Bound State Energies using Phase Integral Analysis

The study of asymptotic properties of solutions to differential equations has a long and arduous history, with the most significant advances having been made in the development of quantum mechanics. A very powerful method of analysis is that of Phase Integrals, described by Heading. Key to this analysis are the Stokes constants and the rules for analytic continuation of an asymptotic solution through the complex plane. These constants are easily determined for isolated singular points, by analytically continuing around them and, in the case of analytic functions, requiring the asymptotic solution to be single valued. However, most interesting problems of mathematical physics involve several singular points. By examination of analytically tractable problems and more complex bound state problems involving multiple singular points, we show that the method of Phase Integrals can greatly improve the determination of bound state energies over the simple WKB values. We also find from these examples that in the limit of large separation the Stokes constant for a first order singular point approaches the isolated singular point value.