Spectrum of the Laplacian on Regular Polyhedra

TL;DR

This paper investigates the eigenvalues and eigenfunctions of the Laplacian on the surfaces of regular polyhedra, revealing two types of eigenfunctions and providing numerical evidence for asymptotic eigenvalue estimates.

Contribution

It classifies eigenfunctions into smooth and singular types on regular polyhedra and introduces an enlargement phenomenon affecting eigenvalues.

Findings

Identification of two eigenfunction types: smooth and singular.

Numerical evidence supporting asymptotic eigenvalue estimates.

Discovery of an eigenfunction enlargement phenomenon on the octahedron.

Abstract

We study eigenvalues and eigenfunctions of the Laplacian on the surfaces of four of the regular polyhedrons: tetrahedron, octahedron, icosahedron and cube. We show two types of eigenfunctions: nonsingular ones that are smooth at vertices, lift to periodic functions on the plane and are expressible in terms of trigonometric polynomials; and singular ones that have none of these properties. We give numerical evidence for conjectured asymptotic estimates of the eigenvalue counting function. We describe an enlargement phenomenon for certain eigenfunctions on the octahedron that scales down eigenvalues by a factor of $\frac{1}{3}$.