On multiplicity bounds for eigenvalues of the clamped round plate

TL;DR

This paper investigates the multiplicity of eigenvalues for the clamped round plate, showing that eigenfunctions are nearly separated and establishing bounds on eigenvalue multiplicities using advanced Bessel function analysis.

Contribution

It proves that eigenfunctions can be expressed as sums of at most two separated functions and establishes that no eigenvalue exceeds multiplicity four in two dimensions.

Findings

Eigenfunctions can be decomposed into at most two separated functions.

Eigenvalue multiplicities are bounded by four in two dimensions.

The proof uses a novel recursion relation for cross-product Bessel functions.

Abstract

We ask whether the only multiplicities in the spectrum of the clamped round plate are trivial, i.e., whether all existing multiplicities are due to the isometries of the sphere, or, equivalently, whether any eigenfunction is separated. We prove that any eigenfunction can be expressed as a sum of at most two separated ones, by showing that otherwise the corresponding eigenvalue is algebraic, contradicting the Siegel-Shidlovskii theory. In two dimensions it follows that no eigenvalue is of multiplicity greater than four. The proof exploits a linear recursion of order two for cross-product Bessel functions with coefficients which are not even algebraic functions, though they do satisfy a non-linear algebraic recursion.