标题： 贝塞尔函数和球体及球壳的韦尔定律 显示英文标题

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

摘要： 本文的目的有两方面。 一方面是研究贝塞尔函数交叉乘积的零点或超球贝塞尔函数导数的零点的性质，以及第一类超球贝塞尔函数导数的零点的性质。 我们研究的性质包括渐近性（具有均匀和非均匀余项估计）、上下界等。 此外，我们给出了某个交叉乘积在大圆内的零点个数，并证明了其所有零点都是实数且简单。 这些结果可能具有独立的兴趣。 另一方面是研究在$\mathbb{R}^d$ ($d\geq 2$) 中球体和球壳上的狄利克雷/诺伊曼拉普拉斯算子以及相关韦尔定律的余项。 我们在所有维度上都得到了新的上界，包括狄利克雷和诺伊曼情况。 证明依赖于我们对贝塞尔函数的研究以及高斯圆问题的最新发展，这由调和分析的新兴解耦理论的应用所推动。 显示英文摘要

摘要： 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.