 4 |

DOIs | |

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