Outer Approximation of the Spectrum of a Fractal Laplacian

TL;DR

This paper introduces a novel outer approximation method for the spectrum of fractal Laplacians, enabling numerical eigenvalue estimation on complex fractals and revealing new spectral structure insights through experimental and theoretical analysis.

Contribution

The paper proposes a new outer approximation technique for fractal Laplacian spectra, supported by experimental validation and theoretical findings on eigenfunction miniaturization.

Findings

Method accurately approximates spectra of known fractals like the Sierpinski Gasket.

Experimental evidence supports the method's applicability to complex fractals.

New theoretical results on eigenfunction miniaturization and spectral structure.

Abstract

We present a new method to approximate the Neumann spectrum of a Laplacian on a fractal K in the plane as a renormalized limit of the Neumann spectra of the standard Laplacian on a sequence of domains that approximate K from the outside. The method allows a numerical approximation of eigenvalues and eigenfunctions for lower portions of the spectrum. We present experimental evidence that the method works by looking at examples where the spectrum of the fractal Laplacian is known (the unit interval and the Sierpinski Gasket (SG)). We also present a speculative description of the spectrum on the standard Sierpinski carpet (SC), where existence of a self-similar Laplacian is known, and also on nonsymmetric and random carpets and the octagasket, where existence of a self-similar Laplacian is not known. At present we have no explanation as to why the method should work. Nevertheless, we are able to prove some new results about the structure of the spectrum involving "miniaturization" of eigenfunctions that we discovered by examining the experimental results obtained using our method.