[Paper Review] WKB-method for the 1D Schrödinger equation in the semi-classical limit: enhanced phase treatment

Anton Arnold, C. Klein. B. Ujvari|arXiv (Cornell University)|Aug 6, 2018

Quantum Mechanics and Non-Hermitian Physics25 references3 citations

TL;DR

This paper enhances a second-order WKB method for solving the 1D Schrödinger equation in the semi-classical limit by refining the error analysis and replacing numerical quadrature with spectral methods for phase computation. The approach achieves uniformly accurate, high-order convergence on coarse grids by eliminating the $1/\varepsilon$-dependent numerical error in phase integration, significantly improving robustness and accuracy for small $\varepsilon$.

ABSTRACT

This paper is concerned with the efficient numerical computation of solutions to the 1D stationary Schrödinger equation in the semiclassical limit in the highly oscillatory regime. A previous approach to this problem based on explicitly incorporating the leading terms of the WKB approximation is enhanced in two ways: first a refined error analysis for the method is presented for a not explicitly known WKB phase, and secondly the phase and its derivatives will be computed with spectral methods. The efficiency of the approach is illustrated for several examples.

Motivation & Objective

• Address the inefficiency of standard numerical schemes in resolving highly oscillatory solutions of the 1D Schrödinger equation when $\varepsilon \ll 1$.
• Overcome the $1/\varepsilon$-dependence in numerical phase integration errors that limit accuracy in prior WKB-based schemes.
• Improve the asymptotic accuracy of the second-order WKB approximation by refining the error analysis to include numerical errors in phase and transformation steps.
• Replace low-order quadrature (e.g., Simpson's rule) with spectral methods for phase computation to achieve exponential convergence and reduced conditioning issues.
• Demonstrate that spectral computation of the phase enables uniformly accurate solutions across a wide range of $\varepsilon$ values, even down to machine precision limits.

Proposed method

• Apply a second-order WKB approximation to transform the original highly oscillatory Schrödinger equation into a smoother system via analytic phase and amplitude decomposition.
• Use the WKB ansatz $\varphi(x) \sim \exp\left(\frac{1}{\varepsilon}\sum_{p=0}^{\infty} \varepsilon^p \phi_p(x)\right)$ to derive the leading-order phase and amplitude corrections.
• Implement spectral collocation (Clenshaw-Curtis with barycentric interpolation) for high-accuracy integration of the phase function $\tilde{\phi}(x) = \int_0^x \left(\sqrt{a(\tau)} - \varepsilon^2 \beta(\tau)\right) d\tau$.
• Introduce a refined error analysis that accounts for both the WKB approximation error and the numerical integration error in the phase, showing that the latter is no longer $\mathcal{O}(1/\varepsilon)$ when spectral methods are used.
• Solve the transformed system for the reduced variable $Z(x)$ using standard finite differences on a coarse mesh, leveraging the smoothness induced by phase removal.
• Compare spectral phase computation with low-order quadrature (Simpson's rule) to demonstrate superior conditioning and reduced numerical error, especially for small $\varepsilon$.

Experimental results

Research questions

• RQ1 Can the $1/\varepsilon$-dependent numerical error in phase computation be eliminated or significantly reduced in WKB-based schemes for the 1D Schrödinger equation?
• RQ2 How does the choice of numerical integration method (spectral vs. quadrature) affect the convergence rate and conditioning of the WKB method in the semi-classical limit?
• RQ3 Does a refined error analysis that includes numerical errors in phase and transformation steps lead to more accurate and reliable convergence estimates?
• RQ4 Can spectral methods for phase computation enable uniformly accurate solutions across a wide range of $\varepsilon$ values, including those approaching machine precision?
• RQ5 Is the reduced system for $Z(x)$ amenable to further spectral treatment, given its oscillatory but low-amplitude behavior?

Key findings

• The spectral computation of the phase eliminates the $\mathcal{O}(1/\varepsilon)$ error term that plagued earlier WKB schemes using low-order quadrature, enabling uniformly accurate solutions.
• For $\varepsilon = 10^{-5}$, the error in the spectral method dropped below machine precision for $h \in [10^{-5}, 10^{-2}]$, indicating that the method achieves optimal accuracy in double precision.
• The method achieves second-order convergence in $h$ for $h$ above the saturation level of the WKB error, as confirmed by slope analysis in the error plots.
• The right plot in Figure 5 (spectral phase) shows no artificial error increase at very small $h$, unlike the Simpson rule case, due to better conditioning and reduced rounding errors.
• The error in the transformed variable $Z$ satisfies bounds (3.51) that do not include the problematic $E/\varepsilon$ term, confirming the robustness of the spectral approach.
• The method outperforms the first-order WKB scheme in both accuracy and convergence rate, with the second-order WKB error reaching saturation only at $h \leq 10^{-4} \text{--} 10^{-3}$, while the spectral method maintains high accuracy at smaller $h$.