Berezin-Li-Yau Inequality

Updated 24 January 2026

Berezin-Li-Yau inequality is a universal semiclassical spectral bound for Dirichlet eigenvalues on bounded domains, capturing the leading term of Weyl’s law.
The method incorporates geometric features like the moment of inertia to derive explicit two-term improvements and refined eigenvalue estimates.
Recent advancements extend the inequality to Neumann cases and mixed operators, offering new insights for spectral and geometric applications.

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: $- \Delta$ 0 and for all $- \Delta$ 1,

$- \Delta$ 2

These bounds are sharp in the semiclassical limit ( $- \Delta$ 3), coinciding with Weyl's law in leading order (Gan et al., 27 Jul 2025).

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,

$- \Delta$ 4

and $- \Delta$ 5, the improved Riesz mean (Theorem 2.1, (Gan et al., 27 Jul 2025)) is: $- \Delta$ 6 where $- \Delta$ 7 has explicit values, e.g., $- \Delta$ 8, and for $- \Delta$ 9,

$0 < \lambda_1(\Omega) \leq \lambda_2(\Omega) \leq \cdots$ 0

This in turn yields a refined eigenvalue counting inequality for all $0 < \lambda_1(\Omega) \leq \lambda_2(\Omega) \leq \cdots$ 1: $0 < \lambda_1(\Omega) \leq \lambda_2(\Omega) \leq \cdots$ 2 As $0 < \lambda_1(\Omega) \leq \lambda_2(\Omega) \leq \cdots$ 3 (so $0 < \lambda_1(\Omega) \leq \lambda_2(\Omega) \leq \cdots$ 4), 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 $0 < \lambda_1(\Omega) \leq \lambda_2(\Omega) \leq \cdots$ 5 and its gradient: $0 < \lambda_1(\Omega) \leq \lambda_2(\Omega) \leq \cdots$ 6 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 ((Gan et al., 27 Jul 2025), Lemmas 2.1–2.3).

4. Consequences for Pólya's Conjecture and Neumann Spectrum

Pólya conjectured that for any domain $0 < \lambda_1(\Omega) \leq \lambda_2(\Omega) \leq \cdots$ 7,

$0 < \lambda_1(\Omega) \leq \lambda_2(\Omega) \leq \cdots$ 8

While this conjecture remains open in full generality, the refined two-term result proves that for $0 < \lambda_1(\Omega) \leq \lambda_2(\Omega) \leq \cdots$ 9, there exist infinitely many $|\Omega|$ 0 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 $|\Omega|$ 1, an analogous result holds for infinitely many Neumann eigenvalues, subject to the analogous inequality (Cor. 3.3, (Gan et al., 27 Jul 2025)).

5. Improved Kröger Inequality and Neumann Case

For the Neumann spectrum $|\Omega|$ 2, Kröger's original bound is: $|\Omega|$ 3 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

$|\Omega|$ 4

where $|\Omega|$ 5 ((Gan et al., 27 Jul 2025), 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 (Gan et al., 27 Jul 2025).
The explicit form of the correction involves domain geometry through $|\Omega|$ 6 and, via $|\Omega|$ 7, the relationship between volume and moment of inertia.