[Paper Review] Using a computer algebra system to simplify expressions for Titchmarsh-Weyl m-functions associated with the Hydrogen Atom on the half line

Cecilia Knoll, Charles T. Fulton|ArXiv.org|Dec 29, 2008

Quantum chaos and dynamical systems1 references3 citations

TL;DR

This paper simplifies complex expressions for Titchmarsh-Weyl m-functions in the radial hydrogen atom problem on the half-line using computer algebra systems. By decomposing the m-function into real and imaginary parts, it derives a polynomial representation for the real component, proving its reality via two independent Mathematica programs that verify consistency up to ℓ=30.

ABSTRACT

In this paper we give simplified formulas for certain polynomials which arise in some new Titchmarsh-Weyl m-functions for the radial part of the separated Hydrogen atom on the half line and two independent programs for generating them using the symbolic manipulator Mathematica.

Motivation & Objective

• To simplify the expression for the Titchmarsh-Weyl m-function associated with the radial hydrogen atom on (0, ∞).
• To decompose the m-function into real and imaginary parts, isolating a polynomial component rℓ(λ)/(2ℓ+1).
• To verify the reality of the derived polynomial rℓ(λ) using symbolic computation.
• To provide two independent computational implementations in Mathematica to validate the simplification.
• To establish a foundation for further analytic proof of the polynomial’s reality for all ℓ ≥ 1.

Proposed method

• Uses Frobenius series solutions and Whittaker functions to express the m-function mℓ(λ) in terms of special functions and Pochhammer symbols.
• Applies the identity mℓ(λ) = −akℓ(λ){⋯} + (i√λ)²ℓ⁺¹/((2ℓ+1)!) × ∑(⋯) to separate real and imaginary parts.
• Introduces a decomposition mℓ(λ) = i√λ kℓ(λ) + rℓ(λ)/(2ℓ+1), defining rℓ(λ) as the real part.
• Employs symbolic manipulation in Mathematica to compute and verify the reality of rℓ(λ)/(2ℓ+1) for ℓ = 1 to 4 and up to ℓ = 30.
• Derives polynomial representations of Pochhammer symbols (−ℓ−t)k and products ∏(λ + a²/(4j²)) to express coefficients explicitly.
• Uses combinatorial identities and coefficient extraction (α(k,n), γ(m,n)) to construct the real polynomial component.

Experimental results

Research questions

• RQ1 Can the complex expression for the Titchmarsh-Weyl m-function of the radial hydrogen atom be simplified into real and imaginary parts?
• RQ2 Is the real part of the m-function expression representable as a polynomial rℓ(λ) of degree ℓ?
• RQ3 Can the reality of rℓ(λ) be computationally verified for arbitrary ℓ using symbolic algebra?
• RQ4 Do two independent computational implementations of the real part yield identical results across multiple ℓ values?
• RQ5 Can a general analytic proof of the reality of rℓ(λ) be constructed, or is it limited to computational verification?

Key findings

• The m-function for the radial hydrogen atom is decomposed into mℓ(λ) = i√λ kℓ(λ) + rℓ(λ)/(2ℓ+1), where rℓ(λ) is a real polynomial of degree ℓ.
• For ℓ=1, rℓ(λ)/(2ℓ+1) = −a t / 36; for ℓ=2, it is −a³t/7200 − 13a t²/7200; for ℓ=3, −a⁵t/8467200 − 23a³t²/4233600 − a t³/21168.
• Two independent Mathematica programs compute rℓ(λ)/(2ℓ+1) and produce identical results for ℓ = 1 to 4 and up to ℓ = 30.
• The imaginary part of the expression for rℓ(λ)/(2ℓ+1) vanishes identically in all tested cases, confirming its reality.
• The coefficients α(k,n) and γ(m,n) required for the polynomial representation remain analytically intractable, leaving a general proof as an open problem.
• Despite the lack of a closed-form expression for the coefficients, the computational verification provides strong evidence for the validity of the decomposition.