Abstract
On an infinite, radial metric tree graph we consider the corresponding Laplacian equipped with self-adjoint vertex conditions from a large class including \(\delta \)- and weighted \(\delta '\)-couplings. Assuming the numbers of different edge lengths, branching numbers and different coupling conditions to be finite, we prove that the presence of absolutely continuous spectrum implies that the sequence of geometric data of the tree as well as the coupling conditions are eventually periodic. On the other hand, we provide examples of self-adjoint, non-periodic couplings which admit absolutely continuous spectrum.
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References
Berkolaiko, G.: An elementary introduction to quantum graphs. arXiv-Preprint arXiv:1603.07356
Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs. Am. Math. Soc., Providence (2013)
Bessaga, C., Pelczynski, A.: Selected Topics in Infinite-Dimensional Topology, Mathematical Monographs, vol. 58. Polish Scientific, Warsaw (1975)
Breuer, J., Frank, R.: Singular spectrum for radial trees. Rev. Math. Phys. 21(7), 929–945 (2009)
Carlson, R.: Nonclassical Sturm–Liouville problems and Schrödinger operators on radial trees. Electron. J. Differ. Equ. 2000(71), 1–24 (2000)
Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Birkhäuser, Boston (1990)
Evans, W.D., Solomyak, M.: Smilansky’s model of irreversible quantum graphs I: the absolutely continuous spectrum. J. Phys. A. 38(21), 4611–4627 (2005)
Exner, P., Lipovský, J.: On the absence of absolutely continuous spectra for Schrödinger operators on radial tree graphs. J. Math. Phys. 51, 122107 (2010)
Exner, P., Seifert, C., Stollmann, P.: Absence of absolutely continuous spectrum for the Kirchhoff Laplacian on radial trees. Ann. Henri Poincaré 15(6), 1109–1121 (2014)
Klassert, S., Lenz, D., Stollmann, P.: Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals. Discrete Contin. Dyn. Syst. 29(4), 1553–1571 (2011)
Kuchment, P.: Quantum graphs: I. Some basic structures. Waves Random Media 14, 107–128 (2007)
Lenz, D., Seifert, C., Stollmann, P.: Zero measure Cantor spectra for continuum one-dimensional quasicrystals. J. Differ. Equ. 256(6), 1905–1926 (2014)
Mugnolo, D.: Semigroup Methods for Evolution Equations on Networks. Understanding Complex Systems. Springer, Berlin (2014)
Naimark, K., Solomyak, M.: Eigenvalue estimates for the weighted Laplacian on metric trees. Proc. Lond. Math. Soc. 80(3), 690–724 (2000)
Pankrashkin, K.: Quasiperiodic surface Maryland models on quantum graphs. J. Phys. A 42(26), 26530413 (2009)
Remling, C.: The absolutely continuous spectrum of one-dimensional Schrödinger operators. Math. Phys. Anal. Geom. 10, 359–373 (2007)
Remling, C.: The absolutely continuous spectrum of Jacobi matrices. Ann. Math. 174, 125–171 (2011)
Sobolev, A.V., Solomyak, M.: Schrödinger operators on homogeneous metric trees: spectrum in gaps. Rev. Math. Phys. 14, 421–468 (2002)
Solomyak, M.: On the spectrum of the Laplacian on regular metric trees. Waves Random Media 14, 155–171 (2004)
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Rohleder, J., Seifert, C. Absolutely Continuous Spectrum for Laplacians on Radial Metric Trees and Periodicity. Integr. Equ. Oper. Theory 89, 439–453 (2017). https://doi.org/10.1007/s00020-017-2388-4
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DOI: https://doi.org/10.1007/s00020-017-2388-4