Absence of Absolutely Continuous Spectrum for the Kirchhoff Laplacian on Radial Trees
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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.
KeywordsContinuous Spectrum Tree Graph Quantum Graph Atomic Measure Radial Tree
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