Laplace Operators on Fractals and Related Functional Equations

We give an overview over the application of functional equations, namely the classical Poincare 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 Sierpinski gasket.