Beylkin, Anderson and Harrison (arXiv:2107.02252, 2021) recast the computation of bound states of the non-relativistic and relativistic (Dirac) equations of quantum chemistry as auxiliary eigenvalue problems for parameter-dependent compact integral operators, solved by a power iteration combined with an outer parameter-tuning root-find that drives the auxiliary eigenvalue to one. Compactness guarantees convergence of the iteration for a fixed potential, and the paper demonstrates robust numerical convergence for the hydrogen atom. However, in self-consistent mean-field models (Kohn–Sham, Hartree–Fock, Dirac–Fock), where the potential is a nonlinear functional of the eigenfunctions being computed, the authors explicitly assume — without proof — that this nonlinear dependence does not spoil convergence and that solutions exist for all parameter values. In the relativistic case the auxiliary eigenvalue λ(κ) is moreover not guaranteed to be monotone in the parameter κ, so the outer root-find λ(κ)=1 may involve multiple roots with nontrivial basins of attraction. The open problem is a rigorous convergence theory for this combined scheme in the fully self-consistent nonlinear setting — existence of nonlinear solutions across the parameter range, convergence conditions and basins for the iteration, and parameter-update rules with guaranteed global convergence — or a rigorous operator-level counterexample exhibiting failure. A 2026 follow-up building on the same framework reports that damping is empirically required for two-electron and heavy-nucleus systems, evidence that the question is live and unresolved.

Background

In electronic-structure theory, a bound state is a square-integrable eigenfunction of a one-particle Hamiltonian with energy below the continuum threshold. Mean-field models such as Kohn–Sham density functional theory and Hartree–Fock theory replace the interacting many-electron problem by N coupled one-particle eigenvalue equations, (−12Δ+V)ψi=Eiψi(-\tfrac{1}{2}\Delta + V)\psi_i = E_i\psi_i, where the effective potential VV contains the electron–nuclear attraction together with Coulomb and exchange (or exchange-correlation) components. In these models VV is itself a functional of the sought eigenfunctions — through the electron density or the orbitals — so the system is a nonlinear eigenvalue problem, conventionally solved by the self-consistent field (SCF) iteration, in which one repeatedly diagonalizes the Hamiltonian built from the current orbitals and updates the orbitals until they reproduce the potential that generated them.

Beylkin, Anderson and Harrison (2021) reformulated bound-state computation, for both the Schrödinger-type (non-relativistic) and Dirac (relativistic) equations, as auxiliary eigenvalue problems for a family of parameter-dependent compact integral operators. A compact operator is a bounded operator that maps the unit ball onto a precompact set; its spectrum is discrete and bounded with zero as the only possible accumulation point, so its dominant eigenvalue is isolated and the power iteration (repeatedly applying the operator to a vector, with re-orthogonalization against eigenfunctions already found) converges from an arbitrary initial guess. In the non-relativistic case they introduce the Yukawa Green's function GμG_\mu of −Δ+μ2-\Delta+\mu^2 and consider λμψ=−2GμVψ\lambda_\mu\psi=-2G_\mu V\psi; when the parameter satisfies μ2=−2E\mu^2=-2E for a true bound-state energy EE, the auxiliary eigenvalue equals one, λμ=1\lambda_\mu=1, and the eigenfunction solves the original equation. The scheme therefore alternates an inner power iteration at fixed parameter (convergent by compactness) with an outer parameter tuning that drives λ(μ)\lambda(\mu) to one, using an explicit derivative formula: for fixed VV, dλμ/dμ<0d\lambda_\mu/d\mu<0, so λ(μ)\lambda(\mu) is monotone and the root-find is well behaved. In the relativistic case the free Dirac Hamiltonian — the wave equation for spin-1/2 particles, whose spectrum is unbounded both above and below — makes plain eigenvalue iteration unusable; factoring the resolvent through the squared operator yields an auxiliary problem of the same compact form with parameter κ=m2c4−E2/(cℏ)\kappa=\sqrt{m^2c^4-E^2}/(c\hbar) and E(κ)<mc2E(\kappa)<mc^2, but there the derivative dλ(κ)/dκd\lambda(\kappa)/d\kappa is not sign-definite, so λ(κ)\lambda(\kappa) need not be monotone and the equation λ(κ)=1\lambda(\kappa)=1 may have several roots. The compactness proofs for the exact, singular Coulomb potential are carried out in momentum-weighted Hilbert spaces Hκ,δ\mathscr{H}_{\kappa,\delta} with δ>0\delta>0, which suffices for all nuclear charges 1≤Z≤1181\le Z\le 118.

All convergence guarantees in this framework are proved for a fixed potential VV; the paper's numerical evidence for arbitrary initializations is the hydrogen atom. For the nonlinear Kohn–Sham / Hartree–Fock / Dirac–Fock (Dirac–Fock being the relativistic analog of Hartree–Fock) setting, the authors state explicitly that they assume the nonlinear dependence of the potential on the eigenfunctions does not affect convergence and that solutions exist for all parameter values, noting that a mean-field theory would be unusable if the resulting nonlinear eigenvalue problem had no solution or could not be reached by iteration. For the standard diagonalization-based SCF iteration there is a neighboring convergence literature: Liu, Wang, Wen and Yuan (2014) proved global convergence from arbitrary starting points for Kohn–Sham models under a sufficiently large gap between occupied and unoccupied states plus bounded second derivatives of the exchange-correlation functional — assumptions they note are almost never satisfied in practice — and Cancès, Kemlin and Levitt (2021) derived convergence rates for damped SCF and direct minimization while exhibiting chaotic behavior of the undamped iteration for a non-quadratic functional. That literature concerns a different iteration map and leaves the convergence of the compact-operator power iteration with outer parameter tuning, in the self-consistent setting, unanalyzed.

Problem Statement

Establish a rigorous convergence theory — or refute by a rigorous counterexample — for the combined iterative scheme of Beylkin, Anderson and Harrison (arXiv:2107.02252) applied to self-consistent mean-field bound-state problems (Kohn–Sham, Hartree–Fock, or Dirac–Fock models), in which the potential is a nonlinear functional of the eigenfunctions being computed. For the auxiliary eigenproblems λμψ=−2GμV[{ψj}]ψ\lambda_\mu\psi=-2G_\mu V[\{\psi_j\}]\psi (non-relativistic) and λ(κ)ψ=−1ℏ2c2(H0+E(κ)I)G(κ)V[{ψj}]ψ\lambda(\kappa)\psi=-\frac{1}{\hbar^2c^2}(\mathcal{H}_0+E(\kappa)\mathcal{I})\mathcal{G}(\kappa)\mathcal{V}[\{\psi_j\}]\psi (relativistic), where GμG_\mu and G(κ)\mathcal{G}(\kappa) are the Green's-function kernels and H0\mathcal{H}_0 the free Dirac Hamiltonian of the source formulation, the question comprises three linked parts:

  1. Existence: determine whether solutions of the nonlinear auxiliary eigenproblem exist for all values of the parameter μ\mu (respectively κ\kappa), as assumed without proof in the source.

  2. Convergence: prove, under explicit and verifiable conditions on the potential functional (for example spectral-gap, boundedness, or damping conditions), that the inner power iteration on the compact auxiliary operator combined with the outer parameter-tuning update converges from arbitrary initializations (the class treated in the source) to a self-consistent solution; or exhibit a physically admissible self-consistent model for which the scheme provably fails (non-existence of a solution for some parameter value, divergence of the iteration, or convergence to a spurious cycle).

  3. Relativistic root structure: in the Dirac case, where dλ(κ)/dκd\lambda(\kappa)/d\kappa is not sign-definite and λ(κ)\lambda(\kappa) may be non-monotone, characterize the existence, multiplicity, and basins of attraction of the roots λ(κ)=1\lambda(\kappa)=1 for the self-consistent problem, and either analyze the Newton-type parameter update proposed in the source or design a parameter-update rule with a guaranteed global-convergence property in the non-monotone setting.

An admissible answer consists of precise theorem statements with complete proofs covering the self-consistent (nonlinear-potential) setting — not merely the fixed-potential case — for at least one of the two formulations, addressing the parts of the question applicable to that formulation (Part 3 arises only in the relativistic case); or an explicit counterexample within a physically admissible mean-field model class — that is, a standard member of the Kohn–Sham, Hartree–Fock, or Dirac–Fock classes with physical Coulomb interactions, not an artificially engineered functional — rigorously established at the operator level rather than as an artifact of a particular discretization.

References

  1. Gregory Beylkin, Joel Anderson, Robert J. Harrison, "On computing bound states of the Dirac and Schrödinger Equations", arXiv:2107.02252 (2021). DOI: 10.48550/arXiv.2107.02252, https://arxiv.org/abs/2107.02252
  2. J. Anderson, B. Sundahl, R. Harrison, G. Beylkin, "Dirac-Fock calculations on molecules in an adaptive multiwavelet basis", The Journal of Chemical Physics 151, 234112 (2019). DOI: 10.1063/1.5128908, https://doi.org/10.1063/1.5128908
  3. Jacopo Masotti, Roberto Di Remigio Eikås, Christian Tantardini, Luca Frediani, "Four-component relativistic calculations in a multiwavelet basis with improved convergence", Theoretical Chemistry Accounts 145, 62 (2026). DOI: 10.1007/s00214-026-03307-8, https://doi.org/10.1007/s00214-026-03307-8 (preprint arXiv:2311.03290, https://arxiv.org/abs/2311.03290)
  4. Xin Liu, Xiao Wang, Zaiwen Wen, Yaxiang Yuan, "On the Convergence of the Self-Consistent Field Iteration in Kohn–Sham Density Functional Theory", SIAM Journal on Matrix Analysis and Applications 35(2), 546–558 (2014). DOI: 10.1137/130911032, https://doi.org/10.1137/130911032
  5. Éric Cancès, Gaspard Kemlin, Antoine Levitt, "Convergence Analysis of Direct Minimization and Self-Consistent Iterations", SIAM Journal on Matrix Analysis and Applications 42(1), 243–274 (2021). DOI: 10.1137/20m1332864, https://doi.org/10.1137/20m1332864