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arXiv · 2610.05636

Planar Dirichlet eigenvalue bounds from a trace identity

Romain Speciel
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Abstract

We prove an asymptotically sharp lower bound for the eigenvalues $λ_k$ 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 $(2/3)\cdot 4πk/A\leq λ_k$ for every $k$, which is two thirds of the bound conjectured by Pólya.

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Romain Speciel. 2026-10-05. Planar Dirichlet eigenvalue bounds from a trace identity. https://arxiv.org/abs/2610.05636

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