The zeta function of the Laplacian on certain fractals
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- by Gregory Derfel, Peter J. Grabner and Fritz Vogl PDF
- Trans. Amer. Math. Soc. 360 (2008), 881-897 Request permission
Abstract:
We prove that the zeta function $\zeta _\Delta$ of the Laplacian $\Delta$ on self-similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane. We characterise the poles, compute their residues, and give expressions for some special values of the zeta function. Furthermore, we discuss the presence of oscillations in the eigenvalue counting function, thereby answering a question posed by J. Kigami and M. Lapidus for this class of fractals.References
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Additional Information
- Gregory Derfel
- Affiliation: Department of Mathematics and Computer Science, Ben Gurion University of the Negev, Beer Sheva 84105, Israel
- MR Author ID: 242762
- Email: derfel@math.bgu.ac.il
- Peter J. Grabner
- Affiliation: Institut für Analysis und Computational Number Theory, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria
- Email: peter.grabner@tugraz.at
- Fritz Vogl
- Affiliation: Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria
- Email: fvogl@osiris.tuwien.ac.at
- Received by editor(s): June 23, 2005
- Received by editor(s) in revised form: November 4, 2005
- Published electronically: September 24, 2007
- Additional Notes: The second author was supported by the Austrian Science Fund project S9605
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 881-897
- MSC (2000): Primary 30B50; Secondary 11M41, 37F10
- DOI: https://doi.org/10.1090/S0002-9947-07-04240-7
- MathSciNet review: 2346475