On the multiplicity of the eigenvalues of discrete tori

TL;DR

This paper establishes that the maximum multiplicity of nonzero Laplacian eigenvalues on 2D discrete tori is 24, and characterizes eigenvalue multiplicities in higher dimensions, using roots of unity theory.

Contribution

It proves the optimal bound of 24 for 2D discrete tori eigenvalue multiplicities and characterizes large multiplicities in higher dimensions.

Findings

Maximum eigenvalue multiplicity in 2D discrete tori is 24.

Eigenvalues with large multiplicities are characterized in higher dimensions.

Provides bounds and characterizations using roots of unity theory.

Abstract

It is well known that the standard flat torus $\mathbb{T}^2=\mathbb{R}^2/\Z^2$ has arbitrarily large Laplacian-eigenvalue multiplicities. We prove, however, that $24$ is the optimal upper bound for the multiplicities of the nonzero eigenvalues of a $2$-dimensional discrete torus. For general higher dimension discrete tori, we characterize the eigenvalues with large multiplicities. As consequences, we get uniform boundedness results of the multiplicity for a long range and an optimal global bound for the multiplicity. Our main tool of proof is the theory of vanishing sums of roots of unity.