[Paper Review] Estimating Coefficients of Frobenius Series by Legendre Transform and WKB Approximation

Amna Noreen, Olaussen, Kåre|arXiv (Cornell University)|May 10, 2012

Quantum Mechanics and Non-Hermitian Physics3 references3 citations

TL;DR

This paper presents a method to accurately estimate the coefficients of Frobenius series solutions to second-order linear ODEs with regular singular points by combining WKB approximation with Legendre transformation. The approach predicts the logarithmic magnitude of coefficients with high precision, enabling efficient high-accuracy computation of series solutions by estimating required computational precision and number of terms.

ABSTRACT

The Frobenius method can be used to represent solutions of ordinary differential equations by (generalized) power series. It is useful to have prior knowledge of the coefficients of this series. In this contribution we demonstrate that the magnitude of the coefficients can be predicted to surprisingly high accuracy by a Legendre transformation of WKB approximated solutions to the differential equations.

Motivation & Objective

• To predict the magnitude of coefficients in Frobenius series solutions to second-order ODEs with regular singular points.
• To reduce computational cost and roundoff errors in high-precision evaluations of power series by estimating required precision and number of terms in advance.
• To develop a practical method for setting computational parameters (precision, number of terms) before running high-accuracy solvers.
• To validate the accuracy of coefficient predictions using WKB approximation combined with Legendre transformation, especially for anharmonic and double-well oscillators.

Proposed method

• Apply the Frobenius method to represent solutions as generalized power series: $\psi(z) = \sum_{m=0}^\infty a_m z^{m+\nu}$.
• Use the WKB approximation to estimate the solution $\psi(z)$, particularly for large $|z|$, incorporating Langer correction for regular singular points.
• Define $x = e^u$ and $|a_m| = e^{s(m)}$, then relate the logarithmic magnitude of coefficients $s(m)$ to the maximum modulus of $\psi(z)$ via $S(u) = \max_\varphi |\psi(e^{u+i\varphi})|$.
• Perform a Legendre transformation between $s(m)$ and $S(u)$, using the relation $S_0(u) = S(u) - \frac{1}{2}\log(2\pi S''(u))$ to account for curvature corrections.
• Derive $s(m)$ from $S(u)$ via $s(m) = S_0(u) - u S_0'(u)$, with $m = S_0'(u)$, enabling coefficient magnitude prediction.
• Apply the method numerically to model equations (anharmonic and double-well oscillators), comparing predictions with numerically computed coefficients.

Experimental results

Research questions

• RQ1 Can the magnitude of Frobenius series coefficients be predicted with high accuracy using asymptotic methods?
• RQ2 How well does the combination of WKB approximation and Legendre transformation reproduce the logarithmic growth of coefficients in power series solutions?
• RQ3 To what extent can this method estimate the number of terms and required precision for high-accuracy evaluation of ODE solutions?
• RQ4 How do logarithmic corrections in the WKB approximation affect the accuracy of coefficient predictions?
• RQ5 Does the method remain effective when the maximum contribution to the series comes from complex $z$-values (i.e., $\varphi \neq 0$)?

Key findings

• The method predicts the logarithmic magnitude of Frobenius coefficients $|a_m|$ with high accuracy, especially when logarithmic corrections are included.
• For the anharmonic oscillator ($c=0$), the corrected prediction $\log|a_m| = \frac{1}{3}\left(2m + \frac{5}{2}\right)\left(1 - \log\left(2m + \frac{5}{2}\right)\right)$ improves accuracy by a factor of $m^{-5/6}$ compared to the crude WKB estimate.
• In the double-well oscillator case, the predicted maximum coefficient location $m \approx \frac{1}{2}x(x + c^2)^{1/2}$ matches the numerically observed peak position within the oscillatory envelope.
• The predicted coefficient magnitude scales as $|a_{\bar{m}}| \sim \exp\left(\frac{1}{3}(x + c^2)^{3/2}\right)$, consistent with the asymptotic behavior of the solution.
• The method accurately estimates the number of terms $\mathcal{M}$ required for convergence to a desired precision $P$, with strong agreement between prediction and actual computation.
• Including the $\log S''(u)$ correction term significantly improves the fit to numerical coefficients, especially at moderate to large $m$.