A Study on Measure-Geometric Laplacians on the Real Line

Abstract

In this thesis, we consider measure-geometric differential operators on the real line as they were introduced by Freiberg and Zähle in 2002. We define derivatives $ nabla mu $ and Laplace operators $ Delta mu $ with respect to different types of compactly supported finite measures $ mu$. We first discuss the class of continuous measures $ nu$. We deduce a harmonic calculus for $ nabla nu $ and $ Delta nu $ from the classical (weak) analysis. It is shown that the eigenvalues of $ Delta nu $ do not depend on the chosen measure and the eigenfunctions are given explicitly. The results are illustrated through examples of (fractal) measures. In the following, the framework of Freiberg and Zähle is extended by dropping the condition that the measures are atomless. We consider distributions $ delta$ which have finite support and define analogous operators $ nabla delta $ and $ Delta delta $. Using that they act on finite-dimensional function spaces, we give matrix representations for both operators and obtain analytic properties for them. General observations on the spectral properties of $ Delta delta $ are discussed and compared to the atomless case. For uniform discrete probability distributions, we determine the eigenvalues and eigenfunctions of the associated Laplacian. We then study measures $ eta$ which have a continuous and an atomic part. Again we define the operators $ nabla eta $ and $ Delta eta $ and obtain properties similar to those for weak first and second order derivatives. We give a systematic way to calculate the eigenvalues and eigenfunctions of $ Delta eta $ and determine them for two leading examples. Differences and similarities to the previous cases are discussed and problems for the general solution of the eigenvalue problem are indicated.