Laplace operators on fractals and related functional equations

Gregory Derfel; Peter J. Grabner; Fritz Vogl

doi:10.1088/1751-8113/45/46/463001

Laplace operators on fractals and related functional equations

Gregory Derfel, Peter J. Grabner, Fritz Vogl

Department of Mathematics

Research output: Contribution to journal › Review article › peer-review

33 Scopus citations

Abstract

We give an overview over the application of functional equations, namely the classical Poincaré and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the Euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to compute the Casimir energy of fractals. We give numerical values for this energy for the Sierpiski gasket.

Original language	English
Article number	463001
Journal	Journal of Physics A: Mathematical and Theoretical
Volume	45
Issue number	46
DOIs	https://doi.org/10.1088/1751-8113/45/46/463001
State	Published - 23 Nov 2012

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Statistics and Probability
Modeling and Simulation
Mathematical Physics
General Physics and Astronomy

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Cite this

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AB - We give an overview over the application of functional equations, namely the classical Poincaré and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the Euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to compute the Casimir energy of fractals. We give numerical values for this energy for the Sierpiski gasket.

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M3 - Review article

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Laplace operators on fractals and related functional equations

Abstract

ASJC Scopus subject areas

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