Asymptotic relation for zeros of cross-product of Bessel functions and applications

Abstract

Let $a_{\nu ,k}$ be the $k$-th positive zero of the cross-product of Bessel functions $J_{\nu }(Rz)Y_{\nu }(z)-J_{\nu }(z)Y_{\nu }(Rz)$, where $\nu \ge 0$ and $R>1$. We derive an initial value problem for a first order differential equation whose solution $\alpha (x)$ characterizes the limit behavior of $a_{\nu ,k}$ in the following sense: $$\underset{k\to \infty }{\lim }\frac{a_{kx,k}}{k}=\alpha (x),x\ge 0.$$ Moreover, we show that $$a_{\nu ,k}<\frac{\pi k}{R-1}+\frac{\pi \nu }{2R}.$$ We use $\alpha (x)$ to obtain an explicit expression of the Pleijel constant for planar annuli and compute some of its values.

arXiv comment16 pages, 2 figures. Remark 3.2 is added; some proofs are expanded and figures are improved according to referee's suggestions. Accepted to Journal of Mathematical Analysis and Applications

References in corpus (4)

• Nodal domains statistics - a criterion for quantum chaos
• On some unexpected properties of radial and symmetric eigenvalues and eigenfunctions of the $p$-Laplacian on a disk
• Nodal domains, spectral minimal partitions, and their relation to Aharonov-Bohm operators
• On exact Pleijel's constant for some domains

Cited by in corpus (4)

• Quantum-information theory of a Dirichlet ring with Aharonov-Bohm field
• The Weyl formula for planar annuli
• Two-parameter localization and related phase transition for a Schrödinger operator in balls and spherical shells
• Zeros of Bessel cross-products coming from oblique derivative boundary value problems