Unexpected features of relativistic models

Abstract

Several unexpected properties of the Dirac equation, related to its multi-component structure and unboundedness from below of the Dirac Hamiltonian are discussed. In particular, a pathological behaviour of an exact density functional derived from either Dirac or Levy-Leblond equation is presented. For a oneelectron atom, as one should expect, the variational minimum of this functional gives the exact ground state energy and corresponds to the correct density. However, the same minimum is also reached by an infinite set of densities which do not correspond to the exact wave-function. Some of the wave-functions corresponding to the correct minimum are orthogonal to the exact one. It also appears that imposing the correct boundary conditions does not remove the fake solutions. As it is known, the two-electron Dirac-Coulomb equation does not have square-integrable solutions if the interaction term is present in the Hamiltonian. As a result, all the states of two-electron atoms, including the ground states are, from the formal point of view, auto-ionizing and should be treated using theoretical methods appropriate for the description of resonances. Some consequences of this facts are discussed. Transformations of the Dirac equation in the spinor space supply another degree of freedom in structuring methods of solving this equation. It appears that some non-standard representations of the Dirac equation may be more convenient in numerical applications and result in more stable variational procedures. Several examples are discussed