---
title: Berezin-Li-Yau Inequality
url: https://www.emergentmind.com/topics/berezin-li-yau-inequality
type: topic
---

# Berezin-Li-Yau Inequality

The Berezin-Li-Yau inequality refers to sharp semiclassical spectral bounds governing the distribution of Dirichlet (and related) eigenvalues of elliptic operators, most notably the Laplacian, on bounded domains in Euclidean space. Originating from the works of Berezin (1972) and Li and Yau (1983), the inequality provides a universal lower bound on Riesz means and eigenvalue sums that reflects the leading term of Weyl's law. Recent research has yielded explicit quantitative improvements including second-order remainder terms, results for mixed local-nonlocal operators, and extensions to the Neumann case and more general settings.

## 1. Classical Berezin-Li-Yau Inequality

Given a bounded open domain $\Omega \subset \mathbb{R}^n$, consider the Dirichlet Laplacian $- \Delta$ with eigenvalues $0 < \lambda_1(\Omega) \leq \lambda_2(\Omega) \leq \cdots$. Define $|\Omega|$ as the Lebesgue measure of $\Omega$ and $\omega_n = |B(0,1)|$ as the volume of the unit $n$-ball. For any $\Lambda > 0$, the classical Berezin-Li-Yau (BLY) Riesz mean bound for order one is:
\[
\sum_{j: \lambda_j < \Lambda} (\Lambda - \lambda_j) \leq \frac{2}{n+2} \, \omega_n \, |\Omega| \, (2\pi)^{-n} \, \Lambda^{1 + n/2}
\]
Choosing $\Lambda = (1+2/n)\lambda_k$ and rearranging yields the eigenvalue bound:
\[
\lambda_k(\Omega) \geq \frac{n}{n+2} (2\pi)^2 \omega_n^{-2/n} |\Omega|^{-2/n} k^{2/n}
\]
and for all $k \geq 1$,
\[
\sum_{j=1}^k \lambda_j(\Omega) \geq \frac{n}{n+2} (2\pi)^2 \omega_n^{-2/n} |\Omega|^{-2/n} k^{1+2/n}
\]
These bounds are sharp in the semiclassical limit ($k \to \infty$), coinciding with Weyl's law in leading order [2507.20330].

## 2. Explicit Two-Term Improvements

The BLY inequality has been substantially refined with the introduction of explicit two-term lower bounds incorporating geometric information. Introducing the moment of inertia,
\[
I(\Omega) = \inf_{a \in \mathbb{R}^n} \int_\Omega |x-a|^2 \, dx,
\]
and $J_1 = |\Omega|^{1/2} /(2 I(\Omega)^{1/2})$, the improved Riesz mean (Theorem 2.1, [2507.20330]) is:
\[
\sum_{j: \lambda_j < \Lambda} (\Lambda - \lambda_j) \leq \frac{2}{n+2}\, \omega_n |\Omega| (2\pi)^{-n} \Lambda^{1 + n/2} \left[ 1 - C_1(n) J_1^2 / \Lambda \right]^{-1}_+
\]
where $C_1(n) > 0$ has explicit values, e.g., $C_1(2) = 157/480$, and for $n \geq 3$,
\[
C_1(n) = \frac{5 n (n+2)}{96} (1 - A_n)^{n-1} (2 - A_n), \quad
A_n = \left( \frac{\sqrt{n+2}}{8 \sqrt{n} \, \Gamma(1+n/2)} \right)^{2/n} < 1.
\]
This in turn yields a refined eigenvalue counting inequality for all $k \geq 1$:
\[
k \leq (1+2/n)^{n/2} \lambda_k^{n/2} |\Omega| \omega_n (2\pi)^{-n}
\left[1 - \frac{nC_1(n)}{4(n+2)} \frac{|\Omega|}{I(\Omega)} \frac{1}{\lambda_k}\right]^{-1}
\]
As $k \to \infty$ (so $\lambda_k \to \infty$), the correction term vanishes, recovering the leading Weyl asymptotic.

## 3. Methodological Innovations

The proof of the improved BLY inequality introduces refined control over the phase-space density $F_\Lambda(\xi) = \sum_{j: \lambda_j < \Lambda} |\widehat{\varphi}_j(\xi)|^2 \leq |\Omega| (2\pi)^{-n}$ and its gradient:
\[
|\nabla F_\Lambda(\xi)| \leq 2 |\Omega|^{1/2} I(\Omega)^{1/2} (2\pi)^{-n}
\]
Using symmetrization techniques and block rearrangement—a replacement of the Dirac-mass ansatz in the Li-Yau argument—the method involves optimizing over functions with fixed mass and slope constraints, leading to the second-term correction ([2507.20330], Lemmas 2.1–2.3).

## 4. Consequences for Pólya's Conjecture and Neumann Spectrum

Pólya conjectured that for any domain $\Omega$,
\[
|\Omega| \omega_n (2\pi)^{-n} \lambda_k^{n/2} \geq k
\]
While this conjecture remains open in full generality, the refined two-term result proves that for $n \geq 3$, there exist infinitely many $k_j \to \infty$ such that this bound holds, i.e., on infinitely many parts of the Dirichlet spectrum the conjectured lower bound is achieved. For the Neumann Laplacian with discrete spectrum and $n \geq 5$, an analogous result holds for infinitely many Neumann eigenvalues, subject to the analogous inequality (Cor. 3.3, [2507.20330]).

## 5. Improved Kröger Inequality and Neumann Case

For the Neumann spectrum $(\gamma_j)$, Kröger's original bound is:
\[
\sum_{j=1}^{k+1} \gamma_j(\Omega) \leq \frac{n}{n+2}(2\pi)^2\omega_n^{-2/n}|\Omega|^{-2/n}k^{1+2/n}
\]
Weidl posed the question of whether a positive correction term could be added. The present quantitative improvement provides a two-term lower bound for the Neumann Riesz mean, yielding
\[
k \geq |\Omega| \omega_n (2\pi)^{-n} \gamma_k^{n/2} + C^n(\Omega) \gamma_k^{(n/2)-2}
\]
where $C^n(\Omega) = \frac{n(n+2)}{6(2\pi)^n} \omega_n J_1^4 > 0$ ([2507.20330], Theorem 3.1). This resolves Weidl's 2006 open question by providing explicit correction terms enhancing Kröger's inequality.

## 6. Geometric, Spectral, and Operator-Theoretic Applications

The improved bounds yield further applications:
- For product domains, separating variables with the two-term BLY bound refines Weyl remainders for eigenvalue sums.
- The corrections provide uniform gains in Faber-Krahn and Sperner-type inequalities under shape perturbations.
- The phase-space gradient techniques adapt to Schrödinger operators (with potential) and to the bi-Laplacian (clamped-plate problems), recovering and sharpening Melas-type improvements previously known ([2507.20330]).
- The explicit form of the correction involves domain geometry through $I(\Omega)$ and, via $J_1$, the relationship between volume and moment of inertia.

## 7. Summary Table of Principal Explicit Bounds

| Context                   | Main Inequality                                                                                                   | Correction/Second Term                    |
|---------------------------|-------------------------------------------------------------------------------------------------------------------|-------------------------------------------|
| Dirichlet Laplacian       | $\sum_{j: \lambda_j < \Lambda} (\Lambda - \lambda_j) \leq c_1 |\Omega| \Lambda^{1 + n/2}$                        | $[1 - C_1(n) J_1^2/\Lambda]^{-1}_+$       |
| Counting Function         | $k \leq c_2 \lambda_k^{n/2} |\Omega|$                                                                             | $[1 - c_3 |\Omega|/(I(\Omega) \lambda_k)]^{-1}$ |
| Neumann Laplacian         | $\sum_{j: \gamma_j < \Gamma} (\Gamma - \gamma_j) \geq c_4 |\Omega| \Gamma^{1 + n/2}$                             | $ \big[1 + c_5 J_1^4/(\Gamma^{1/2}(4J_1^2 + \Gamma)^{3/2})\big]$   |

Constants $c_i$ are given explicitly in [2507.20330].

## References

- For explicit two-term improvements, geometric dependence, and resolution of Weidl's question: [2507.20330]
- Extensions to mixed local-nonlocal operators and the fractional Laplacian: [2506.17780], [1009.4270], [1406.5407]
- Related foundational and sharp inequalities: [2507.20330], [2512.08966], [1010.2683], [1509.06705], [0909.4132]

Source: https://www.emergentmind.com/topics/berezin-li-yau-inequality