Outer Approximation of the Spectrum of a Fractal Laplacian

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 in which the spectrum of the fractal Laplacian is known (the unit interval and the Sierpiński gasket). We also present a speculative description of the spectrum on the standard Sierpiński carpet, 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.