Abstract
The spectrum of the fully symmetric Laplacian on the fractal pentagasket is studied by theoretical and experimental methods. We show how to construct derived eigenspaces of high multiplicity for both Dirichlet and Neumann spectrum starting from primitive Neumann eigenspaces. We prove that both spectra may be parceled into groups of five dimensional spaces which decompose in a prescribed way under the D 5 symmetry group. We show that D 5 invariant eigenfunctions possess additional local symmetries that in particular force them to be constant along the Cantor set bordering the inner deleted pentagon. Numerical approximations for eigenfunctions and eigenvalues obtained using the finite element method are reported. We formulate several conjectures based on this data. More data can be found at http://www.mathlab.cornell.edu/~sas60/
Research supported by the National Science Foundation through the Research Experiences for Undergraduates Program at Cornell, and the VIGER grant to Cornell.
Research supported in part by the National Science Foundation, grant DMS 9970337.
National Science Foundation Postdoctoral Fellow.
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Adams, B., Smith, S.A., Strichartz, R.S., Teplyaev, A. (2003). The Spectrum of the Laplacian on the Pentagasket. In: Grabner, P., Woess, W. (eds) Fractals in Graz 2001. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8014-5_1
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DOI: https://doi.org/10.1007/978-3-0348-8014-5_1
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