On the Positivity of the Jansen-hess Operator for Arbitrary Mass

acting on the Hilbert space L(R) ⊗ C, where γ := Ze, Z the nuclear charge number, e = (137.04) the fine structure constant, α and β the Dirac matrices and x := |x|. It is well-known that H is not bounded from below. As long as pair creation is neglected, the conventional way to circumvent this deficiency is the introduction of the semibounded operator P+HP+ where P+ projects onto the positive spectral subspace of H (Furry picture, see Sucher [10] and [11] for a review). Jansen and Heß [8], based on work by Douglas and Kroll [3] suggested an approximate operator which is derived from a Foldy-Wouthuysen-type transformation scheme. It is a second-order operator in the potential strength γ. It can be written in the form Λ+(D0 + V + i 2 [W1, B1]) Λ+ on L (R) ⊗ C where Λ+ projects onto the positive spectral subspace of the free Dirac operator D0 while W1 and B1 are operators linear in γ [7]. Alternatively, as in [4, 2], it can be reduced to an operator acting on two-spinors φ ∈ L(R) ⊗ C