Asymptotics of radial wave equations

The Langer modification is an improvement in the WKB analysis of the radial Schrödinger equation. We derive a generalization of the Langer modification to any radial operator. For differential operators we write the modified classical symbols explicitly and show that the WKB wavefunctions with the modification have the exact limiting behavior for small radius. Unlike in the Schrödinger case, generally the modified radial analysis is not equivalent to the WKB analysis of the full problem before reduction by the spherical symmetry.