Resummed Wentzel-Kramers-Brillouin series: Quantization and physical interpretation

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 closed-form formula that exactly resums WKB series to all orders two turning point problems. elegantly interpreted as action evaluated using product of spatially varying wave number and coefficient related transmissivity; unit transmissivity yields Bohr-Sommerfeld quantization.