# Laplacian, on the Sierpinski tetrahedron

## TL;DR

This paper provides a detailed spectral analysis of the Laplacian on the Sierpinski tetrahedron, including explicit eigenvalues and estimates of the spectral counting function, extending the understanding of fractal Laplacians to three dimensions.

## Contribution

It explicitly determines the spectrum of the Laplacian on the Sierpinski tetrahedron and estimates its spectral counting function, a significant extension from the well-studied Sierpinski gasket.

## Key findings

- Explicit spectrum of the Laplacian on the Sierpinski tetrahedron
- First eigenvalues characterized in detail
- Spectral counting function estimated, analogous to Weyl's law

## 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)