Observations on the JWKB treatment of the quadratic barrier

Historically, the “lowest-order” JWKB “parabolic” connection formula between the left and right classically allowed regions for tunneling through a parabolic barrier involved a rather non-JWKB-like square-root $$ \sqrt {1 + e^{ - 2\pi \left( { - E} \right)/\hbar } } $$ and a non-JWKB phase factor that had been extracted from the asymptotic expansion of the parabolic cylinder function. Generalization to higher order was not obvious. We show how the usual JWKB connection formulas at the linear turning points, combined with matching in a common Stokes region when ħ is complex, lead to the historical formula and its generalization. The limit of real ħ is tricky, because Stokes lines coalesce, and the common Stokes region that joins the two turning points disappears. Certain Stokes lines are thus “doubled,” but which ones depend on the sign of arg ħ → ±0. The square root and phase arise from the Borel sum of the “normalization factors” $$ e^{ \pm i\sum\nolimits_{n = 1}^\infty {S_R^{\left( n \right)} \left( \infty \right)\hbar ^{2n - 1} } } $$ , which as conjectured by Sato are summed by a gamma function. Real ħ is a Stokes line for these factors, causing the Borel sums of a single JWKB wave function even in the classically allowed region to be different for arg ħ → ±0. A proof of Sato’s conjecture is given in an Addendum.