Large-order perturbation theory of linear eigenvalue problems

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 equitorially-trapped Rossby waves, and two simplified models based on quasinormal modes of Reissner-Normstrom de Sitterblack holes.

• The paper introduces a systematic framework for capturing large-order divergence in eigenvalue perturbation series via regular expansions and matched asymptotic techniques.
• The study applies boundary layer analysis and higher-order Stokes phenomena to precisely characterize the factorial and oscillatory divergence of series coefficients.
• The approach combines analytic derivations with numerical validation, informing improved resummation methods and eigenvalue computations in various physical contexts.

Large-Order Perturbation Theory of Linear Eigenvalue Problems

Introduction and Motivation

This paper presents a systematic framework for analyzing the large-order divergence of perturbation series in linear eigenvalue problems parameterized by a small parameter $\epsilon$. Such problems are prevalent in quantum mechanics, fluid dynamics, and general relativity, where the eigenvalue often represents a physically meaningful quantity (energy, frequency, growth rate). The perturbation series for the eigenvalue in powers of $\epsilon$ is frequently divergent, and understanding the precise nature of this divergence is essential for both theoretical insight and practical computation.

The classical approach, exemplified by Bender and Wu's work on the quantum anharmonic oscillator, involves analytic continuation and matched asymptotic expansions, often requiring intricate calculations. The method developed in this paper provides a more direct and generalizable route to determining the asymptotic behavior of the series coefficients, with explicit treatment of exponential asymptotics, Stokes phenomena, and boundary layer analysis in the late terms.

General Framework

The analysis proceeds by:

1. Regular Perturbation Expansion: The eigenfunction and eigenvalue are expanded in powers of $\epsilon$, yielding recurrence relations for the coefficients.
2. Outer Expansion: The expansion is nonuniform for large $x$; rescaling yields an outer solution with its own asymptotic series, whose late terms exhibit factorial/power divergence due to singularities in the leading-order solution.
3. Boundary Layer Analysis in Late Terms: The late-term approximation of the outer expansion is itself nonuniform as $n \to \infty$. Introducing a local variable in the equation for the late terms generates a new inner expansion, coupling the two sources of divergence and allowing determination of the eigenvalue's large-order behavior.

This approach is illustrated through four examples: a simplified black hole model, the quantum anharmonic oscillator, a Rossby wave model, and a problem with divergence driven by two singularities.

Example 1: Simplified Black Hole Model

The eigenvalue problem is constructed to mimic quasinormal modes of Reissner-Nordström de Sitter black holes. The perturbation series for the eigenvalue $\omega$ is shown to diverge factorially, with the late terms given by

$$\omega_n \sim \frac{(-1)^n \Gamma(n)}{2\sqrt{2}\,\pi}$$

as $n \to \infty$. The analysis reveals that the divergence is driven by a singularity in the outer solution, and the boundary layer analysis near $X=0$ is essential to couple the divergent terms and fix the asymptotic form. The presence of logarithmic corrections in higher-order terms slows the convergence of Richardson extrapolation in numerical comparisons.

Example 2: Quantum Anharmonic Oscillator

The classical quartic oscillator is revisited, with the eigenvalue expansion coefficients $\lambda_n$ exhibiting factorial divergence: $$\lambda_n \sim \frac{(-1)^{n+1}\sqrt{6}}{\pi^{3/2}\, 3^{n}\Gamma(n+1/2)}$$ The analysis follows the same framework, but the boundary layer near $X=0$ requires a different scaling ($X = \xi/n^{1/2}$) due to the structure of the recurrence relations. The matching conditions involve careful treatment of branch cuts and Stokes lines, with the higher-order Stokes phenomenon playing a central role. Numerical results confirm the accuracy of the asymptotic prediction.

Example 3: Simplified Rossby Wave Model

This example demonstrates the method in a fluid dynamics context, where the eigenvalue expansion coefficients $\mu_n$ satisfy

$\mu_n \sim \frac{\Gamma(n-1/2)}{\sqrt{\pi}}$

The divergence is again driven by a singularity in the outer solution, but only one singularity contributes due to the structure of the problem and the presence of a higher-order Stokes line that suppresses the contribution from the other potential singularity. The boundary layer analysis and matching conditions are analogous to the previous examples, with logarithmic corrections affecting the rate of convergence in numerical comparisons.

Example 4: Divergence Driven by Two Singularities

A model problem is constructed to illustrate the case where the divergence of the eigenvalue expansion is driven by two distinct singularities in the outer solution. The late terms exhibit oscillatory behavior due to the interference between the two contributions: $$\omega_n \sim \frac{i b}{\sqrt{2}\, \pi(c+i b)} \frac{\Gamma(n)}{\chi_0^{n}} - \frac{i b}{\sqrt{2}\, \pi(c-i b)}\frac{\Gamma(n)}{\bar{\chi}_0^{n}}$$ where $\chi_0$ and $\bar{\chi}_0$ are determined by the locations of the singularities. The resulting oscillations in the coefficients are confirmed numerically, and the analysis highlights the necessity of treating multiple singularities and their associated Stokes phenomena.

Implications and Future Directions

The framework developed in this paper provides a robust and generalizable method for determining the large-order behavior of divergent perturbation series in linear eigenvalue problems. The explicit treatment of exponential asymptotics and higher-order Stokes phenomena enables precise characterization of the divergence, which is essential for optimal truncation, resummation, and understanding the analytic structure of solutions.

Practically, these results inform the design of numerical algorithms for eigenvalue problems in quantum mechanics, fluid dynamics, and general relativity, where naive perturbation theory fails at large order. Theoretically, the approach clarifies the interplay between singularities, Stokes lines, and the structure of divergent series, with potential extensions to nonlinear problems, PDEs, and systems with more complex singularity structure.

Future work may focus on:

• Extending the method to nonlinear eigenvalue problems and PDEs.
• Developing automated symbolic/numeric tools for extracting large-order asymptotics from recurrence relations.
• Investigating the resurgent structure and Borel summability of the divergent series in more general settings.
• Applying the framework to problems in quantum field theory, hydrodynamic stability, and black hole physics where large-order behavior is critical.

Conclusion

This paper establishes a systematic procedure for analyzing the large-order divergence of perturbation series in linear eigenvalue problems, with explicit application to several canonical examples. The method combines regular and outer expansions, boundary layer analysis in the late terms, and matching via higher-order Stokes phenomena to yield precise asymptotic formulas for the series coefficients. The approach is broadly applicable and provides both theoretical insight and practical tools for tackling divergent series in applied mathematics and theoretical physics.