Bessel functions and Weyl's law for balls and spherical shells

TL;DR

This paper studies zeros of Bessel functions and their derivatives, providing asymptotics and bounds, and applies these results to improve Weyl's law estimates for Laplacians on balls and spherical shells.

Contribution

It offers new bounds and zero properties of Bessel functions and applies these findings to refine Weyl's law remainder estimates for Laplacians in spherical domains.

Findings

Zeros of Bessel function cross-products are real and simple.

New upper bounds for Weyl's law remainder in all dimensions.

Asymptotic properties of Bessel function zeros with uniform estimates.

Abstract

The purpose of this paper is twofold. One is to investigate the properties of the zeros of cross-products of Bessel functions or derivatives of ultraspherical Bessel functions, as well as the properties of the zeros of the derivative of the first-kind ultraspherical Bessel function. The properties we study include asymptotics (with uniform and nonuniform remainder estimates), upper and lower bounds and so on. In addition, we provide the number of zeros of a certain cross-product within a large circle and show that all its zeros are real and simple. These results may be of independent interest. The other is to investigate the Dirichlet/Neumann Laplacian on balls and spherical shells in $\mathbb{R}^d$ ($d\geq 2$) and the remainder of the associated Weyl's law. We obtain new upper bounds in all dimensions, both in the Dirichlet and Neumann cases. The proof relies on our studies of Bessel functions and the latest development in the Gauss circle problem, which was driven by the application of the emerging decoupling theory of harmonic analysis.