Eigenmodes of the Laplacian on some Laakso spaces

Abstract

We analyse the spectrum of a self-adjoint operator on a Laakso space using the projective limit construction originally given by Barlow and Evans. We will use the hierarchical cell structure induced by the choice of approximating quantum graphs to calculate the spectrum with multiplicities. We also extend the method for using the hierarchical cell structure to more general projective limits beyond Laakso spaces.

Spectrum and Heat Kernel Asymptotics on General Laakso Spaces

“…Then in [16], a more in depth description of the projective limit construction was used to calculate the spectrum of the Laplacian constructed in [17]. We will be using the construction in [16] as it is also well-suited for our calculations.…”

“…In Section 3.1 we provide an analytical proof by examining, as in [16], the different "shapes" that make up the space. Since each shape has a unique contribution to the spectrum counting the number of shapes allows us to calculate the spectrum with multiplicities.…”

“…We begin by reviewing the construction of the Laakso spaces as presented in [11,16,17] in Section 2. This section also contains background information on the Hausdorff dimension, its calculation for Laakso spaces, and some specific values for certain Laakso spaces.…”

“…This paper will be concerned instead with the irregular domain being a fractal itself. Some notable works with this type of domain include [3,6,8,16,17,19] among others. Laakso's spaces were introduced in [11].…”

“…They are a family of fractals with an arbitrary Hausdorff dimension greater than one and were considered originally for their nice analytic properties. Constructions of the Laakso spaces are given in [11,16,17] as well as in Section 2 of this paper. Theorem 6.1 in [16] gives the spectrum of the Laplacian operator on any given Laakso space, in Theorem 3.1 we give the multiplicities.…”