Correction for the Uniform WKB Wave Functions and the WKB Phase Shifts

A method of improving the uniform WKB solution for single turning point problems is discussed and the correction formulas for the WKB phase shifts are obtained. The method involves the equation of motion which is in a form similar to that used by Fröman and Fröman to obtain the connection formulas in the WKB approximation. A sequence of solutions to the equation of motion is obtained in a remarkably compact form—and in a novel way—by utilizing the involutionality of certain 2 × 2 matrices appearing in the equations. A numerical test of the result thus obtained is presented in comparison with the corresponding numerical solution result. Unlike the ordinary WKB method, the present method is completely free of the connection formulas of the WKB approximation.