New Phase-Integral Method Platform Function

TL;DR

This paper introduces a new platform function for the phase-integral method that simplifies higher-order calculations and clarifies the distinction between physical and unphysical contributions in solving differential equations.

Contribution

A novel platform function is proposed to streamline third-order calculations in the phase-integral method, enhancing its applicability and accuracy.

Findings

Simplifies third-order phase-integral calculations

Provides directly integrable conditions for phase integrals

Distinguishes physical from unphysical contributions

Abstract

The phase-integral method (PIM) is an asymptotic method of the geometrical optics or semi-classical type for solving approximately, but in many cases very accurately, a wide class of differential equations in physics. Unlike the related (J)WKB method, the higher-order corrections in the PIM can be generated from a generic, unspecified base function, providing added symmetry and flexibility. However, with the conventional approach of using the next-to-lowest (third) order correction to the integrand in the phase integral as a platform for calculating higher (fifth, seventh, ninth,...) order corrections, the higher-order calculations very often become quite complicated. We therefore introduce a new platform function, which considerably simplifies the calculation of the third-order contribution for a wide range of problems. We also present directly integrable conditions for the phase integral, which so far seem to have gone unnoticed. For a large number of observables, our analysis makes a clearer distinction between physical and, in a sense, unphysical contributions.