pith. sign in

arxiv: 2605.25390 · v1 · pith:ADTRP5EXnew · submitted 2026-05-25 · ⚛️ physics.atom-ph · math-ph· math.MP· nucl-th· physics.comp-ph

The Pseudospectral Method for the Dirac Equation with Confining Potential

Dengshan Liu , Huihui Xie , Pengxiang Du , Jian Li , Tomoya Naito This is my paper

Pith reviewed 2026-06-29 19:54 UTC · model grok-4.3

classification ⚛️ physics.atom-ph math-phmath.MPnucl-thphysics.comp-ph
keywords Dirac equationconfining potentialpseudospectral methodkinetically balancedenergy eigenvalueswave functionsnumerical convergence
0
0 comments X

The pith

The mono-kinetically-balanced generalized pseudospectral method yields converged energy eigenvalues and smooth wave functions for the Dirac equation with confining potentials.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper identifies that the standard generalized pseudospectral method produces deteriorating convergence and oscillatory wave functions for the Dirac equation as confinement radius decreases. This problem arises from the first-order differentiation in the method. By using the mono-kinetically-balanced version that adds the kinetically-balanced condition, the approach achieves convergence and smooth functions. This marks the first successful application of the method to confined potentials, validated specifically for small radii.

Core claim

Solving the Dirac equation for confined potentials with the generalized pseudospectral method leads to deteriorating convergence of energy eigenvalues and highly oscillatory wave functions as the confinement radius decreases because of the first-order differentiation formulation. Adopting the mono-kinetically-balanced generalized pseudospectral method incorporates the kinetically-balanced condition and produces converged energy eigenvalues along with smooth, continuous wave functions. This is the first application to confined potentials with effectiveness shown for small confinement radii.

What carries the argument

The mono-kinetically-balanced generalized pseudospectral (MKB-GPS) method that incorporates the kinetically-balanced condition into the GPS method to overcome differentiation-related instabilities.

If this is right

  • Energy eigenvalues converge reliably even as confinement radius decreases.
  • Wave functions become smooth and continuous rather than oscillatory.
  • The method works effectively for small confinement radii where standard GPS fails.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Kinetic balance conditions may stabilize pseudospectral discretizations in other relativistic equations.
  • The approach could be applied to different confining potential shapes beyond those tested.

Load-bearing premise

The deteriorating convergence and oscillatory wave functions in the standard GPS method stem from the first-order differentiation formulation employed in the GPS method.

What would settle it

If applying the MKB-GPS method to the Dirac equation with small confinement radii still results in non-converged energy eigenvalues or highly oscillatory wave functions, the claim would be falsified.

Figures

Figures reproduced from arXiv: 2605.25390 by Dengshan Liu, Huihui Xie, Jian Li, Pengxiang Du, Tomoya Naito.

Figure 2.1
Figure 2.1. Figure 2.1: Relative errors of the bound-state energies of the hydrogen atom calculated by GPS method [PITH_FULL_IMAGE:figures/full_fig_p005_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Relative errors of the bound-state energies of the hydrogen atom calculated by GPS method [PITH_FULL_IMAGE:figures/full_fig_p005_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: The radial wave function of the 1s1/2 state of the hydrogen atom calculated by GPS method with the confinement radius Rmax = 1 a.u. (a) Large component and (b) small component. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_2_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Radial wave functions of the 1s1/2 state calculated by GPS method (blue hollow circle and line) and Eqs. (3.2a) and (3.2b) (red solid circle and line). 10 4 10 3 10 2 10 1 100 10 2 10 1 100 101 102 103 104 105 106 107 r(a.u.) PL PR (a) 10 4 10 3 10 2 10 1 10 0 10 4 10 2 100 102 104 106 108 r(a.u.) QL QR (b) [PITH_FULL_IMAGE:figures/full_fig_p007_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: Functions on the left- and right-hand sides of the second set of equations to be satisfied [Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p007_3_5.png] view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Relative errors in the 1s1/2 state energies of the confined hydrogen atom calculated by the GPS and MKB-GPS methods with different confinement radii Rmax. The second-order derivative of a function u (x) at collocation points can be approximated by [1, 22] d 2 dx2 u (x) [PITH_FULL_IMAGE:figures/full_fig_p010_4_6.png] view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: Relative errors in the 2s1/2 state energies of the confined hydrogen atom calculated by the GPS and MKB-GPS methods with different confinement radii Rmax. 0 10 20 30 40 50 60 70 80 90 100 10-21 10-17 10-13 10-9 0 10 20 30 40 50 60 70 80 90 100 10-20 10-17 10-14 10-11 10-8 Relative error Rmax 1s1/2 N = 100 N = 200 N = 300 N = 400 Relative error Rmax 2s1/2 N = 100 N = 200 N = 300 N = 400 [PITH_FULL_IMAGE:… view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: Relative errors of the bound-state energies of the hydrogen atom calculated by MKB-GPS method [PITH_FULL_IMAGE:figures/full_fig_p011_4_8.png] view at source ↗
Figure 4.9
Figure 4.9. Figure 4.9: The radial wave function of the 1s1/2 state of the hydrogen atom calculated by MKB-GPS method with the confinement radius Rmax = 1 a.u. method successfully resolves the oscillation issues in the small-component wave functions, yielding smooth and continuous wave functions [PITH_FULL_IMAGE:figures/full_fig_p012_4_9.png] view at source ↗
Figure 4.10
Figure 4.10. Figure 4.10: Functions on the left- and right-hand sides of the second set of equations to be satisfied. [PITH_FULL_IMAGE:figures/full_fig_p012_4_10.png] view at source ↗
read the original abstract

We observe that solving the Dirac equation for confined potentials using the generalized pseudospectral (GPS) method leads to deteriorating convergence of energy eigenvalues and highly oscillatory in wave functions as the confinement radius decreases. It is found that this issue stems from the first-order differentiation formulation employed in GPS method. Motivated by this insight, we adopt the kinetically balanced generalized pseudospectral method, which incorporates the kinetically-balanced condition into the GPS method. Numerical results demonstrate that the mono-kinetically-balanced generalized pseu dospectral (MKB-GPS) method yields converged energy eigenvalues and generates smooth, continuous wave functions. This is the first application of the MKB-GPS method to confined potentials, and its effectiveness is validated for small confinement radii.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces the mono-kinetically-balanced generalized pseudospectral (MKB-GPS) method for the Dirac equation with confining potentials. It reports that the standard GPS method exhibits deteriorating eigenvalue convergence and oscillatory wave functions as the confinement radius decreases, attributes this behavior to the first-order differentiation formulation, and presents numerical results claiming that MKB-GPS produces converged energies and smooth wave functions. The work positions itself as the first application of MKB-GPS to confined potentials and validates effectiveness for small radii.

Significance. If the numerical demonstrations are robust, the MKB-GPS approach could supply a practical route to accurate solutions of the Dirac equation under hard-wall confinement, a setting relevant to atomic physics models of confined systems. The paper supplies the first reported use of this variant for such potentials.

major comments (2)
  1. [Abstract] Abstract: The statement that the observed deterioration 'stems from the first-order differentiation formulation employed in GPS method' is presented without an isolating numerical test that rules out alternative explanations such as mismatch in boundary-condition enforcement at the hard wall or aliasing on the pseudospectral grid near the discontinuity. Because the MKB construction is introduced specifically to address this diagnosed cause, the attribution is load-bearing for the central claim.
  2. [Abstract] The abstract asserts that 'numerical results demonstrate' converged eigenvalues and smooth wave functions, yet supplies no concrete error metrics, convergence tables, basis-size scaling, or comparison against known analytic limits or other methods. This absence prevents independent assessment of the claimed effectiveness.
minor comments (1)
  1. [Abstract] Abstract contains a typographical spacing error: 'pseu dospectral' should read 'pseudospectral'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each major comment below and indicate the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The statement that the observed deterioration 'stems from the first-order differentiation formulation employed in GPS method' is presented without an isolating numerical test that rules out alternative explanations such as mismatch in boundary-condition enforcement at the hard wall or aliasing on the pseudospectral grid near the discontinuity. Because the MKB construction is introduced specifically to address this diagnosed cause, the attribution is load-bearing for the central claim.

    Authors: We agree that the abstract's attribution would benefit from stronger supporting evidence in the form of an isolating test. In the full manuscript, the conclusion is drawn from the contrast in performance between the standard GPS and the MKB-GPS methods, along with analysis of the formulation. To directly address this concern, we will add a new subsection in the revised manuscript presenting additional numerical tests designed to isolate the effect of the first-order differentiation, for example by examining the behavior under modified boundary conditions and grid resolutions. This will provide a more rigorous basis for the claim. revision: yes

  2. Referee: [Abstract] The abstract asserts that 'numerical results demonstrate' converged eigenvalues and smooth wave functions, yet supplies no concrete error metrics, convergence tables, basis-size scaling, or comparison against known analytic limits or other methods. This absence prevents independent assessment of the claimed effectiveness.

    Authors: The abstract is constrained by length requirements and thus summarizes the findings without detailed metrics. The full manuscript contains extensive numerical results, including convergence with respect to the number of grid points, error estimates for the eigenvalues, and plots demonstrating the smoothness of the wave functions. To enhance the abstract, we will include brief quantitative statements, such as the achieved precision for small radii and the basis size used. Regarding comparisons to analytic limits, since this is the first application to confined potentials, no direct analytic solutions are available for the tested cases; however, we compare against the unconfined limit and other numerical methods where applicable. We will clarify this in the revision. revision: partial

Circularity Check

0 steps flagged

No circularity: numerical validation of method modification stands independently

full rationale

The paper reports an observed numerical failure mode in standard GPS for small confinement radii, attributes it observationally to first-order differentiation, introduces the MKB-GPS variant, and demonstrates via direct computation that the variant produces converged eigenvalues and smooth wave functions. No mathematical derivation is presented whose output is forced by construction to equal its inputs; no parameters are fitted and then relabeled as predictions; no self-citations are invoked as load-bearing uniqueness theorems; and the central claims rest on reproducible numerical benchmarks rather than self-referential definitions or ansatzes smuggled through prior work. The attribution of the failure mode is an untested diagnostic assumption, but that is a question of evidential strength, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities used in the method.

pith-pipeline@v0.9.1-grok · 5671 in / 1030 out tokens · 25511 ms · 2026-06-29T19:54:13.259934+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

22 extracted references · 16 canonical work pages

  1. [1]

    Canuto et al

    C. Canuto et al. Spectral Methods: Fundamentals in Single Domains . Berlin, Heidelberg: Springer- Verlag, 2006

    2006

  2. [2]

    Beyond the Floquet theorem: generalized Floquet formalisms and quasienergy methods for atomic and molecular multiphoton processes in intense laser fields

    Shih-I Chu and Dmitry A. Telnov. “Beyond the Floquet theorem: generalized Floquet formalisms and quasienergy methods for atomic and molecular multiphoton processes in intense laser fields” . In: Phys. Rep. 390.1 (2004), p. 1. issn: 0370-1573. doi: 10 . 1016 / j . physrep . 2003 . 10 . 001. url: https : //www.sciencedirect.com/science/article/pii/S0370157303003946

    2004

  3. [3]

    Solution to the Dirac equation using the finite difference method

    Ji-Yu Fang, Shou-Wan Chen, and Tai-Hua Heng. “Solution to the Dirac equation using the finite difference method” . In: Nucl. Sci. Tech. 31.2 (2020), p. 15. doi: 10.1007/s41365-020-0728-6

    work page doi:10.1007/s41365-020-0728-6 2020

  4. [4]

    A fresh computational approach to atomic structures, processes and cascades

    Stephan Fritzsche. “A fresh computational approach to atomic structures, processes and cascades” . In: Comput. Phys. Commun. 240 (2019), p. 1. issn: 0010-4655. doi: 10.1016/j.cpc.2019.01.012. url: https://www.sciencedirect.com/science/article/pii/S0010465519300189

    work page doi:10.1016/j.cpc.2019.01.012 2019

  5. [5]

    Kinetically Balanced B-Spline Expansions in Radial Dirac Equation

    Akinori Igarashi. “Kinetically Balanced B-Spline Expansions in Radial Dirac Equation” . In: J. Phys. Soc. Jpn. 76.5 (2007), p. 054301. doi: 10.1143/JPSJ.76.054301 . url: https://doi.org/10.1143/ JPSJ.76.054301

    work page doi:10.1143/jpsj.76.054301 2007

  6. [6]

    Development of the kinetically and atomically balanced generalized pseudospec- tral method

    Li Guang Jiao et al. “Development of the kinetically and atomically balanced generalized pseudospec- tral method” . In: Phys. Rev. A 104 (2 Aug. 2021), p. 022801. doi: 10.1103/PhysRevA.104.022801 . url: https://link.aps.org/doi/10.1103/PhysRevA.104.022801

    work page doi:10.1103/physreva.104.022801 2021

  7. [7]

    CI-MBPT: A package of programs for relativistic atomic calculations based on a method combining configuration interaction and many-body perturbation theory

    M. G. Kozlov et al. “CI-MBPT: A package of programs for relativistic atomic calculations based on a method combining configuration interaction and many-body perturbation theory” . In: Comput. Phys. Commun. 195 (2015), p. 199. issn: 0010-4655. doi: 10 . 1016 / j . cpc . 2015 . 05 . 007. url: https : //www.sciencedirect.com/science/article/pii/S001046551500185X

    2015

  8. [8]

    Spectral pollution and how to avoid it

    Mathieu Lewin and Éric Séré. “Spectral pollution and how to avoid it” . In: Proc. London Math. Soc. 100.3 (2010), p. 864. doi: 10 . 1112 / plms / pdp046. url: https : / / londmathsoc . onlinelibrary . wiley.com/doi/abs/10.1112/plms/pdp046. 13

    work page doi:10.1112/plms/pdp046 2010

  9. [9]

    Finite element formulation of the Dirac equation and the prob- lem of fermion doubling

    C. Müller, N. Grün, and W. Scheid. “Finite element formulation of the Dirac equation and the prob- lem of fermion doubling” . In: Phys. Lett. A 242.4 (1998), p. 245. issn: 0375-9601. doi: 10 . 1016 / S0375 - 9601(98 ) 00218 - 7. url: https : / / www . sciencedirect . com / science / article / pii / S0375960198002187

    1998

  10. [10]

    Relativistic all-order pair functions from a discretized single-particle Dirac Hamiltonian

    Sten Salomonson and Per Öster. “Relativistic all-order pair functions from a discretized single-particle Dirac Hamiltonian” . In: Phys. Rev. A 40 (10 Nov. 1989), p. 5548. doi: 10.1103/PhysRevA.40.5548 . url: https://link.aps.org/doi/10.1103/PhysRevA.40.5548

    work page doi:10.1103/physreva.40.5548 1989

  11. [11]

    A fully relativistic approach for calculating atomic data for highly charged ions

    Douglas H. Sampson, Hong Lin Zhang, and Christopher J. Fontes. “A fully relativistic approach for calculating atomic data for highly charged ions” . In: Phys. Rep. 477.4 (2009), p. 111. issn: 0370-1573. doi: 10.1016/j.physrep.2009.04.002. url: https://www.sciencedirect.com/science/article/ pii/S037015730900129X

    work page doi:10.1016/j.physrep.2009.04.002 2009

  12. [12]

    Dual kinetic balance approach to basis-set expansions for the Dirac equation

    V. M. Shabaev et al. “Dual kinetic balance approach to basis-set expansions for the Dirac equation” . In: Phys. Rev. Lett. 93.13 (2004), p. 130405. doi: 10.1103/PhysRevLett.93.130405

    work page doi:10.1103/physrevlett.93.130405 2004

  13. [13]

    Finite-difference method for Dirac electrons in circular quantum dots

    Bartłomiej Szafran, Alina Mreńca-Kolasińska, and Dariusz Żebrowski. “Finite-difference method for Dirac electrons in circular quantum dots” . In: Phys. Rev. B 99 (19 May 2019), p. 195406. doi: 10. 1103/PhysRevB.99.195406. url: https://link.aps.org/doi/10.1103/PhysRevB.99.195406

    work page doi:10.1103/physrevb.99.195406 2019

  14. [14]

    3D mesh calculations for covariant density functional theory

    Y. Tanimura, K. Hagino, and H. Z. Liang. “3D mesh calculations for covariant density functional theory” . In: Prog. Theor. Exp. Phys. 2015.7 (July 2015), p. 073D01. issn: 2050-3911. doi: 10.1093/ ptep/ptv083. url: https://doi.org/10.1093/ptep/ptv083

    work page doi:10.1093/ptep/ptv083 2015

  15. [15]

    Exterior complex scaling method in time-dependent density-functional theory: Multiphoton ionization and high-order-harmonic generation of Ar atoms

    Dmitry A. Telnov et al. “Exterior complex scaling method in time-dependent density-functional theory: Multiphoton ionization and high-order-harmonic generation of Ar atoms” . In: Phys. Rev. A 87 (5 May 2013), p. 053406. doi: 10.1103/PhysRevA.87.053406 . url: https://link.aps.org/doi/10.1103/ PhysRevA.87.053406

    work page doi:10.1103/physreva.87.053406 2013

  16. [16]

    Spurious states of the dirac equation in a finite basis set

    I. I. Tupitsyn and V. M. Shabaev. “Spurious states of the dirac equation in a finite basis set” . In: Opt. Spectrosc. 105.2 (2008), p. 183. doi: 10.1134/S0030400X08080043

    work page doi:10.1134/s0030400x08080043 2008

  17. [17]

    A deep neural network approach to solve the Dirac equation

    Chuanxin Wang et al. “A deep neural network approach to solve the Dirac equation” . In: Eur. Phys. J. A 61.7 (July 2025), p. 162. issn: 1434-601X. doi: 10 . 1140 / epja / s10050 - 025 - 01630 - 5. url: https://doi.org/10.1140/epja/s10050-025-01630-5

    work page doi:10.1140/epja/s10050-025-01630-5 2025

  18. [18]

    Generalized pseudospectral methods with mappings for bound and resonance state problems

    Guanhua Yao and Shih-I Chu. “Generalized pseudospectral methods with mappings for bound and resonance state problems” . In: Chem. Phys. Lett. 204.3 (1993), p. 381. issn: 0009-2614. doi: 10 . 1016/0009- 2614(93)90025- V . url: https://www.sciencedirect.com/science/article/pii/ 000926149390025V

    1993

  19. [19]

    DBSR_HF: A B-spline Dirac–Hartree–Fock program

    Oleg Zatsarinny and Charlotte Froese Fischer. “DBSR_HF: A B-spline Dirac–Hartree–Fock program” . In: Comput. Phys. Commun. 202 (2016), p. 287. issn: 0010-4655. doi: 10.1016/j.cpc.2015.12.023. url: https://www.sciencedirect.com/science/article/pii/S0010465516000047

    work page doi:10.1016/j.cpc.2015.12.023 2016

  20. [20]

    Resolving the spurious-state problem in the Dirac equation with the finite-difference method

    Ying Zhang et al. “Resolving the spurious-state problem in the Dirac equation with the finite-difference method” . In: Phys. Rev. C 106 (5 Nov. 2022), p. L051303. doi: 10.1103/PhysRevC.106.L051303 . url: https://link.aps.org/doi/10.1103/PhysRevC.106.L051303

    work page doi:10.1103/physrevc.106.l051303 2022

  21. [21]

    Spherical Dirac equation on the lattice and the problem of the spurious states

    Bin Zhao. “Spherical Dirac equation on the lattice and the problem of the spurious states” . In: Acta Phys. Sin. 65 (2016), p. 052401. issn: 1000-3290. doi: 10 . 7498 / aps . 65 . 052401. url: https : / / wulixb.iphy.ac.cn//article/id/66841

    2016

  22. [22]

    Revisiting the generalized pseudospectral method: Radial expectation values, fine structure, and hyperfine splitting of confined atom

    Lin Zhu et al. “Revisiting the generalized pseudospectral method: Radial expectation values, fine structure, and hyperfine splitting of confined atom” . In: Int. J. Quantum Chem. 120.14 (2020), e26245. doi: 10.1002/qua.26245. url: https://onlinelibrary.wiley.com/doi/abs/10.1002/qua.26245. 14

    work page doi:10.1002/qua.26245 2020