On the Eigenfunctions of No-pair Operators in Classical Magnetic Fields

We consider a relativistic no-pair model of a hydrogenic atom in a classical, exterior magnetic field. First, we prove that the corresponding Hamiltonian is semi-bounded below, for all coupling constants less than or equal to the critical one known for the Brown-Ravenhall model, i.e., for vanishing magnetic fields. We give conditions ensuring that its essential spectrum equals [1,∞) and that there exist infinitely many eigenvalues below 1. (The rest energy of the electron is 1 in our units.) Assuming that the magnetic vector potential is smooth and that all its partial derivatives increase subexponentially, we finally show that an eigenfunction corresponding to an eigenvalue λ < 1 is smooth away from the nucleus and that its partial derivatives of any order decay pointwise exponentially with any rate a < √ 1− λ, for λ ∈ [0, 1), and a < 1, for λ < 0.