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