Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 057, 7 pages      arXiv:2103.01732      https://doi.org/10.3842/SIGMA.2021.057

Asymptotic Estimation for Eigenvalues in the Exponential Potential and for Zeros of $K_{{\rm i}\nu}(z)$ with Respect to Order

Yuri Krynytskyi and Andrij Rovenchak
Department for Theoretical Physics, Ivan Franko National University of Lviv, Ukraine

Received May 15, 2021, in final form June 01, 2021; Published online June 10, 2021

Abstract
The paper presents the derivation of the asymptotic behavior of $\nu$-zeros of the modified Bessel function of imaginary order $K_{{\rm i}\nu}(z)$. This derivation is based on the quasiclassical treatment of the exponential potential on the positive half axis. The asymptotic expression for the $\nu$-zeros (zeros with respect to order) contains the Lambert $W$ function, which is readily available in most computer algebra systems and numerical software packages. The use of this function provides much higher accuracy of the estimation comparing to known relations containing the logarithm, which is just the leading term of $W(x)$ at large $x$. Our result ensures accuracies sufficient for practical applications.

Key words: quasiclassical approximation; exponential potential; $\nu$-zeros; modified Bessel functions of the second kind; imaginary order; Lambert $W$ function.

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