Recently, I’ve submitted a preprint to arXiv, where I studied some properties of zeros of the cross-product of Bessel functions

\begin{equation} \label{eq:cp} J_\nu(R z) Y_\nu(z) - J_\nu(z) Y_\nu(R z), \quad \nu\geq 0, ~R>1. \end{equation}

This research originated from the previous post, where I studied the explicit expression (and value) of the Pleijel constant for the disk; see also this preprint. Recall that the expression for $Pl(B)$ crucially relies on the result of Elbert & Laforgia  (see also ) on the asymptotic $j_{kx,k}$ as $k\to \infty$ for the zero $j_{kx,k}$ of the Bessel function $J_{kx}$. More precisely, Elbert & Laforgia  shown that

$\lim_{k \to \infty} \frac{j_{kx,x}}{k} = \iota(x), \quad x > -1,$

where $\iota(x)$ is a unique solution of the intial value problem

$\frac{dy}{dx} = \frac{\arccos\left(\frac{x}{y}\right)}{\sqrt{1 - \left(\frac{x}{y}\right)^2}}, \qquad y(0) = \pi.$

In fact, $\iota(x)$ admits a closed-form representation in terms of a solution of a transcendental equation (2.1) in ; see also Section 1.5 in .

However, if one would like to obtain an explicit expression for the Pleijel constant of annuli in a similar fashion as for the disk, then a similar asymptotic relation as of Elbert & Laforgia but for zeros of \eqref{eq:cp} is needed. (Let me recall that eigenvalues of the Dirichlet Laplace operator on the annulus $A_{R} = \{x \in \mathbb{R}^2:~ 1 < |x| < R\}$ are given by $\lambda_{\nu,k} = a_{\nu,k}^2$, where $a_{\nu,k}$ is the $k$-th positive zero of \eqref{eq:cp}, $\nu \in \mathbb{N}$.) Unfortunately, I was not able to find any appropriate result for $a_{\nu,k}$ in the literature. Moreover, it really seems that the theory of zeros of \eqref{eq:cp} is much much less developed than the theory of zeros of $J_{\nu}$, and only few bounds and asymptotic results are known. (But, from eigenbusiness point of view, zeros of \eqref{eq:cp} are no less important than zeros of $J_\nu$.)

So, in the submitted manuscript, I tried to fill in the absence of results for $a_{\nu,k}$. First, the following theorem is proved:

Theorem. Let $R> 1$. Then

$\lim_{k \to \infty} \frac{a_{kx,k}}{k} = \alpha(x), \quad x \geq 0 ,$

where $\alpha(x)$ is a unique solution of the initial value problem

$\frac{dy}{dx} = \frac{\arccos\left(\frac{x}{Ry}\right) - \mathrm{Ac}\left(\frac{x}{y}\right)}{R\,\sqrt{1 - \left(\frac{x}{Ry}\right)^2} - \mathrm{Sr}\bigg(1-\Big(\frac{x}{y}\Big)^2\bigg)}, \qquad y(0) = \frac{\pi}{R-1},$

for $x \geq 0$. Here, the functions $\mathrm{Ac}$ and $\mathrm{Sr}$ are zero extensions of $\arccos$ and square root, respectively, defined as

\mathrm{Ac}(t) = \left\{ \begin{aligned} \arccos t \quad \mathrm{for} \quad &|t| \leq 1,\\ 0 \quad \mathrm{for} \quad &|t| > 1, \end{aligned} \right. \qquad \mathrm{Sr}(t) = \left\{ \begin{aligned} \sqrt{t} \quad \mathrm{for} \quad &t \geq 0,\\ 0 \quad \mathrm{for} \quad &t < 0. \end{aligned} \right.

Using this theorem, we get the following explicit expression for the Pleijel constant for the annuli:

$Pl(A_{R}) = \frac{8}{R^2-1} \, \sup_{x>0} \left\{\frac{x}{\alpha(x)^2}\right\},$

provided any sufficiently large eigenvalue $\lambda_k$ on $A_R$ has the multiplicity at most two. In the case of a higher multiplicity, $Pl(A_{R})$ is estimated by a given expression from below. (Note that a higher multiplicity can hypothetically occur, see the previous post.)

There is one main ingredient for the proof of Theorem. It is the formula of Willis  for the derivative of $a_{\nu,k}$ with respect to $\nu$:

\begin{align} \notag \frac{da_{\nu,k}}{d\nu} &= \frac{2a_{\nu,k}(J_\nu^2(a_{\nu,k})+Y_\nu^2(a_{\nu,k}))\int_0^\infty K_0(2Ra_{\nu,k} \sinh t) e^{-2\nu t} \, dt}{(J_\nu^2(a_{\nu,k})+Y_\nu^2(a_{\nu,k})) - (J_\nu^2(Ra_{\nu,k})+Y_\nu^2(Ra_{\nu,k}))} \newline \notag &-\frac{2a_{\nu,k} (J_\nu^2(Ra_{\nu,k})+Y_\nu^2(Ra_{\nu,k}))\int_0^\infty K_0(2a_{\nu,k} \sinh t) e^{-2\nu t} \, dt}{(J_\nu^2(a_{\nu,k})+Y_\nu^2(a_{\nu,k})) - (J_\nu^2(Ra_{\nu,k})+Y_\nu^2(Ra_{\nu,k}))}, \end{align}

where $K_0$ is the modified Bessel function of the second kind and zero order.

The idea of the proof of Theorem consists, roughly speaking, on the passage to the limit in Willis’ formula as $\nu \to \infty$, and it was inspired by the approach of Elbert & Laforgia . It is interesting to mention that, according to the biographical obituary of A. Elbert written by M. Muldoon, it was exactly the idea of Elbert to use the formulas for the derivative of zeros with respect to the order to obtain various important properties of these zeros. Then, this approach was developed in the collaboration with A. Laforgia.

Using the formula of Willis, the following upper bound is also proved:

$a_{\nu,k} < \frac{\pi k}{R-1} + \frac{\pi \nu}{2 R}.$

In fact, much more can be said about $a_{\nu,k}$ via Willis’ formula. Hopefully, this will be the subject of a further research.