Abstract
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.
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Communicated by Jean Bellissard.
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Exner, P., Seifert, C. & Stollmann, P. Absence of Absolutely Continuous Spectrum for the Kirchhoff Laplacian on Radial Trees. Ann. Henri Poincaré 15, 1109–1121 (2014). https://doi.org/10.1007/s00023-013-0274-4
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DOI: https://doi.org/10.1007/s00023-013-0274-4