Improved leading coefficient in the individual Berezin-Li-Yau bound via energy orthogonality

Abstract

Let $λ_k$ be the $k$th eigenvalue of the Dirichlet Laplacian on an open set $Ω\subset\mathbb R^n$ of finite positive measure. The direct individual consequence of the Berezin--Li--Yau sum inequality has leading coefficient $n/(n+2)$ relative to the Weyl term. The main contribution of this paper is a strict improvement of this coefficient. Energy orthogonality gives a frequency-dependent cap on the Fourier density of the first $k$ eigenfunctions. Combining this cap with the standard Bessel bound and a bathtub principle for the radial capacity yields \[ λ_k\geq c_n(2π)^2ω_n^{-2/n} |Ω|^{-2/n}k^{2/n}, \qquad \frac{n}{n+2}<c_n<1, \] for every $k\geq1$ and every $n\geq2$, without boundary regularity. The constants are characterized by explicit scalar equations; in dimension two, $c_2=0.5383068077\ldots$, giving a $7.66\%$ improvement over the individual Li--Yau coefficient. We emphasize that this improves the leading coefficient in the individual eigenvalue bound and the new constant $c_n$ is independent of the geometry and index $k$.