Resummed Wentzel-Kramers-Brillouin Series: Quantization and Physical Interpretation

TL;DR

This paper introduces an exact resummation formula for the divergent WKB series in two turning point problems, providing a clear physical interpretation and improving the predictive power of wave analysis.

Contribution

It presents a novel closed-form resummation of the WKB series, linking it to physical quantities like transmissivity and quantization conditions.

Findings

Exact resummation formula for WKB series derived

Physical interpretation in terms of wave transmissivity established

Connection to Bohr-Sommerfeld quantization clarified

Abstract

The Wentzel-Kramers-Brillouin (WKB) perturbative series, a widely used technique for solving linear waves, is typically divergent and at best, asymptotic, thus impeding predictions beyond the first few leading-order effects. Here, we report a closed-form formula that exactly resums the perturbative WKB series to all-orders for two turning point problem. The formula is elegantly interpreted as the action evaluated using the product of spatially-varying wavenumber and a coefficient related to the wave transmissivity; unit transmissivity yields the Bohr-Sommerfeld quantization.