Laplace-Beltrami spectrum of ellipsoids that are close to spheres and analytic perturbation theory

TL;DR

This paper analyzes the Laplace-Beltrami spectrum of ellipsoids close to spheres using analytic perturbation theory, providing eigenvalue estimates, multiplicity characterizations, numerical comparisons, and a conjecture on nodal domains.

Contribution

It introduces an analytic perturbation approach to estimate eigenvalues of near-spherical ellipsoids and characterizes eigenvalue multiplicities and simplicity for biaxial and triaxial cases.

Findings

Eigenvalues estimated up to second order for near-spherical ellipsoids.

First N eigenvalues of biaxial ellipsoids have multiplicity at most two.

At least the first sixteen eigenvalues of certain triaxial ellipsoids are simple.

Abstract

We study the spectrum of the Laplace-Beltrami operator on ellipsoids. For ellipsoids that are close to the sphere, we use analytic perturbation theory to estimate the eigenvalues up to two orders. We show that for biaxial ellipsoids sufficiently close to the sphere, the first $N$ eigenvalues have multiplicity at most two, and characterize those that are simple. For the triaxial ellipsoids sufficiently close to the sphere that are not biaxial, we show that at least the first sixteen eigenvalues are all simple. We also give the results of various numerical experiments, including comparisons to our results from the analytic perturbation theory, and approximations for the eigenvalues of ellipsoids that degenerate into infinite cylinders or two-dimensional disks. We propose a conjecture on the exact number of nodal domains of near-sphere ellipsoids.