Weyl's law

Remarks

For a cube the eigenvalues are explicit, and counting them amounts to counting lattice points in a ball of radius $\sqrt{\lambda}$, which gives the formula. For a general set one fills $\Omega$ from inside and covers it from outside by small cubes. Then one compares eigenvalues with Dirichlet and Neumann conditions on the cubes; this is Dirichlet–Neumann bracketing. Example: for $\Omega = (0, L) \subseteq \R$, the eigenvalues are $(k\pi / L)^2$ for $k \ge 1$, so $N(\lambda) = \floor{L \sqrt{\lambda} / \pi}$. This agrees with the formula, since $\omega_1 = 2$ and $\frac{2L}{2\pi} = \frac{L}{\pi}$. Equivalently, $\lambda_k \sim \frac{(2\pi)^2}{(\omega_d \abs{\Omega})^{2/d}} \, k^{2/d}$ as $k \to \infty$. Hermann Weyl proved the law in 1911.

Going deeper

For a student

Weyl's law shows that the volume of a domain can be read off from its spectrum. So it gives a first positive answer to the question whether one can hear the shape of a drum. The same leading term holds for the Neumann condition when the boundary is regular enough, and an analogous law holds on compact Riemannian manifolds.