[Paper Review] Bound State Energies using Phase Integral Analysis

Ruth B. White, Anton Kutlin|arXiv (Cornell University)|Apr 4, 2017

Quantum and Classical Electrodynamics3 citations

TL;DR

This paper proposes an improved method for calculating bound state energies in quantum mechanical systems using Phase Integral methods, which extend the WKB approximation by incorporating Stokes constants and analytic continuation through complex singularities. The key contribution is that this approach reduces energy error from 6% (WKB) to 0.6% for anharmonic oscillators, with Stokes constants asymptotically approaching the isolated singularity value of $ i $ as singularities separate.

ABSTRACT

The study of asymptotic properties of solutions to differential equations has a long and arduous history, with the most significant advances having been made in the development of quantum mechanics. A very powerful method of analysis is that of Phase Integrals, described by Heading. Key to this analysis are the Stokes constants and the rules for analytic continuation of an asymptotic solution through the complex plane. These constants are easily determined for isolated singular points, by analytically continuing around them and, in the case of analytic functions, requiring the asymptotic solution to be single valued. However, most interesting problems of mathematical physics involve several singular points. By examination of analytically tractable problems and more complex bound state problems involving multiple singular points, we show that the method of Phase Integrals can greatly improve the determination of bound state energies over the simple WKB values. We also find from these examples that in the limit of large separation the Stokes constant for a first order singular point approaches the isolated singular point value.

Motivation & Objective

• To improve the accuracy of bound state energy calculations beyond standard WKB approximation for quantum mechanical potentials.
• To investigate how multiple singularities in the complex plane affect Stokes constants and energy eigenvalues.
• To determine whether the asymptotic behavior of Stokes constants approaches the isolated singularity value $ i $ as singularities separate.
• To assess the effectiveness of Phase Integral methods in non-Hermitian and anharmonic potentials where WKB fails.

Proposed method

• The method uses Phase Integral techniques to globally connect WKB solutions across the complex plane via Stokes and anti-Stokes lines emanating from singularities of $ Q(z,E) $.
• It applies analytic continuation rules to track changes in dominant and subdominant solutions across Stokes lines, branch cuts, and anti-Stokes lines.
• Stokes constants $ S $ are computed by traversing closed loops around singularities, enforcing single-valuedness of the WKB approximation.
• For isolated first-order singularities, $ S = 2i\cos(\pi/(n+2)) $, and the paper shows $ S \to i $ as separation increases.
• The method connects solutions between singular points using phase factors $ [b,a] = e^{i\int_b^a Q^{1/2} dz} $, enabling global solution construction.
• Numerical validation is performed by comparing Phase Integral results with exact and WKB energies for $ Q = E - iz^3 $, $ E - z^2 $, $ E - z^4 $, and $ E - z^6 $.

Experimental results

Research questions

• RQ1 How do multiple singularities in the complex plane affect the accuracy of WKB-based bound state energy calculations?
• RQ2 What is the asymptotic behavior of Stokes constants as the separation between singularities increases?
• RQ3 Can Phase Integral methods significantly reduce energy error compared to standard WKB in anharmonic and non-Hermitian potentials?
• RQ4 Does the Stokes constant for a first-order singularity approach $ i $, the isolated singularity value, in the large separation limit?
• RQ5 Are there higher-order corrections to the Stokes constant that can be systematically computed in perturbation theory?

Key findings

• For the $ Q = E - iz^3 $ potential, the WKB ground state energy has a 6% error, while the Phase Integral method reduces this to 0.6%.
• The Phase Integral method achieves energy values within 0.006% of exact results for the first few levels of $ Q = E - iz^3 $, with errors decreasing exponentially with quantum number.
• In all examined cases, the asymptotic value of the Stokes constant approaches $ S = i $, matching the isolated first-order zero case.
• The correction to the Stokes constant is found to be second order or higher in the small parameter $ \epsilon = e^{-\sqrt{3}W} $, indicating slow convergence.
• The method shows increasing improvement over WKB as the potential deviates from harmonic oscillator shape, particularly in systems with additional complex singularities.
• The authors conjecture that $ S \to i $ universally for isolated first-order singularities at large separation, though higher-order corrections remain an open question.