Computing stationary states of nonlinear Dirac operators — spinor eigenstates of a stationary nonlinear Dirac equation at fixed charge — is obstructed by variational collapse: the Dirac spectrum is unbounded below, so the energy functional is strongly indefinite and naive minimization or gradient descent diverges into negative-energy states. Cai and Lorin (J. Comput. Phys. 420 (2020) 109679) proposed the first systematic solver family for this problem — Feit–Fleck, Rayleigh–Ritz, and min-max methods for chemical potentials, and normalized gradient flows equipped with balance operators for eigenenergies — but the convergence guarantees established there cover only some of the proposed methods, and no purely discrete mathematical treatment of the schemes — under simultaneous space and time discretization — is established in Cai and Lorin (2020) or in the subsequent literature. The open problem is to supply the fully discrete theory: formulate the space-and-time discretized counterparts of these schemes and prove convergence and error estimates toward stationary states of the nonlinear Dirac operator — in particular whether the discrete schemes retain the balance operators' prevention of variational collapse and avoid spurious spectral-pollution states.

Background

The Soler-type nonlinear Dirac equation describes a relativistic spinor field ψ:Rd→CN\psi:\mathbb{R}^d\to\mathbb{C}^N whose mass density generates a scalar self-interaction, and it arises in relativistic mean-field models, in effective descriptions of condensed-matter Dirac quasiparticles, and as a model relativistic nonlinear wave equation. A stationary state is a solution of the form ψ(x)e−iμt\psi(x)e^{-i\mu t}, where the real number μ\mu is the chemical potential (eigenenergy) and ψ\psi solves a nonlinear eigenvalue problem for the Dirac operator −ic α⋅∇+mc2β-ic\,\boldsymbol{\alpha}\cdot\nabla+mc^2\beta plus the nonlinear potential, typically under a fixed-charge constraint ∥ψ∥L2=ρ\|\psi\|_{L^2}=\rho (the charge plays the role that mass plays for the nonlinear Schrödinger equation). Existence of such stationary states was established analytically by Merle (1988) and by variational constructions of Esteban and Séré (1995).

Two features make the numerical computation of these states genuinely different from the nonlinear Schrödinger/Gross–Pitaevskii case. First, the free Dirac operator has spectrum (−∞,−mc2]∪[mc2,+∞)(-\infty,-mc^2]\cup[mc^2,+\infty), so the associated energy functional is strongly indefinite — unbounded above and below even on the fixed-charge constraint set — and direct minimization collapses toward negative-energy states; this is the variational-collapse phenomenon, and Galerkin discretizations of Dirac operators can additionally produce spurious eigenvalues (spectral pollution). Second, the toolbox that works for the nonlinear Schrödinger side does not transfer: there, normalized gradient flows — gradient descent for the energy projected onto the fixed-mass sphere, introduced for Bose–Einstein condensate ground states by Bao and Du (2004), with convergence analysis of the associated algorithms by Faou and Jézéquel (2018), and extended to gradient flows on nonlinear quantum graphs by Besse, Duboscq, and Le Coz (2022) — rely on the energy being bounded below on the constraint manifold, which is exactly what fails for Dirac. For linear Dirac eigenvalue problems, min–max characterizations of eigenvalues lying in the spectral gap (Dolbeault, Esteban, and Séré, 2000) are the standard structure-preserving route that avoids spectral pollution and underlies variational methods in relativistic quantum chemistry.

Cai and Lorin (2020) combined these ideas into a first systematic attack on the nonlinear stationary Dirac problem: the Feit–Fleck scheme (which extracts eigenvalues from long-time propagation of the associated time-dependent equation), Rayleigh–Ritz approximations (eigenvalue estimation within a finite trial subspace), and min-max schemes for computing chemical potentials; normalized gradient flow schemes for computing eigenenergies; and so-called balance operators — corrections of the relative weights of the spinor components that maintain the proper normalization balance so that iterates do not fall into the negative-energy spectral region. The convergence guarantees established there cover only some of the proposed methods, and the paper's numerical experiments validate some but not all of them. In particular, a purely discrete-level treatment of the schemes — with both space and time discretization imposed simultaneously — is not established in Cai and Lorin (2020), and none appears in the subsequent literature. Since 2020, the numerical Dirac literature has concentrated on time-dependent propagation (energy-preserving exponential integrators by Li (2022), splitting spectral methods for the nonlinear Dirac–Poisson system by Wang, Zhang, and Wang (2023), semi-nonrelativistic limit solvers by Jahnke and Kirn (2023), and pseudospectral wavefunction dynamics in photonic graphene by Lorin and Yang (2024)) — so the discrete-level theory of the stationary-state schemes has remained untouched.

Problem Statement

Provide the purely discrete-level mathematical treatment of the computational schemes proposed by Cai and Lorin (2020) for stationary states of nonlinear Dirac operators. Specifically: (i) formulate the fully discrete counterparts of these schemes — the normalized gradient flows (including the discrete action of the balance operators on the spinor components), and the Rayleigh–Ritz, Feit–Fleck, and min-max methods for the chemical potential — after simultaneous space and time discretization; and (ii) prove convergence and error estimates for these fully discrete schemes toward stationary states of the nonlinear Dirac operator, treating the normalized gradient flow scheme (including the discrete action of the balance operators) and each of the Rayleigh–Ritz, Feit–Fleck, and min-max methods as four separately required sub-goals. For each sub-goal, resolve, under explicit assumptions on the mesh size, time step, and initial data, whether the fully discrete scheme converges to a stationary state without variational collapse — that is, whether its iterates avoid diverging along the unbounded-below part of the discrete spectrum and avoid converging to spurious states induced by spectral pollution — either by proving the corresponding convergence-and-error estimate together with the absence of collapse, or by exhibiting a rigorous counterexample in a specified discrete scheme and parameter regime. All four sub-goals are required; a counterexample stands in place of the convergence-and-error proof for the family it resolves.

References

  1. Y. Cai, E. Lorin, Stationary state computation for nonlinear Dirac operators, Journal of Computational Physics 420 (2020) 109679, DOI 10.1016/j.jcp.2020.109679, https://doi.org/10.1016/j.jcp.2020.109679
  2. E. Lorin, X. Yang, Computational wavefunction dynamics in photonic graphene with symmetry breaking, Applied Numerical Mathematics 199 (2024) 85–104, DOI 10.1016/j.apnum.2023.05.022, https://doi.org/10.1016/j.apnum.2023.05.022
  3. W. Bao, Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM Journal on Scientific Computing 25 (2004) 1674–1697, DOI 10.1137/S1064827503422956, https://doi.org/10.1137/S1064827503422956
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  5. C. Besse, R. Duboscq, S. Le Coz, Gradient flow approach to the calculation of stationary states on nonlinear quantum graphs, Annales Henri Lebesgue 5 (2022) 387–428, DOI 10.5802/ahl.126, https://doi.org/10.5802/ahl.126
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  9. J. Li, Energy-preserving exponential integrator Fourier pseudo-spectral schemes for the nonlinear Dirac equation, Applied Numerical Mathematics 172 (2022) 1–26, DOI 10.1016/j.apnum.2021.09.006, https://doi.org/10.1016/j.apnum.2021.09.006
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