Higher orders of semiclassical expansion for the one-dimensional Schrödinger equation

The divergence of a semiclassical series of wave functions due to the presence of one or two turning points is investigated. The asymptotic expansion of higher semiclassical corrections ${\phi }_n$ with respect to the inverse power of the order number n is obtained. Amazingly, the expansion terms are subject to the same recurrence relations as the terms of the initial expansion with respect to Planck’s constant ħ. Quantitative criteria for obtaining maximally accurate semiclassical wave functions are derived.