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

TL;DR

This paper derives highly accurate asymptotic formulas for the zeros of the modified Bessel function of imaginary order, using the Lambert W function, based on quasiclassical analysis of the exponential potential.

Contribution

It introduces a new asymptotic expression for the zeros of $K_{i u}(z)$ involving Lambert W, improving accuracy over previous logarithmic approximations.

Findings

Asymptotic formula for zeros involving Lambert W function

Higher accuracy compared to previous logarithmic estimates

Practical applicability for computational purposes

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.