Bounding Eigenvalues with Packing Density

We prove a lower bound on the eigenvalues ${\lambda }_k$, $k\in \mathbb{N}$, of the Dirichlet Laplacian of a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ of volume $V$: $${\lambda }_k\ge C_n(\delta \frac{k}{V}{)}^{2\mathrm{/}n}$$ where $\delta$ is a constant that measures how efficiently $\mathrm{\Omega }$ can be packed into ${\mathbb{R}}^n$ and $C_n$ is the constant found in Weyl’s law. This generalizes a result of Urakawa in 1984. If ${\delta }^{2\mathrm{/}n}>n\mathrm{/}(n+2)$, this bound is stronger than the eigenvalue bound proven by Li and Yau in 1983. For example, in the case of convex planar domains, we have for all $k\in \mathbb{N}$, $${\lambda }_k\ge \frac{2\sqrt{3}\pi k}{V}\mathrm{.}$$