Large-order perturbation theory of linear eigenvalue problems

TL;DR

The paper develops a systematic exponential-asymptotics framework for large-order perturbation theory in linear eigenvalue problems with a small parameter $\epsilon$, where the eigenvalue expansion is divergent. By combining inner-outer matching with a novel boundary-layer analysis in the late-term expansion, it couples nonuniform outer behavior to the eigenvalue’s divergence and yields precise large-$n$ asymptotics, including signs and constants, and reveals higher-order Stokes phenomena. The method is illustrated on four models—simplified black holes, the anharmonic oscillator, simplified Rossby waves, and a two-singularity divergence—producing explicit late-term formulas such as $\omega_n \sim (-1)^n\Gamma(n)/(2\sqrt{2}\pi)$ and $\lambda_n \sim (-1)^{n+1}\frac{\sqrt{6}}{\pi^{3/2}}3^n\Gamma(n+1/2)$. These results inform optimal truncation, potential resummation, and provide a template for analysing similar divergences in other linear eigenvalue problems with nonperturbative corrections.

Abstract

We consider a class of linear eigenvalue problems depending on a small parameter epsilon in which the series expansion for the eigenvalue in powers of epsilon is divergent. We develop a new technique to determine the precise nature of this divergence. We illustrate the technique through its application to four examples: the anharmonic oscillator, a simplified model of equatorially-trapped Rossby waves, and two simplified models based on quasinormal modes of Reissner-Nordstrom-de Sitter black holes.

Figures (5)

• Figure 1: Divergence of the coefficients in the asymptotic expansion of $\omega$. (a) coefficients determined numerically from (\ref{['rec1']})-(\ref{['rec2']}). The linear growth is consistent with factorial divergence. (b) The ratio of the numerical value to the asymptotic prediction (\ref{['ex2omegan']}). Blue is the base series, while orange, green and red correspond to enhanced convergence using Richardson extrapolation on two, three and four terms respectively. The black line is the asymptote, included to aid the eye. The convergence is slower than expected because of the presence of log terms in the higher-order corrections, unaccounted for in the extrapolation.
• Figure 2: The ratio of the numerical value found by iterating (\ref{['ex3rec1']})-(\ref{['ex3rec2']}) to the asymptotic prediction (\ref{['ex3lambdan']}). Blue is the base series, while orange and green correspond to enhanced convergence using Richardson extrapolation on two and three terms respectively. The black line is the asymptote, included to aid the eye.
• Figure 3: The ratio of the numerical value found by iterating (\ref{['ex4:rec1']})-(\ref{['ex4:rec2']}) to the asymptotic prediction (\ref{['ex4omegan']}). Blue is the base series, while orange, green and red correspond to enhanced convergence using Richardson extrapolation on two, three and four terms respectively. The black line is the asymptote, included to aid the eye. The convergence is slower than expected because of the presence of logarithmic terms in the higher-order corrections, unaccounted for in the extrapolation.
• Figure 4: Divergence of the coefficients in the asymptotic expansion of $\omega$, determined numerically from (\ref{['ex5:rec']}). The linear growth is consistent with factorial divergence.
• Figure 5: A comparison of the asymptotic approximation \ref{['ex5:ans']} with $\omega_n$ found by numerically iterating \ref{['ex5:rec']}, for various values of $b$ and $c$. We normalise by $\Gamma(n)/|\chi_0|^{n}$ to remove the exponential growth. The solid curve shows \ref{['ex5:ans']} as a continuous function of $n$, which is a sinusoidal oscillation of period $\arg(\chi_0)/2\pi$. The green dots show \ref{['ex5:ans']} evaluated at integer $n$. The red dots are the numerical values.