The Dirac Operator; From Numerics to the Theory of G-convergence

Licentiate thesis, 2010

We consider two main issues concerning the Dirac operator, the first is widely known as the appearance of spurious eigenvalues within the spectrum. The second is the study of the asymptotic behavior of the eigenvalues for a family of Dirac operators with oscillatory potential added to the Coulomb-Dirac Hamiltonian. In the first problem a stable Finite element scheme is used to treat the problem of spuriousity of the radial Dirac operator with a Coulomb potential. The numerical accuracy depends strongly on the derivation of a fine-intrinsic stability parameter. In the second problem we consider the Coulomb-Dirac operator with addition to an abstract oscillating potential. Using the spectral measure of operators we project into the positive part of the perturbed Hamiltonian. By using G-convergence theory of positive self-adjoint operators in Hilbert spaces and ¡- convergence of the associated quadratic functionals we prove G-compactness for a family of positive Dirac operators under certain assumptions on the potentials.