Improved higher order phase-integral approximations of the JWKB type in the vicinity of zeros and singularities of the “wave number”

This paper treats the local behaviour of certain phase-integral approximations (PIA) of arbitrary order. This is done for the vicinities of zeros and poles of the coefficient Q²(z) in the wave-equation d²ψ/dz²+Q²(z)ψ = 0. This coefficient can be complex on the real axis. A simple formula relating the cumulative error of PIA (μ-integral) to the first truncated correction in the phase-integral expansion is derived. For a simple model of a zero or pole of Q²(z), PIA of arbitrary order are determined explicitly. Their phase-integrals exhibit certain symmetry properties which justify the “loop integrals” often used in the JWKB formulae when applied in higher orders. We demonstrate the asymptotic character of the phase-integral expansion explicitly and determine its optimum order and accuracy. The well-known connection formulae are shown to retain their one-directional character in higher orders. Modifications which improve PIA at their critical points are also discussed in detail both at the modification point and elsewhere. Some quantum mechanical applications are also discussed.