Resolving the spurious-state problem in Dirac equation by using the staggered-grid method

TL;DR

This paper introduces a staggered-grid finite-difference scheme for the Dirac equation that effectively eliminates spurious states, improves convergence, and maintains simplicity, enabling accurate relativistic bound-state and scattering calculations.

Contribution

The paper presents a novel staggered-grid discretization method for the Dirac equation that suppresses spurious states without additional filtering or Wilson terms.

Findings

Successfully suppresses spurious states in Dirac equation discretization.

Demonstrates rapid convergence and reduced sensitivity to box size.

Retains simplicity and extendability to higher dimensions.

Abstract

Discretizing the Dirac equation on a uniform grid with the central difference formula often generates spurious states. We propose a staggered-grid scheme in the framework of the finite-difference method that suppresses these spurious states without introducing Wilson terms or ad-hoc filtering. In this approach, the large and small components of the Dirac equation are placed on interlaced nodes, and the first-order derivatives are evaluated between staggered points, yielding a Hamiltonian that breaks the unitary transformation between $H_\kappa$ and $H_{-\kappa}$. Benchmarks with the nuclear Woods-Saxon potentials demonstrate one-to-one agreement with the eigenvalues obtained from shooting method and asymmetric finite-difference method, rapid convergence for weakly bound states, and reduced box-size sensitivity. The method retains the simplicity of central differences and standard matrix diagonalization, while naturally extending to higher-order and multi-dimension systems. It provides a compact and efficient tool for relativistic bound-state and scattering calculations.