Analysis of the Laplacian and Spectral Operators on the Vicsek Set

TL;DR

This paper investigates the spectral properties of the Laplacian on Vicsek fractals, providing explicit eigenfunctions, solutions to PDEs, and analyzing spectral convergence as the fractal parameter grows.

Contribution

It extends spectral analysis techniques to Vicsek sets, including explicit eigenfunction computation, PDE solutions, and spectral convergence analysis.

Findings

Explicit eigenfunctions and eigenvalues computed

Solutions to heat and wave equations derived

Spectral convergence to crossed lines demonstrated

Abstract

We study the spectral decomposition of the Laplacian on a family of fractals $\mathcal{VS}_n$ that includes the Vicsek set for $n=2$, extending earlier research on the Sierpinski Gasket. We implement an algorithm [24] for spectral decimation of eigenfunctions of the Laplacian, and explicitly compute these eigenfunctions and some of their properties. We give an algorithm for computing inner products of eigenfunctions. We explicitly compute solutions to the heat equation and wave equation for Neumann boundary conditions. We study gaps in the ratios of eigenvalues and eigenvalue clusters. We give an explicit formula for the Green's function on $\mathcal{VS}_n$. Finally, we explain how the spectrum of the Laplacian on $\mathcal{VS}_n$ converges as $n \to \infty$ to the spectrum of the Laplacian on two crossed lines (the limit of the sets $\mathcal{VS}_n$.)