Improved semiclassical eigenvalue estimates for the Laplacian and the Landau Hamiltonian

TL;DR

The paper proves improved semiclassical bounds for the Riesz means of eigenvalues of the Dirichlet/Neumann Laplacian on finite-measure domains and, in two dimensions, the magnetic Landau Hamiltonian. An uncertainty-principle approach, built on thickness-based spectral inequalities and the regularized inradius $\rho_\theta(\Omega)$, yields multiplicative exponential improvements to the classical Berezin–Li–Yau and Kröger bounds, valid for all open $\Omega$ of finite measure. The core method uses abstract Dirichlet/Neumann trace identities and quantitative bounds on the high-energy leakage of eigenfunctions, then extends the γ=1 results to all γ>1 via the Aizenman–Lieb formula. In the magnetic 2D setting, the results involve the Landau levels $B(2k-1)$ and retain a similar exponential improvement with an additional magnetic term $\rho_\theta(\Omega)^2 B$ in the exponent. Collectively, the work delivers a unified framework for improved semiclassical estimates applicable to a broad class of domains and, notably, to the Landau Hamiltonian in dimension two.

Abstract

The Berezin--Li--Yau and the Kröger inequalities show that Riesz means of order $\geq 1$ of the eigenvalues of the Laplacian on a domain $Ω$ of finite measure are bounded in terms of their semiclassical limit expressions. We show that these inequalities can be improved by a multiplicative factor that depends only on the dimension and the product $\sqrtΛ|Ω|^{1/d}$, where $Λ$ is the eigenvalue cut-off parameter in the definition of the Riesz mean. The same holds when $|Ω|^{1/d}$ is replaced by a generalized inradius of $Ω$. Finally, we show similar inequalities in two dimensions in the presence of a constant magnetic field.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Remark 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Lemma 10