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

Abstract

The paper presents the derivation of the asymptotic behavior of $\nu$-zeros of the modified Bessel function of imaginary order $K_{\mathrm{i}\nu }(z)$. 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.

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