Large-order perturbation theory of linear eigenvalue problems

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.

💡 Research Summary

The paper addresses linear eigenvalue problems of the form Lε g = λ g that depend on a small parameter ε, where the formal perturbation series λ = ∑ₙ εⁿ λₙ is divergent. The authors develop a systematic three‑stage method—inner region, outer region, and a late‑term boundary layer—to extract the precise asymptotic behaviour of the coefficients λₙ as n → ∞.

In the inner region the solution is expanded in regular powers of ε, satisfying the original boundary conditions at the origin (or another regular point). This yields recursion relations for the coefficients λₙ and the associated eigenfunctions gₙ, but the large‑n growth is hidden in these recursions.

The outer region is introduced by rescaling the spatial variable (x → X = ε x) so that the expansion becomes singular for large X. Two distinct sources of divergence appear in gₙ: (i) the familiar factorial/power divergence driven by the singularity of the leading‑order outer solution, which is captured by an ansatz gₙ ∼ G Γ(n+γ) χⁿ with χ′ = 2(1‑X) leading to χ = −(1‑X)², and (ii) a divergence directly proportional to λₙ itself, giving a term gₙ ∼ Q λₙ where Q contains logarithmic factors (e.g., (1‑X)¹ᐟ² log X).

The crucial new observation is that the large‑n outer approximation is non‑uniform near X = 0. To resolve this, the authors introduce a boundary‑layer scaling X = ξ/n, which yields a coupled asymptotic problem for the late‑term amplitude H(ξ) and the overall growth factor Ω defined by λₙ ∼ Ω (−1)ⁿ Γ(n). Substituting the scaled forms into the governing equation leads to a second‑order ODE for H(ξ): −(ξ H′)′ + 2ξ H′ = 2. Its solution involves the exponential integral Ei(2ξ) and a logarithmic term. The presence of Ei(2ξ) signals a higher‑order Stokes phenomenon: as ξ crosses the positive real axis, the exponentially small contribution e^{2ξ}/ξ switches on, producing a rapid change in the late‑term behaviour. By matching the inner‑limit of the outer solution (including the logarithmic piece) with the boundary‑layer solution, the constants Ω and the prefactor Λ are determined.

The method is applied to four illustrative models:

1. Simplified Reissner‑Nordström‑de Sitter black‑hole QNM – The eigenvalue expansion yields λₙ ∼ (−1)ⁿ Γ(n) 2 √(2π). Numerical iteration of the recursion confirms the prediction, with Richardson extrapolation improving convergence.

2. Anharmonic oscillator – Re‑deriving the classic Bender‑Wu result, the analysis shows λₙ ∼ (−1)ⁿ Γ(n) (3/2)/π · 2ⁿ, reproducing the known factorial growth and the associated exponentially small splitting.

3. Equatorially trapped Rossby wave model – The divergent real series determines an exponentially small imaginary part of the eigenvalue, interpreted as a growth rate. The boundary‑layer analysis captures the Stokes switching that generates this imaginary component.

4. Second simplified black‑hole QNM model – Similar to the first case, but with a different complex prefactor, illustrating how the method accommodates various boundary conditions and logarithmic contributions.

Across all examples the large‑order behaviour is governed by a universal factorial growth multiplied by a model‑dependent constant, while the exponentially small non‑perturbative pieces arise from the boundary‑layer Stokes phenomenon. The approach bypasses the intricate analytic continuation and Liouville‑Green constructions of the traditional Bender‑Wu technique, offering a more transparent and broadly applicable framework.

By providing explicit formulas for the optimal truncation point, the minimal achievable error, and the non‑perturbative corrections, the paper equips researchers with practical tools for handling divergent perturbation series in quantum mechanics, fluid dynamics, and black‑hole physics. The identification of higher‑order Stokes lines in the late‑term expansion is a novel contribution, highlighting subtle transition structures that are invisible in the original ε‑expansion but crucial for accurate asymptotics.