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Spectral analysis of Laplacians on the Vicsek set. (English) Zbl 1177.28029
The author obtains a formula, expressed in terms of Chebyshev polynomials, of the spectral decimation function for the standard Laplacian on the $n$-branch Vicsek set, and determines all the forbidden eigenvalues for the Laplacian, where the spectral decimation function is in the sense of {\it T. Shima} [Japan J. Ind. Appl. Math. 13, No. 1, 1--23 (1996; Zbl 0861.58047)]. The author then shows that there exists a gap in the spectrum of the standard Laplacian by verifying the conditions for the criterion for gaps which was obtained earlier by the author. Moreover, he determines the order of the eigenvalues for the Laplacian and describes their asymptotic behavior.

MSC:
 28A80 Fractals 42C99 Non-trigonometric Fourier analysis 31C25 Dirichlet spaces
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