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On the Essential Spectrum of Schrödinger Operators on Trees

  • Jonathan BreuerEmail author
  • Sergey Denisov
  • Latif Eliaz
Article

Abstract

It is known that the essential spectrum of a Schrödinger operator H on \(\ell ^{2}\left (\mathbb {N}\right )\) is equal to the union of the spectra of right limits of H. The natural generalization of this relation to \(\mathbb {Z}^{n}\) 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.

Keywords

Right limits Schrödinger operators Graph Laplacian Essential spectrum 

Mathematics Subject Classification (2010)

Primary 34L05 Secondary 35J10 

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA

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