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Multiplicities of eigenvalues and quadratic representations of integers

This paper characterizes the set of multiplicities of non-zero eigenvalues for the Laplace operator on two-dimensional rectangles and tori by establishing precise conditions on side length ratios and angles that determine whether these multiplicities encompass all positive integers, specific even multiples, or remain finite, utilizing the study of integral lattice points on dilated ellipses.

Original authors: Siqi Fu, Andrew Pendleton

Published 2026-03-17

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Siqi Fu, Andrew Pendleton

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a magical drum. When you hit it, it doesn't just make one sound; it vibrates in many different patterns at once. Some patterns are simple (just one way to vibrate), while others are complex (many different ways to vibrate at the exact same pitch).

In mathematics, these "ways to vibrate" are called eigenvalues, and the number of different ways to vibrate at a specific pitch is called the multiplicity.

This paper is a detective story about two types of drums:

Rectangular Drums: Shaped like a standard piece of paper.
Toroidal Drums: Shaped like a donut (or a video game screen where if you go off the right edge, you appear on the left).

The authors, Siqi Fu and Andrew Pendleton, wanted to answer a simple question: What determines how many different vibration patterns a drum can have?

They discovered that the answer isn't about the size of the drum, but about the mathematical "rationality" of its shape.

The Rectangular Drum: The Perfect Square vs. The Weird Rectangle

Think of a rectangular drum as a grid. The vibrations are like waves bouncing off the walls.

The Rule: If the ratio of the rectangle's width to its height is a "nice" number (specifically, if the square of that ratio is a fraction like 3/4 or 5/2), the drum is chaotic in a good way. It can vibrate in any number of ways. You can find a pitch that vibrates in 1 way, another that vibrates in 100 ways, another in 1,000 ways, and so on. The set of possibilities is infinite and covers every positive integer.
The Weird Case: If the ratio is a "weird" number (like the square root of 2, which cannot be written as a simple fraction), the drum is boring. It almost always vibrates in only one way. There are no complex patterns; every pitch is unique.

The Analogy: Imagine a dance floor.

If the floor is a perfect grid (rational ratio), dancers can form lines, circles, and squares in endless combinations.
If the floor is a weird, skewed shape (irrational ratio), the dancers can only stand in a single line. They can't form complex groups.

The Donut Drum (Torus): The Angle and the Ratio

Now, imagine a drum shaped like a donut. This is trickier because the shape depends on two things:

The ratio of the lengths of the two main loops (like the inner circle vs. the outer circle).
The angle between them (is it a perfect square donut, or a slanted, stretched one?).

The authors found that for a donut drum to have infinite possibilities for complex vibrations, both the length ratio and the angle must be "nice" numbers (rational).

The "Goldilocks" Zones:
- If the shape is just right (specifically related to the number -3 or -4 in a special math formula), the drum vibrates in groups of 6 or 4.
- If it's a "standard" nice shape, it vibrates in groups of 2.
The "Broken" Zones: If either the angle or the length ratio is a "weird" number, the drum becomes very limited. It can only vibrate in 2 ways, or maybe 2 and 4 ways. It loses its ability to create complex, high-multiplicity patterns.

The Secret Ingredient: Integer Lattice Points

How did they figure this out? They didn't just listen to the drums; they looked at numbers.

They realized that counting the vibration patterns is exactly the same as counting dots on a grid that fit inside a stretched oval (an ellipse).

Imagine a grid of dots (like graph paper).
Stretch the paper into an oval shape.
How many dots are inside?

If the shape of the oval is "rational" (based on fractions), you can stretch it just right to catch exactly 100 dots, or 1,000 dots, or any number you want.
If the shape is "irrational" (based on weird numbers), the dots just won't line up. You'll mostly catch 2 dots, and rarely 4.

Why Does This Matter?

This paper connects two worlds that usually don't talk to each other:

Physics/Geometry: How things vibrate (sound, light, quantum mechanics).
Number Theory: How numbers can be built from squares and fractions.

The Big Takeaway:
The "music" a shape makes is dictated by the "arithmetic" of its geometry.

Simple shapes (Rational): Can make complex, rich music with many harmonies.
Weird shapes (Irrational): Can only make simple, sparse music.

The authors proved that if you want a drum that can produce a sound with any level of complexity, you must build it with dimensions that are based on simple fractions. If you use "weird" numbers, the drum will be mathematically limited, no matter how big or small you make it.

1. Problem Statement

The paper investigates the set $M$ of all multiplicities of non-zero eigenvalues for the Laplace operator on two specific geometric domains:

Rectangles: $\Omega_{a,b} = [0, a] \times [0, b]$ .
2D Tori: $T = \mathbb{R}^2 / L$ , where $L$ is a lattice generated by vectors $\vec{a}$ and $\vec{b}$ .

The central question is how the arithmetic properties of the geometric parameters (side length ratios for rectangles, and length ratios/angles for tori) determine the structure of the set $M$ . Specifically, the authors aim to characterize when $M$ is the set of all positive integers ( $\mathbb{N}$ ), an infinite subset of even integers, or a finite set.

2. Methodology

The authors bridge spectral geometry and algebraic number theory. Their methodology relies on the following steps:

Spectral Reduction to Lattice Point Counting:
- For rectangles, the eigenvalues are given by $\lambda_{m,n} = \pi^2(m^2/a^2 + n^2/b^2)$ . The multiplicity of an eigenvalue corresponds to the number of integer lattice points $(m,n)$ in the first quadrant lying on a specific ellipse.
- For tori, the eigenvalues are determined by the dual lattice $L^*$ . The multiplicity corresponds to the number of integer solutions to a binary quadratic form $f(x,y) = ax^2 + bxy + cy^2 = \text{const}$ .
Quadratic Form Theory:
- The problem is transformed into studying the proper representation counting function $r^+_Q(n)$ , which counts the number of ways an integer $n$ can be represented by a primitive positive-definite binary quadratic form $Q$ , up to proper automorphisms.
- The authors utilize the Ideal Class Group of imaginary quadratic fields. They establish a correspondence between representations of integers by quadratic forms and ideals in the ring of integers of $K = \mathbb{Q}(\sqrt{\Delta})$ , where $\Delta$ is the discriminant.
Number Theoretic Tools:
- Weber's Theorem: Used to guarantee the existence of primes represented by specific quadratic forms.
- Unique Factorization of Ideals: Used in Dedekind domains to count the number of distinct ideals with a given norm, which translates directly to the number of distinct solutions (multiplicities).
- Ambiguous Forms: Special attention is paid to forms where the class group element is of order $\le 2$ (ambiguous forms), as these dictate the specific structure of the multiplicity set (e.g., multiples of 2, 4, or 6).

3. Key Contributions and Results

A. Rectangles

Theorem 1.1: The rectangle $\Omega_{a,b}$ satisfies property (M) (i.e., $M = \mathbb{N}$ , the set of all positive integers) if and only if the square of the side length ratio, $(a/b)^2$ , is rational.

Significance: This resolves an open problem left by previous works [7, 9]. If $(a/b)^2$ is irrational, all eigenvalues are simple ( $M=\{1\}$ ). If rational, the authors prove that for any integer $k$ , there exists an eigenvalue with multiplicity $k$ .
Proof Strategy: Instead of deriving explicit formulas for lattice points on general ellipses, they prove the existence of an eigenvalue with arbitrary multiplicity $k$ by constructing a specific prime $p$ and using Theorem 1.2.

Theorem 1.2: For any coprime $m, n \in \mathbb{N}$ with $mn > 3$, there exists a prime $p$ such that the equation $p^{2k-1} = mx^2 + ny^2$ has exactly $k$ distinct solutions in $\mathbb{N}^2$ . This is proven using ideal class group properties in $\mathbb{Q}(\sqrt{-mn})$ .

B. Tori

Let $r = |\vec{a}|/|\vec{b}|$ and $\theta$ be the angle between the generating vectors.
Theorem 1.3: The multiplicity set $M(T)$ is infinite if and only if both $r^2 \in \mathbb{Q}$ and $r \cos \theta \in \mathbb{Q}$ .

Infinite Case: If the conditions are met, the set $M(T)$ $M (T)$ is a subset of even integers determined by the discriminant $\Delta$ $Δ$ of the associated quadratic form:
- $M(T) = 6\mathbb{N}$ if $\Delta = -3$ .
- $M(T) = 4\mathbb{N}$ if $\Delta = -4$ .
- $M(T) = 2\mathbb{N}$ if $\Delta \neq -3, -4$ .
Finite Case: If at least one of $r^2$ $r^{2}$ or $r \cos \theta$ $r cos θ$ is irrational, $M(T)$ $M (T)$ is finite.
- $M(T) = \{2\}$ under specific linear independence conditions or if $r^2 \in \mathbb{Q}$ but $r \notin \mathbb{Q}$ .
- $M(T) = \{2, 4\}$ if specific algebraic dependencies exist between $r \cos \theta$ and $r^2$ (specifically when $r \cos \theta = \alpha r^2 + \beta$ and $\alpha^2 + \beta$ is a rational square).

Theorem 1.4: A foundational number-theoretic result stating that for any primitive positive-definite binary quadratic form $Q$ , the proper representation counting function $r^+_Q$ is surjective onto $\mathbb{Z}_{\ge 0}$ . This means every non-negative integer can be realized as the number of representations of some integer by $Q$ (up to automorphism).

4. Significance

Resolution of Open Problems: The paper definitively answers the question of when a rectangle satisfies property (M), a problem that had been open since the works of [7] and [9].
Unification of Fields: It provides a rigorous link between the spectral geometry of the Laplacian (a PDE/physics problem) and the arithmetic of quadratic forms and ideal class groups (number theory).
Generalization of Legendre's Formula: While previous results relied on specific cases (like squares or specific ratios 2, 3, 7), this work provides a complete characterization for all rational and irrational ratios.
Methodological Innovation: The authors avoid the difficulty of finding explicit formulas for lattice points on arbitrary ellipses. Instead, they use the existence of primes with specific representation properties (via Weber's theorem and ideal class groups) to prove the existence of eigenvalues with arbitrary multiplicities.
Complete Classification for Tori: The paper offers the first complete classification of multiplicity sets for 2D tori based on the arithmetic nature of the lattice parameters, distinguishing clearly between infinite sets (multiples of 2, 4, or 6) and finite sets ({2} or {2, 4}).

In summary, the paper demonstrates that the "spectrum" of a drum (rectangle or torus) is deeply governed by the rationality of its geometric parameters, with the set of eigenvalue multiplicities acting as a direct reflection of the underlying arithmetic structure of quadratic forms.