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Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees

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Abstract

We present examples of rooted tree graphs for which the Laplacian has singular continuous spectral measures. For some of these examples we further establish fractional Hausdorff dimensions. The singular continuous components, in these models, have an interesting multiplicity structure. The results are obtained via a decomposition of the Laplacian into a direct sum of Jacobi matrices.

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Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel

    Jonathan Breuer

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  1. Jonathan Breuer
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Correspondence to Jonathan Breuer.

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Communicated by B. Simon

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Breuer, J. Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees. Commun. Math. Phys. 269, 851–857 (2007). https://doi.org/10.1007/s00220-006-0121-2

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  • DOI: https://doi.org/10.1007/s00220-006-0121-2

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