[Paper Review] Programs in Mathematica relevant to Phase Integral Approximation for coupled ODEs of the Schrödinger type

A. A. Skorupski|ArXiv.org|Oct 29, 2007

Numerical methods for differential equations3 citations

TL;DR

This paper presents three Mathematica programs—Y2n, bvm, and Ymsvm—that automate the computation of phase integral approximation (PIA) corrections for one or two coupled Schrödinger-type ordinary differential equations. The tools generate symbolic expressions for higher-order corrections $Y_{2n}(x)$, vectors $<\mathbf{b}_m(x)\u0026gt;$, and auxiliary quantities, using direct summation of recurrence relations to ensure accuracy and minimize programming errors.

ABSTRACT

Three programs in Mathematica are presented, which produce expressions for the lowest order and the higher order corrections of the Phase Integral Approximation. First program is pertinent to one ordinary differential equation of the Schrödinger type. The remaining two refer to a set of two such equations.

Motivation & Objective

• To automate the symbolic computation of phase integral approximation (PIA) corrections for Schrödinger-type ODEs.
• To provide reliable, error-minimized code for higher-order corrections in scalar and coupled systems.
• To support both general and specific input cases, including trigonometric and complex potentials.
• To enable verification through consistency checks, such as vanishing odd-order corrections and agreement between Hermitian and non-Hermitian formulations.
• To facilitate numerical evaluation and result simplification via customizable output formats (TeX, Fortran, OutputForm).

Proposed method

• Implement three Mathematica programs: Y2n for scalar equations, bvm and Ymsvm for coupled systems.
• Use direct nested Do-loops to compute multiple sums from recurrence relations (e.g., Eq. 40, 54), preserving mathematical clarity.
• Allow user input via external .dat files or default parameters, with interactive dialog for output format and simplification options.
• Support symbolic manipulation with Expand, Simplify, Together, and Apart, with optional Trig expansion for trigonometric functions.
• Preserve full symbolic integrity by avoiding premature simplification of integrals (e.g., Eq. 105), favoring correctness over speed.
• Enable appending custom scripts (e.g., ap.mat) for targeted numerical evaluation and result verification.

Experimental results

Research questions

• RQ1 How can symbolic computation of phase integral corrections for coupled ODEs be automated with high reliability?
• RQ2 What is the role of multiple-sum recurrence relations in computing higher-order corrections $Y_{2n}(x)$ and associated vectors?
• RQ3 How can consistency checks—such as vanishing odd-order corrections—validate the correctness of the implementation?
• RQ4 In what way do the programs ensure agreement between Hermitian and non-Hermitian formulations of the same problem?
• RQ5 How can user-defined inputs and output formats be flexibly integrated into a symbolic computation workflow?

Key findings

• The programs successfully compute symbolic expressions for $Y_{2n}(x)$, $<\mathbf{b}_m(x)\u0026gt;$, and $<\mathbf{s}_m(x)\u0026gt;$ up to desired order, with results saved in multiple formats.
• The vanishing of odd-order corrections ($Y_{2n-1}(x) \equiv 0$) was confirmed through cancellation of many terms, validating the code’s correctness.
• In Hermitian cases, identical results were obtained from both simplified and non-simplified formulations, confirming consistency.
• The use of direct summation (e.g., nested Do loops) minimized programming errors and preserved mathematical fidelity.
• Numerical evaluation via appended scripts (e.g., ap.mat) confirmed agreement with published results in Table I of the reference paper.
• The framework supports complex and trigonometric potentials (e.g., Example C.4 with $\cos \phi \pm i \sin \phi$), enabling broad applicability.