Eigenvalues and Eigenfunctions of One-Dimensional Fractal Laplacians

Abstract

We study the eigenvalues and eigenfunctions of one-dimensional weighted fractal Laplacians. These Laplacians are defined by self-similar measures with overlaps. We first prove the existence of eigenvalues and eigenfunctions. We then set up a framework for one-dimensional measures to discretize the equation defining the eigenvalues and eigenfunctions, and obtain numerical approximations of the eigenvalue and eigenfunction by using the finite element method. Finally, we show that the numerical eigenvalues and eigenfunctions converge to the actual ones and obtain the rate of convergence.