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We consider a relativistic hydrogenic atom in a strong magnetic field. Theground state level depends on the strength of the magnetic field and reachesthe lower end of the spectral gap of the Dirac-Coulomb operator for a certaincritical value, the critical magnetic field. We also define a critical magneticfield in a Landau level ansatz. In both cases, when the charge Z of the nucleusis not too small, these critical magnetic fields are huge when measured inTesla, but not so big when the equation is written in dimensionless form. Whencomputed in the Landau level ansatz, orders of magnitude of the critical fieldare correct, as well as the dependence in Z. The computed value is howeversignificantly too big for a large Z, and the wave function is not wellapproximated. Hence, accurate numerical computations involving the Diracequation cannot systematically rely on the Landau level ansatz. Our approach isbased on a scaling property. The critical magnetic field is characterized interms of an equivalent eigenvalue problem. This is our main analytical result,and also the starting point of our numerical scheme.

Characterization of the critical magnetic field in the Dirac-Coulomb equation