Recently I've learned something about the spectra of the Laplacians. Given a bounded domain $\Omega \subset \mathbb{R}^n$ with $\partial \Omega$ smooth, we can consider eigenfunctions of Dirichlet type, i.e. $u \in C^2(\Omega)\cap C(\partial \Omega)$ s.t. $-\triangle u=\lambda u$ and $u|_{\partial \Omega}=0$. By standard results in functional analysis, $-\triangle$ has a discrete spectrum $0<\lambda_1 \leq \lambda_2 \leq \cdots$ with $\lambda_k \to +\infty$.

A well-known asymptotic formula by Weyl says $\displaystyle \lambda_k \sim W_n (\frac{k}{V(\Omega)})^{2/n}$. We refer $W_n$ as the Weyl constant. And Pólya conjectured that $\displaystyle \lambda_k \geq W_n (\frac{k}{V(\Omega)})^{2/n}$ holds.

As far as I know, the best known result is due to Li and Yau. They proved the conjecture in the sense of "average": $\displaystyle \sum_{j=1}^k \lambda_j \geq \frac{nW_n}{n+2}{k}^{(n+2)/n}{V(\Omega)}^{-n/2}$.

I find their argument is elementary, only employing some standard Fourier tricks. And the big picture is quite clear if put into the quantum framework. My question is in some sense "soft", but it does make me feel absurd: what makes it so difficult to estimate the eigenvalues one by one while their average is so well understood? Does anyone work on this problem by carrying further Li & Yau's analysis? I do know one instance: Kröger has transplanted their proof to Neumann settings, but how about the original Dirichlet problem?