Evaluation of higher order JWKB phase integrals involving two, three or four turning points

Abstract

A method for evaluating the higher order phase integrals involving two, three and four turning points is derived and tested with integrals encountered in the symmetric double well problem. The techniques have the form of gaussian-like integration formulae requiring only values of the integrand on the real axis and a straightforward evaluation of weights. Their development allows calculations based on the higher order JWKB quantization conditions to be done with virtually the same ease now associated with use of the first order approximation.