Laplace Operators on Fractals and Related Functional Equations

TL;DR

This paper reviews how functional equations like Poincaré and renewal equations are used to analyze the spectrum of Laplace operators on fractals, and explores the concept of Casimir energy in this context.

Contribution

It provides a comparative overview of spectral analysis techniques on fractals versus Euclidean spaces and introduces the calculation of Casimir energy for fractals.

Findings

Spectral zeta functions are used to define Casimir energy on fractals.

Numerical values for Casimir energy are provided for the Sierpiński gasket.

The paper compares analytical techniques between fractal and Euclidean geometries.

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.