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Asymptotic relation for zeros of cross-product of Bessel functions and applications. (English) Zbl 1416.33008
The cross-product of Bessel functions given by \[ f_{\nu,R}(x)=J_\nu(Rx) Y_\nu(x)-J_\nu(x) Y_\nu(Rx), \] where \(\nu, x\geq0\) and \(R>1\), has an infinite number of simple zeros. The \(k\/\)th positive zero of \(f_{\nu,R}(x)\) is denoted by \(a_{\nu,k}\), \(k=1, 2, \ldots\,\). By using a known expression for \(da_{\nu,k}/d\nu\), the author shows that \[ \lim_{k\to\infty}\frac{a_{kx,k}}{k}=\alpha(x)\qquad x\geq0, \] where \(\alpha(x)\) is the unique solution of an initial value problem for a first-order differential equation. In addition, the upper bound \(a_{\nu,k}<\frac{\pi k}{R-1}+\frac{\pi\nu}{2R}\) is established.
These results are applied to the determination of the Pleijel constant Pl for planar annular regions \(A_R:=\{x\in \mathbb{R}^2,1<|x|<R\}\), \(R>1\). This quantity describes the nodal domain statistics for the eigenfunctions of the Dirichlet problem \(\Delta u=-\lambda u\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), where \(\Omega\in\mathbb{R}^2\) is a bounded domain. It is shown that \[ \mathrm{Pl}(A_R)\geq\frac{8}{R^2-1} \sup_{x>0}\,\biggl\{\frac{x}{\alpha(x)^2}\biggr\}. \] Some numerical estimates of Pl(\(A_R)\) for different \(R>1\) are given.

MSC:
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
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