Planar Dirichlet eigenvalue bounds from a trace identity

Abstract

We prove an asymptotically sharp lower bound for the eigenvalues $\lambda_k$ of the Dirichlet Laplacian on bounded open sets in the plane with area $A>0$. The proof employs a trace identity for the commutator between a certain multiplication operator and its adjoint. As a corollary, we obtain $(2/3)\cdot 4\pi k/A\leq \lambda_k$ for every $k$, which is two thirds of the bound conjectured by P\'olya.