On the eigenfunctions of no-pair operators in classical magnetic fields

Details On the eigenfunctions of no-pair operators in classical magnetic fields On the eigenfunctions of no-pair operators in classical magnetic fields

2008-10-27

arxiv.org/abs/0810.4897v1

Integral Equations Operator Theory 65 (2): 255--283, 2009

Physics

Mathematical Physics

Scientific paper

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,\infty) 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 \lambda<1 is smooth away from the nucleus and that its partial derivatives of any order decay pointwise exponentially with any rate a<(1-\lambda^2)^{1/2}, for \lambda\in[0,1), and a<1, for \lambda<0.

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