Spurious Modes in Dirac Calculations and How to Avoid Them

Abstract In this paper we consider the problem of the occurrence of spurious modes when computing the eigenvalues of Dirac operators, with the motivation to describe relativistic electrons in an atom or a molecule. We present recent mathematical results which we illustrate by simple numerical experiments. We also discuss open problems. Computing the eigenvalues of an operator on a computer can be a subtle task, in particular when one is interested in those lying in a gap of the spectrum. In this case, spurious modes can sometimes appear and persist when the size of the discretization basis is increased. The phenomenon, called spectral pollution, is well-known and well documented. For instance, it is encountered when dealing with perturbations of periodic Schrödinger operators [6, 8] or Sturm-Liouville operators [1, 30, 31]. It also appears in elasticity, electromagnetism and hydrodynamics [2, 3, 7, 9, 15, 22, 24, 25]. In this paper we are interested in relativistic computations based on the Dirac operator, like those used in quantum chemistry and atomic physics. The spectrum of the free Dirac operator is (−∞,−mc2] ∪ [mc2,∞) and adding an external potential usually creates eigenvalues in the gap (−mc2, mc2). Computing them might lead to spurious modes. Practical solutions to overcome this problem have been proposed