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

TL;DR

This paper investigates the asymptotic behavior of zeros of the cross-product of Bessel functions, deriving a differential equation for their limit distribution, and applies this to compute the Pleijel constant for planar annuli.

Contribution

It introduces a differential equation characterizing the asymptotic distribution of zeros of the cross-product of Bessel functions and applies it to explicit geometric constants.

Findings

Derived an initial value problem for the limit behavior of zeros

Established an upper bound for the zeros in terms of k, R, and ν

Computed explicit values of the Pleijel constant for annuli

Abstract

Let $a_{\nu,k}$ be the $k$-th positive zero of the cross-product of Bessel functions $J_\nu(R z) Y_\nu(z) - J_\nu(z) Y_\nu(R z)$, where $\nu\geq 0$ 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: $$ \lim_{k \to \infty} \frac{a_{kx,k}}{k} = \alpha(x), \quad x \geq 0. $$ 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.