The Weyl formula for planar annuli

Abstract

We study the zeros of cross-product of Bessel functions and obtain their approximations, based on which we reduce the eigenvalue counting problem for the Dirichlet Laplacian associated with a planar annulus to a lattice point counting problem associated with a special domain in R 2 . Unlike other lattice point problems, the one arisen naturally here has interesting features that lattice points under consideration are translated by various amount and the curvature of the boundary is unbounded. By transforming this problem into a relatively standard form and using classical van der Corput's bounds, we obtain a two-term Weyl formula for the eigenvalue counting function for the planar annulus with a remainder of size O(µ 2/3 ). If we additionally assume that certain tangent has rational slope, we obtain an improved remainder estimate of the same strength as Huxley's bound in the Gauss circle problem, namely O(µ 131/208 (log µ) 18627/8320 ). As a by-product of our lattice point counting results, we readily obtain this Huxley-type remainder estimate in the two-term Weyl formula for planar disks.

A note on the Weyl formula for balls in $\mathbb{R}^d$

“…In dimension two, there are better bounds of R B (µ). The bound O(µ 2/3 ) was given in Kuznecov and Fedosov [12] and Colin de Verdière [3], which was improved a little bit in [4] and recently to O(µ 131/208 (log µ) 18627/8320 ) in [5]. We guess that O (µ 1/2+ ) may be its true size.…”

“…Proof. When ν is a nonnegative integer, a proof is provided in the appendix of [5]. For real positive ν we recall the integral representation of J ν (x) for x > 0 ([1, p.360])…”

“…Our domain D here is simply a special case of it (with r = 0 and R = 1 there). To prove this lemma, we just need to repeat the computation of N u D 1 (µ) in the proof of [5,Theorem 4.1]. We give a sketch in the appendix.…”

“…For results on other interesting (planar) domains, see [10] for ellipses, [5] for annuli, [11] for domains of separable variables type, etc.…”