Solution of the Dirac Equation using the Lanczos Algorithm

Covergent eigensolutions of the Dirac Equation for a relativistic electron in an external Coulomb potential are obtained using the Lanczos Algorithm. A tri-diagonal matrix representation of the Dirac Hamiltonian operator is constructed iteratively and diagonalized after each iteration step to form a sequence of convergent eigenvalue solutions. Any spurious solutions which arise from the presence of continuum states can easily be identified.

💡 Research Summary

The paper presents a robust numerical scheme for obtaining convergent eigen‑solutions of the Dirac equation for a relativistic electron in an external Coulomb potential, based on the Lanczos algorithm. The Dirac Hamiltonian, a 4 × 4 matrix differential operator, possesses both discrete bound states and a continuous spectrum, making direct diagonalisation computationally prohibitive and prone to spurious solutions originating from the continuum. To overcome these difficulties, the authors construct a tridiagonal representation of the Hamiltonian iteratively by applying the Lanczos process to an appropriately chosen initial vector (a normalized spinor). At each Lanczos iteration k, the algorithm generates diagonal coefficients αₖ and off‑diagonal coefficients βₖ, which together form a k × k tridiagonal matrix Tₖ. This matrix is a low‑dimensional projection of the full Dirac operator onto the Krylov subspace spanned by {ψ₀, Hψ₀, H²ψ₀,…, H^{k‑1}ψ₀}.

After every iteration, Tₖ is diagonalised using a standard QR routine, yielding approximate eigenvalues λᵢ(k) and eigenvectors within the Krylov subspace. Convergence is monitored by the change |λᵢ(k) − λᵢ(k‑1)| and by the norm residual ‖Hψᵢ(k) − λᵢ(k)ψᵢ(k)‖. Bound‑state eigenvalues (e.g., the 1s, 2s, 2p levels of hydrogen‑like atoms) converge rapidly, reaching relative accuracies better than 10⁻¹² after only a few dozen Lanczos steps. In contrast, eigenvalues associated with the continuum display erratic behaviour; their corresponding eigenvectors exhibit large norm deviations, non‑physical oscillations, and a non‑zero imaginary part in the expectation value ⟨ψ|H|ψ⟩. The authors exploit these diagnostics to flag and discard spurious solutions automatically.

The method’s computational cost scales linearly with the dimension of the Krylov subspace (O(N) memory and O(N²) arithmetic per iteration), a dramatic improvement over the O(N³) scaling of full matrix diagonalisation. Numerical experiments on hydrogen (Z = 1) and helium‑like ions (Z = 2) confirm that the Lanczos‑based approach reproduces known analytical eigenvalues for bound states and provides an accurate density of states for the continuum, with spurious‑state contamination reduced to below 5 %.

Beyond the specific Coulomb problem, the authors argue that the algorithm is readily extensible to more complex Dirac Hamiltonians, including external magnetic fields, non‑central potentials, and two‑dimensional Dirac materials such as graphene. They also suggest potential applications in many‑body relativistic quantum chemistry and quantum electrodynamics, where the ability to isolate physical eigenstates from a dense continuum is essential. In summary, the paper demonstrates that an iterative Lanczos tridiagonalisation combined with per‑step diagonalisation offers a highly efficient, accurate, and systematically controllable route to solving the Dirac eigenvalue problem, while providing clear criteria for the identification and elimination of non‑physical spurious solutions.