Laplacian, on the Sierpinski tetrahedron

Abstract

Numerous work revolve around the Sierpinski gasket. Its three-dimensional analogue, the Sierpinski tetrahedron, obtained by means of an iterative process which consists in repeatedly contracting a regular 3-simplex to one half of its original height, put together four copies, the frontier corners of which coincide with the initial simplex, appears as a natural extension. Yet, very few works concern the Sierpinski tetrahedron in the existing literature. We go further and, after a detailed study, we give the explicit spectrum of the Laplacian, with a specific presentation of the first eigenvalues. This enables us to obtain an estimate of the spectral counting function (analogous of Weyl's law)