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High-Precision Computation and PSLQ Identification of Stokes Multipliers for Anharmonic Oscillators

Jian Zhou

Abstract

We present a large-scale computational study combining arbitrary-precision arithmetic, sequence acceleration, and the PSLQ integer relation algorithm to discover exact closed-form expressions for fundamental constants arising in asymptotic analysis. We compute the Stokes multipliers C_M of the one-dimensional anharmonic oscillators H = p^2/2 + x^2/2 + g x^{2M} for M = 2, 3, ..., 11, extracting 17-30 significant digits from up to 1200 perturbation coefficients computed at 300-digit working precision. The computational pipeline consists of three stages: (i) Rayleigh-Schrodinger recursion in the harmonic oscillator basis, (ii) Richardson extrapolation of order 40-100 to accelerate convergence of ratio sequences, and (iii) PSLQ searches over bases of Gamma-function values and algebraic numbers. This pipeline discovers three new exact identities: C_3^2 pi^4 = 32, C_5^4 Gamma(1/4)^4 pi^5 = 2^{12} 3^2, and C_7^6 Gamma(1/3)^9 pi^6 = 2^{20} 3^3, in addition to confirming the known C_2^2 pi^3 = 6. Equally significant is a negative result: exhaustive PSLQ searches at 30-digit precision with coefficient bounds up to 2000 find no closed form for C_4, strongly suggesting the x^8 case introduces a genuinely new transcendental number. A number-theoretic pattern emerges: closed-form existence correlates with Euler's totient function phi(M-1)/2, which counts algebraically independent Gamma-function transcendentals at denominator M-1. We formulate conjectures connecting computational constant recognition to classical number theory, and provide all code and data for full reproducibility.

High-Precision Computation and PSLQ Identification of Stokes Multipliers for Anharmonic Oscillators

Abstract

We present a large-scale computational study combining arbitrary-precision arithmetic, sequence acceleration, and the PSLQ integer relation algorithm to discover exact closed-form expressions for fundamental constants arising in asymptotic analysis. We compute the Stokes multipliers C_M of the one-dimensional anharmonic oscillators H = p^2/2 + x^2/2 + g x^{2M} for M = 2, 3, ..., 11, extracting 17-30 significant digits from up to 1200 perturbation coefficients computed at 300-digit working precision. The computational pipeline consists of three stages: (i) Rayleigh-Schrodinger recursion in the harmonic oscillator basis, (ii) Richardson extrapolation of order 40-100 to accelerate convergence of ratio sequences, and (iii) PSLQ searches over bases of Gamma-function values and algebraic numbers. This pipeline discovers three new exact identities: C_3^2 pi^4 = 32, C_5^4 Gamma(1/4)^4 pi^5 = 2^{12} 3^2, and C_7^6 Gamma(1/3)^9 pi^6 = 2^{20} 3^3, in addition to confirming the known C_2^2 pi^3 = 6. Equally significant is a negative result: exhaustive PSLQ searches at 30-digit precision with coefficient bounds up to 2000 find no closed form for C_4, strongly suggesting the x^8 case introduces a genuinely new transcendental number. A number-theoretic pattern emerges: closed-form existence correlates with Euler's totient function phi(M-1)/2, which counts algebraically independent Gamma-function transcendentals at denominator M-1. We formulate conjectures connecting computational constant recognition to classical number theory, and provide all code and data for full reproducibility.

Paper Structure

This paper contains 47 sections, 36 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Setup and Conventions
  3. Sign convention.
  4. Relation to Bender--Wu conventions.
  5. Computational Method
  6. Rayleigh--Schrödinger Recursion
  7. Extraction of Parameters
  8. Instanton action.
  9. Gamma-function shift.
  10. Stokes multiplier.
  11. Richardson Extrapolation
  12. Error Assessment
  13. PSLQ Search
  14. Instanton Action: Review and Explicit Formula
  15. Derivation from the Classical Bounce (Review)
  16. ...and 32 more sections

Figures (1)

  • Figure 1: Convergence of the Stokes multiplier extraction for $M = 4$. Upper panel: The raw sequence $\widetilde{C}_k$ (blue dots) defined by \ref{['eq:Ck-extraction']}, showing slow convergence due to $O(1/k)$ corrections. Lower panel: Richardson-extrapolated estimates $R_N$ for $N = 20, 40, 60, 80, 100$ starting from $k_0 = 600$. The horizontal dashed line shows the best estimate $C_4 \approx -0.74005\,14983$. Convergence to 25+ digits is achieved by $N = 80$.

Theorems & Definitions (4)

  • Remark 5.1
  • Conjecture 7.1: Totient threshold
  • Remark 7.2
  • Remark 7.3