Variational collapse in gradient-based minimization of the Dirac equation: a general selection mechanism for positive-energy eigenstates

The Dirac Hamiltonian is unbounded from below: its spectrum contains a negative-energy continuum (the Dirac sea), so the physical positive-energy bound states are saddle-type stationary points of the Rayleigh quotient rather than minima. Gradient-based minimization, which converges for the nonrelativistic Schrödinger eigenproblem because that Hamiltonian is bounded below, therefore undergoes variational collapse: descent drives trial states into the negative-energy continuum. All known remedies are problem-specific devices - polynomial spectral filters in lattice nuclear covariant density functional theory, inverse-Hamiltonian transformations, kinetic-balanced or problem-tailored bases in relativistic quantum chemistry, component-imbalance (balance-operator) constraints in gradient flows for nonlinear Dirac operators, compact integral-operator reformulations, and residual-based neural-network training that abandons the variational structure altogether. The open problem is a general selection or constraint mechanism that makes positive-energy Dirac eigenstates local minima of a functional amenable to standard gradient-based minimization, defined without problem-specific bases, exact spectral projectors, or Hamiltonian inversion, together with provable convergence guarantees for the resulting iterations - or a rigorous demonstration that no such mechanism exists. Rigorous minimax characterizations of eigenvalues in spectral gaps partially characterize the spectrum, but as saddle quantities that leave the optimization landscape descent-hostile.

Background

The Dirac equation is the relativistic wave equation for spin-1/2 particles; its stationary eigenproblem $H\psi = E\psi$ has $H = c\,\alpha\cdot p + \beta mc^2 + V(x)$, where $\alpha$ and $\beta$ are $4\times4$ Dirac matrices, $c$ is the speed of light, and $V$ is a (possibly self-consistent) potential. Unlike the Schrödinger Hamiltonian, the Dirac Hamiltonian is not bounded from below: its essential spectrum is $(-\infty, -mc^2]\,\cup\,[mc^2, \infty)$ plus discrete bound states in the gap $(-mc^2, mc^2)$, with bound states of both positive and negative energy. For the nonrelativistic eigenproblem, the lowest eigenvalue is the minimum of the Rayleigh quotient $\langle\psi|H|\psi\rangle/\langle\psi|\psi\rangle$, so imaginary-time propagation, gradient descent, and conjugate-gradient minimization converge to ground states; this underlies practical density-functional-theory solvers. For the Dirac operator the same quotient has no minimum: because of the negative-energy continuum (the Dirac sea), each positive-energy bound state is a stationary point of saddle type - its Morse index equals the number of eigenvalues below it - so any naive descent path lowers the energy without bound and 'falls into the Dirac sea'. This is the variational collapse problem; it is the relativistic cousin of the finite-basis 'spurious state' and Brown-Ravenhall instabilities in four-component quantum chemistry, where the Brown-Ravenhall instability makes the positive-energy-projected ('no-pair') Hamiltonian unbounded below once the projection is realized in an incomplete basis, and where basis sets must be constructed with kinetic balance - matching the large (upper) and small (lower) two-component blocks of the Dirac spinor - to avoid unphysical low states.

A long line of work offers partial remedies. Talman proposed a minimax principle - minimize over the large component of the spinor, maximize over the small one - explored numerically for molecules and the Dirac-Coulomb problem; mathematics later established rigorous minimax characterizations of eigenvalues in spectral gaps of Dirac operators with Coulomb-type potentials. These results characterize the eigenvalues but leave them as saddle (minimax) quantities: they do not produce a functional whose local minima are the positive-energy states. In nuclear lattice covariant density functional theory (CDFT), where the Dirac equation with self-consistent mean fields must be solved on 3D coordinate grids, three families of devices are used: the inverse Hamiltonian method (IHM), which iterates with $(H-E_0)^{-1}$ so that states near the reference energy $E_0$ become extremal for the inverted operator; Wilson-fermion discretizations (a lattice Dirac operator with an added momentum-dependent term that removes fermion doubling, the spurious duplication of low-energy modes at the Brillouin-zone edge) combined with IHM, which enabled the first 3D mesh relativistic density-functional calculations; and the preconditioned conjugate-gradient method with a filtering function (PCG-F), in which a polynomial filter $p(H)$ suppresses negative-energy components so that positive-energy states become minima within the filtered subspace. PCG-F has since carried production 3D CDFT calculations, including fully self-consistent and exact-exchange ones. In numerical analysis, gradient-flow schemes for nonlinear Dirac operators use balance operators - normalizations that preserve the imbalance between large and small spinor components - to damp the unwanted spectral sector without exact spectral projectors; their fully discrete convergence analysis remains open. Other routes sidestep rather than solve the selection problem: recasting the bound-state search as an eigenproblem for a parameterized compact integral operator (whose spectrum is discrete and bounded), reformulating training as residual minimization with physics-informed neural networks (PINNs, networks trained to minimize the equation residual at sample points rather than an energy functional), or importing IHM into deep-neural-network solvers. Notably, the residual functional vanishes on negative-energy eigenstates as well, so residual-based training avoids collapse operationally without providing a mechanism that selects the positive-energy branch, and no convergence guarantees are available for it.

Problem Statement

For the Dirac eigenproblem $H\psi = E\psi$ with $H = c\,\alpha\cdot p + \beta mc^2 + V$, where $V$ is a fixed external or self-consistent mean-field potential, the negative-energy continuum makes the positive-energy bound states saddle-type stationary points of the Rayleigh quotient, so gradient-based minimization undergoes variational collapse into the Dirac sea. Construct - or rigorously rule out - a general selection or constraint mechanism for gradient-based minimization of this Dirac eigenproblem, namely a functional $F$ together with a constraint set (or modified normalization/dynamics) such that:

(i) the positive-energy bound states of $H$ are local minima of $F$ over the constraint set, and negative-energy states are not attractors of the associated gradient-based iteration;

(ii) the mechanism is defined in terms of the generic Dirac structure only - it must not require an exact spectral projector onto the positive-energy subspace, inversion of the Hamiltonian, a spectral filter (a polynomial or other fixed function of $H$) whose form is tuned to the spectrum of the specific potential, or a basis tailored to the specific problem or potential;

(iii) the gradient-based minimization built on the mechanism carries provable convergence guarantees under a stated discretization.

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