Multiplicities of eigenvalues and quadratic representations of integers

TL;DR

This paper characterizes the multiplicities of eigenvalues of the Laplace operator on rectangles and tori based on rationality conditions of geometric ratios, linking spectral properties to quadratic representations of integers.

Contribution

It provides a complete characterization of eigenvalue multiplicities for rectangles and tori in terms of rationality of geometric ratios, connecting spectral theory with quadratic number representations.

Findings

Eigenvalue multiplicities on rectangles are all positive integers if and only if the side ratio squared is rational.

On tori, multiplicities are infinite and specific (2, 4, or 6 times natural numbers) when certain ratios are rational.

When ratios are irrational, the set of multiplicities reduces to {2} or {2, 4}.

Abstract

We study the set $M$ of all multiplicities of non-zero eigenvalues for the Laplace operator on a two-dimensional rectangle or torus. We show that for a rectangle with the side length ratio $r$, $M=\mathbb{N}$, the set of all positive integers, if and only if $r^2$ is rational. For a torus whose generating vectors have a length ratio $r$ and the angle between them $\theta$, we show that $M$ is an infinite set if and only if both $r\cos\theta$ and $r^2$ are rational. In this case, $M=2\mathbb{N}$, $4\mathbb{N}$, or $6\mathbb{N}$, and we obtain a characterization for each of these cases in term of $r\cos\theta$ and $r^2$. In the case when at least one of $r\cos\theta$ or $r^2$ is irrational, we show that $M=\{2\}$ or $\{2, 4\}$, and obtain a characterization for these cases. We prove these results by studying the number of integral lattice points on dilated ellipses.