An improved remainder estimate in the Weyl formula for the planar disk

TL;DR

This paper improves the estimate of the remainder term in the Weyl formula for the eigenvalue counting function of the Dirichlet Laplacian on a planar disk, achieving a sharper bound using exponential sum estimation.

Contribution

It introduces a refined remainder estimate of order $O(\lambda^{2/3-1/495})$, enhancing previous results by Colin de Verdière.

Findings

Sharper remainder estimate $O(\lambda^{2/3-1/495})$

Method combines exponential sum estimation with spectral analysis

Improves understanding of eigenvalue distribution for the disk

Abstract

In \cite{colin}, Y. Colin de Verdi\`ere proved that the remainder term in the two-term Weyl formula for the eigenvalue counting function for the Dirichlet Laplacian associated with the planar disk is of order $O(\lambda^{2/3})$. In this paper, by combining with the method of exponential sum estimation, we will give a sharper remainder term estimate $O(\lambda^{2/3-1/495})$.