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Zbl 1187.35143
Hare, Kathryn E.; Zhou, Denglin
Gaps in the ratios of the spectra of Laplacians on fractals. (English)
[J] Fractals 17, No. 4, 523-535 (2009). ISSN 0218-348X

Summary: In contrast to the classical situation, it is known that many Laplacian operators on fractals have gaps in their spectrum. This surprising fact means there can be no limit in the Weyl counting formula and it is a key ingredient in proving that the convergence of Fourier series on fractals can be better than in the classical setting. Recently, it was observed that the Laplacian on the Sierpinski gasket has the stronger property that there are intervals which contain no ratios of eigenvalues. In this paper we give general criteria for this phenomena and show that Laplacians on many interesting classes of fractals satisfy our criteria.

MSC 2000:: *35P05 General spectral theory of PDE
28A80 Fractals
35J05 Laplace equation, etc.

Keywords: eigenvalues; fractal tree; gaps; Laplacian; Sierpiński gasket; spectrum; spectral decimation; Vicsek set

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