Laplace Operators on Fractals and Related Functional Equations
Spectral Theory
2020-07-27 v1 Mathematical Physics
math.MP
Abstract
We give an overview over the application of functional equations, namely the classical Poincar\'e 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 define the Casimir energy of fractals. We give numerical values for this energy for the Sierpi\'nski gasket.
Keywords
Cite
@article{arxiv.1206.1211,
title = {Laplace Operators on Fractals and Related Functional Equations},
author = {Gregory Derfel and Peter Grabner and Fritz Vogl},
journal= {arXiv preprint arXiv:1206.1211},
year = {2020}
}