An Improved Remainder Estimate in the Weyl Formula for the Planar Disk

Abstract

Y. Colin de Verdière 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 no more than $O(\mu ^{2/3})$. In this paper, we study this problem from spectral geometry by using analysis and number theory. More precisely, by combining with the method of exponential sum estimation, we give a sharper remainder term estimate $O(\mu ^{2/3-1/495})$.

Appendix A. A Useful Lemma

Here is a quantitative version of the inverse function theorem. It is routine to prove by following the standard argument.

Lemma A.1

Suppose that f is a $C^{(k)}$ ($k\geqslant 2$) mapping from an open set $\Omega \subset \mathbb {R}^d$ into $\mathbb {R}^d$ and $b=f(a)$ for some $a\in \Omega$. Assume $|\det (\nabla f(a))|$$\geqslant$c and for any $x\in \Omega$,

$$\begin{aligned} |D^{\nu } f_i(x)|\leqslant C \quad \quad \text {for }|\nu |\leqslant 2, 1\leqslant i\leqslant d. \end{aligned}$$

If $r_0\leqslant \sup \{r>0: B(a, r)\subset \Omega \}$ then f is bijective from $B(a, r_1)$ to an open set containing $B(b, r_2)$ where

$$\begin{aligned} r_1=\min \left\{ \frac{c}{2d^{2} d! C^d}, r_0\right\} \quad \text {and}\quad r_2=\frac{c}{4d!C^{d-1}}r_1. \end{aligned}$$

The inverse mapping $f^{-1}$ is also in $C^{(k)}$.