Absence of Absolutely Continuous Spectrum for the Kirchhoff Laplacian on Radial Trees

In this paper, we prove that the existence of absolutely continuous spectrum of the Kirchhoff Laplacian on a radial metric tree graph together with a finite complexity of the geometry of the tree implies that the tree is in fact eventually periodic. This complements the results by Breuer and Frank i...

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Bibliographic Details
Published in:Annales Henri Poincaré 2014-06, Vol.15 (6), p.1109-1121
Main Authors: Exner, Pavel, Seifert, Christian, Stollmann, Peter
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Dynamical Systems and Ergodic Theory
Elementary Particles
Mathematical and Computational Physics
Mathematical Methods in Physics
Physics
Physics and Astronomy
Quantum Field Theory
Quantum Physics
Relativity Theory
Theoretical
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Summary:In this paper, we prove that the existence of absolutely continuous spectrum of the Kirchhoff Laplacian on a radial metric tree graph together with a finite complexity of the geometry of the tree implies that the tree is in fact eventually periodic. This complements the results by Breuer and Frank in (Rev Math Phys 21(7):929–945, 2009 ) in the discrete case as well as for sparse trees in the metric case.
ISSN:1424-0637
1424-0661
DOI:10.1007/s00023-013-0274-4