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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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##### References:
  Blum, G.; Gnutzmann, S.; Smilansky, U., Nodal domains statistics: a criterion for quantum chaos, Phys. Rev. Lett., 88, 11, Article 114101 pp., (2002)  Bobkov, V., On exact Pleijel’s constant for some domains, Doc. Math., 23, 799-813, (2018) · Zbl 1401.35214  Bobkov, V.; Drábek, P., On some unexpected properties of radial and symmetric eigenvalues and eigenfunctions of the p-Laplacian on a disk, J. Differential Equations, 263, 3, 1755-1772, (2017) · Zbl 1366.35109  Bonnaillie-Noël, V.; Helffer, B.; Hoffmann-Ostenhof, T., Nodal domains, spectral minimal partitions, and their relation to Aharonov-Bohm operators, IAMP News Bull., 3-28, (October 2017)  Cochran, J. A., Remarks on the zeros of cross-product Bessel functions, J. Soc. Ind. Appl. Math., 12, 3, 580-587, (1964) · Zbl 0132.05601  Cochran, J. A.; Pecina, R. G., Mode propagation in continuously curved waveguides, Radio Sci., 1, 6, 679-696, (1966)  Elbert, Á., Some recent results on the zeros of Bessel functions and orthogonal polynomials, J. Comput. Appl. Math., 133, 1-2, 65-83, (2001) · Zbl 0989.33004  Elbert, Á.; Laforgia, A., An asymptotic relation for the zeros of Bessel functions, J. Math. Anal. Appl., 98, 2, 502-511, (1984) · Zbl 0549.33007  Hartman, P., Ordinary Differential Equations, (1982), Birkhäuser: Birkhäuser Boston, Basel, Stuttgart · Zbl 0125.32102  Helffer, B.; Hoffmann-Ostenhof, T., A review on large k minimal spectral k-partitions and Pleijel’s Theorem, (Spectral Theory and Partial Differential Equations. Spectral Theory and Partial Differential Equations, Contemp. Math., vol. 640, (2015), AMS), 39-57 · Zbl 1346.35132  Kerimov, M. K., Studies on the zeros of Bessel functions and methods for their computation, Comput. Math. Math. Phys., 54, 9, 1337-1388, (2014) · Zbl 1326.33008  Kline, M., Some Bessel equations and their applications to guide and cavity theory, Stud. Appl. Math., 27, 1-4, 37-48, (1948) · Zbl 0031.29802  Lewis, J. T.; Muldoon, M., Monotonicity and convexity properties of zeros of Bessel functions, SIAM J. Math. Anal., 8, 1, 171-178, (1977) · Zbl 0365.33004  McCann, R. C., Lower bounds for the zeros of Bessel functions, Proc. Amer. Math. Soc., 64, 1, 101-103, (1977) · Zbl 0364.33009  McMahon, J., On the roots of the Bessel and certain related functions, Ann. of Math., 9, 1/6, 23-30, (1894) · JFM 25.0842.02  Neuman, E.; Sandor, J., On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities, Math. Inequal. Appl., 13, 4, 715-723, (2010) · Zbl 1204.26023  Pleijel, A., Remarks on Courant’s nodal line theorem, Comm. Pure Appl. Math., 9, 3, 543-550, (1956) · Zbl 0070.32604  Polterovich, I., Pleijel’s nodal domain theorem for free membranes, Proc. Amer. Math. Soc., 137, 3, 1021-1024, (2009) · Zbl 1162.35005  Watson, G. N., A Treatise on the Theory of Bessel Functions, (1944), The University Press: The University Press Cambridge · Zbl 0063.08184  Willis, D. M., A property of the zeros of a cross-product of Bessel functions, Math. Proc. Cambridge Philos. Soc., 61, 2, 425-428, (1965) · Zbl 0131.07204
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