🔢 mathematics

On the lower bound and upper bound of the Sum of Eigenvalues of the Fractional-Logarithmic Laplacian

This paper establishes both lower and upper bounds for the sum of eigenvalues of the Fractional-Logarithmic Laplacian, overcoming the analytical challenge posed by the operator's non-monotone Fourier symbol at low frequencies.

Original authors: H. Hajaiej

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: H. Hajaiej

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 are standing in a large, empty room (let's call it $\Omega$ ). This room has a very special property: it's filled with invisible, vibrating strings. When you pluck these strings, they don't just vibrate at one frequency; they vibrate at a whole spectrum of frequencies, from a low hum to a high-pitched squeal.

In mathematics, these "strings" are called eigenvalues, and the room is a "domain." The goal of this paper is to figure out how much "energy" is stored in the first $k$ loudest notes of this room.

The Cast of Characters

The Room ( $\Omega$ ): A bounded space (like a box or a sphere).
The Notes ( $\lambda$ ): The specific frequencies the room can vibrate at. They are ordered from lowest to highest: $\lambda_1, \lambda_2, \lambda_3, \dots$ .
The Sum: The paper asks: "If I add up the energy of the first $k$ notes, how big is that total?"
The Conductor (The Operator): This is the rule that tells the strings how to vibrate.
- The Classic Conductor: The standard Laplacian (like a drum). Its notes grow like a power of the number of notes ( $k^2$ ).
- The Fractional Conductor: A "weird" drum where the notes grow slower ( $k^{2s}$ ).
- The Logarithmic Conductor: A very quiet instrument where notes grow very slowly, like the logarithm of the number of notes ( $\ln k$ ).
- The Star of the Show: The Fractional-Logarithmic Laplacian. This is a hybrid instrument. It combines the "weirdness" of the fractional drum with the "quietness" of the logarithmic one. Its rule is a mix of power and logs: $k^{2s} \times \ln k$ .

The Big Challenge: The "Bumpy Road"

The author, Hichem Hajaieja, faces a tricky problem. To predict how the notes behave, mathematicians usually look at the "map" of the frequencies (called the Fourier symbol).

For the classic drum, the map is a smooth, straight hill. It always goes up as you go further out. This makes it easy to predict the sum.
For this new Fractional-Logarithmic instrument, the map is bumpy. At low frequencies (near the start), it dips down and behaves strangely. It's not a smooth hill; it's a hill with a valley at the bottom.

This "bumpiness" makes it very hard to calculate the total energy (the sum of eigenvalues) because the usual mathematical tools get confused by the dip.

The Solution: Building a Fence and a Bridge

The paper solves this by building two fences around the answer: a Lower Bound and an Upper Bound.

1. The Lower Bound (The "Floor")

The Analogy: Imagine you want to know the minimum amount of water in a pool. You can't measure every drop, but you know the pool is at least this deep.
The Method: The author uses a clever trick involving "slowly varying functions." Think of this as realizing that even though the road is bumpy at the start, once you get far enough down the road (high frequencies), the bumps smooth out, and the road becomes a straight, steep hill again.
The Result: He proves that no matter how weird the low notes are, the total sum of the first $k$ $k$ notes cannot be smaller than a specific formula involving $k$ $k$ raised to a power and multiplied by a log.
- Formula: Total Energy $\ge$ (Constant) $\times k^{1 + \text{fraction}} \times \ln k$ .

2. The Upper Bound (The "Ceiling")

The Analogy: Now, imagine you want to know the maximum amount of water. You can't fill the pool to the brim, but you know it can't overflow a certain height.
The Method: This is the harder part. The author uses "cutoff functions." Imagine taking a flashlight and shining it only on the part of the room that is far away from the walls. He creates "test waves" (like fake notes) that live mostly in the middle of the room and ignore the messy edges.
The Result: He proves that the total sum cannot be larger than a very similar formula.
- Formula: Total Energy $\le$ (Same Constant) $\times k^{1 + \text{fraction}} \times \ln k + \text{a little bit of extra noise}$ .

The "Aha!" Moment

The most beautiful part of the paper is the conclusion.

Even though the Fractional-Logarithmic Laplacian is a complex, hybrid creature, its behavior turns out to be multiplicative.

The Fractional part contributes a power growth ( $k^{2s}$ ).
The Logarithmic part contributes a slow growth ( $\ln k$ ).
When you combine them, the total growth is simply the product of the two: $k^{2s} \times \ln k$ .

It's as if you have a car engine (fractional) and a turbocharger (logarithmic). The paper proves that the speed of the car isn't some complicated, unpredictable mess; it's just the engine speed multiplied by the turbo boost.

Why Does This Matter?

In the real world, these mathematical "rooms" and "notes" model things like:

How heat spreads through a material with memory.
How particles move in a fluid that isn't quite normal (anomalous diffusion).
Quantum mechanics in complex environments.

By finding the precise "floor" and "ceiling" for the energy of these systems, scientists can better predict how these materials will behave without having to simulate every single atom. This paper provides the rulebook for a new, complex type of physical system, showing that even with its "bumpy" start, it follows a surprisingly elegant and predictable pattern in the long run.

1. Problem Statement

The paper addresses the spectral theory of the Fractional-Logarithmic Laplacian, denoted as $(-\Delta)^{s+\ln}$ , defined on a bounded domain $\Omega \subset \mathbb{R}^n$ . This operator is a relatively recent development, introduced as the derivative of the fractional Laplacian $(-\Delta)^s$ with respect to the order parameter $s$ .

The central object of study is the sequence of Dirichlet eigenvalues:
$0 < \lambda_1^{s+\ln}(\Omega) \le \lambda_2^{s+\ln}(\Omega) \le \dots \uparrow +\infty$
The primary goal is to establish rigorous lower and upper bounds for the partial sums of these eigenvalues:
$S_k = \sum_{j=1}^k \lambda_j^{s+\ln}(\Omega)$
Specifically, the authors aim to prove that these sums satisfy sharp asymptotic estimates that combine the power-law growth of the fractional Laplacian with the logarithmic correction characteristic of the logarithmic Laplacian.

The Core Challenge:
Unlike the standard fractional Laplacian (whose symbol $|\xi|^{2s}$ is globally monotone) or the logarithmic Laplacian (symbol $2\ln|\xi|$ ), the symbol of the fractional-logarithmic Laplacian is:
$m(\xi) = |\xi|^{2s} \ln |\xi|^2$
This symbol is not globally monotone in the radial variable $|\xi|$ due to its behavior at low frequencies (where $\ln|\xi|^2 < 0$ for $|\xi| < 1$ ). This non-monotonicity complicates the application of standard spectral estimation techniques (like the Berezin-Li-Yau method) which typically rely on monotone symbols.

2. Methodology

The paper employs a combination of Tauberian theorems, variational principles, and Fourier analysis adapted to non-monotone symbols.

A. Asymptotic Analysis via Tauberian Theorems

The authors first establish the asymptotic behavior of the sum $S_k$ based on the known asymptotic of individual eigenvalues $\lambda_k^{s+\ln} \sim C k^{2s/n} \ln k$ .

They utilize Karamata's Tauberian theorem for power series involving slowly varying functions.
By treating $\ln k$ as a slowly varying function, they rigorously derive that the sum behaves as:
$S_k \sim \frac{2}{n+2s} (2\pi)^{2s} (\omega_n |\Omega|)^{-2s/n} k^{1+2s/n} \ln k$
This confirms the heuristic that the spectral structure is a multiplicative product of the fractional and logarithmic contributions.

B. Lower Bound Estimation (Berezin-Li-Yau Type)

To prove the lower bound, the authors adapt the classical Berezin-Li-Yau method:

Fourier Representation: They define a density function $f(\xi) = \sum_{j=1}^k |\hat{\phi}_j(\xi)|^2$ , where $\phi_j$ are the eigenfunctions.
Moment Estimates: They relate the sum of eigenvalues to the moment $M_2 = \int |\xi|^{2s} \ln |\xi|^2 f(\xi) d\xi$ .
Handling Non-Monotonicity: Since the symbol $|\xi|^{2s} \ln |\xi|^2$ is negative for small $|\xi|$ , they employ a specific optimization lemma (Lemma 2.4). This lemma provides a lower bound for the integral of a non-monotone weight function against a bounded density, separating the contribution of the low-frequency region (where the symbol is negative) from the high-frequency region.
Optimization: By optimizing over a radius $R$ , they derive a sharp inequality that accounts for the "negative" part of the symbol, ensuring the leading term remains positive and matches the asymptotic prediction.

C. Upper Bound Estimation (Kröger Type)

To prove the upper bound, the authors use a Rayleigh-Ritz variational principle combined with cutoff plane waves:

Test Functions: They construct test functions $v_\sigma(z, x) = w_\sigma(x) e^{-ix \cdot z}$ , where $w_\sigma$ is a smooth cutoff function that is 1 deep inside $\Omega$ and 0 near the boundary $\partial \Omega$ .
Orthogonal Projection: They project these test functions onto the orthogonal complement of the first $k$ eigenfunctions to utilize the min-max principle for $\lambda_{k+1}$ .
Boundary Layer Estimates: A critical step involves estimating the error introduced by the cutoff near the boundary. They assume a geometric condition on the domain (boundary layer estimate $|\Omega_t| \le C t$ ) to control the remainder terms.
Symbol Integration: They compute the energy of the test functions using the Fourier symbol. The non-monotonicity is handled by carefully choosing the cutoff radius $r$ and the boundary layer width $\sigma$ to ensure the dominant term comes from the high-frequency region where the symbol is positive and growing.

3. Key Contributions and Results

Main Theorems

The paper establishes the following sharp bounds for $k \ge 1$ (with specific conditions on $k$ for the lower bound):

1. Lower Bound (Theorem 2.5):
$\sum_{j=1}^k \lambda_j^{s+\ln}(\Omega) \ge \frac{2}{n+2s}(2\pi)^{2s}(\omega_n|\Omega|)^{-2s/n} k^{1+2s/n} \left( \ln \frac{(2\pi)^n k}{\omega_n|\Omega|} - \frac{n}{n+2s} \right)$

Significance: This bound has the correct leading-order growth $k^{1+2s/n} \ln k$ . It is the first rigorous lower bound of Berezin-Li-Yau type for this specific operator.

2. Upper Bound (Theorem 3.2):
Under the assumption that the domain satisfies a boundary layer estimate, there exists a constant $C$ such that:
$\sum_{j=1}^k \lambda_j^{s+\ln}(\Omega) \le \frac{2}{n+2s}(2\pi)^{2s}(\omega_n|\Omega|)^{-2s/n} k^{1+2s/n} \ln k + C k^{1+2s/n}$

Significance: This provides a Kröger-type upper bound, confirming that the asymptotic order cannot be exceeded.

Key Technical Insights

Multiplicative Spectral Structure: The results confirm that the eigenvalue sums of the fractional-logarithmic Laplacian inherit a "multiplicative" structure. The leading coefficient is the product of the fractional Laplacian's coefficient and the logarithmic Laplacian's coefficient.
Overcoming Non-Monotonicity: The paper successfully demonstrates that despite the symbol's lack of global monotonicity (due to low-frequency negativity), the high-frequency growth is sufficiently rigid to allow for sharp spectral bounds.
Independence of Volume in Logarithmic Regime: While the fractional part depends on $|\Omega|$ , the logarithmic correction interacts with the volume in a specific way that distinguishes this operator from pure power-law operators.

4. Significance

Bridging Gaps: This work bridges the spectral theory of homogeneous nonlocal operators (fractional Laplacian) and their logarithmic corrections. It fills a gap in the literature regarding eigenvalue bounds for operators defined as derivatives with respect to the order parameter.
Methodological Advancement: The techniques developed to handle the non-monotone symbol $|\xi|^{2s} \ln |\xi|^2$ provide a template for analyzing other singular or mixed-order nonlocal operators where standard monotonicity assumptions fail.
Geometric Spectral Theory: The results deepen the understanding of how domain geometry (via $|\Omega|$ and boundary regularity) influences the spectrum of complex nonlocal operators, extending the classical Weyl law and Li-Yau inequalities to this new class of operators.

In summary, Hajaiej rigorously proves that the sum of eigenvalues for the fractional-logarithmic Laplacian follows a precise asymptotic law, providing both lower and upper bounds that match the predicted growth rate, thereby establishing a solid foundation for the spectral analysis of this emerging class of operators.