 07314 Jean Dolbeault, Maria J. Esteban, Michael Loss
 Characterization of the critical magnetic field in the DiracCoulomb
equation
(8961K, Postscript)
Dec 23, 07

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Abstract. We consider a relativistic hydrogenic atom in a strong magnetic
field. The ground state level depends on the strength of the magnetic
field and reaches the lower end of the spectral gap of the Dirac
Coulomb operator for a certain critical value, the critical magnetic
field. We also define a critical magnetic field in a Landau level
ansatz.
In both cases, when the charge Z of the nucleus is not too small,
these critical magnetic fields are huge when measured in Tesla, but
not so big when the equation is written in dimensionless form. When
computed in the Landau level ansatz, orders of magnitude of the
critical field are correct, as well as the dependence in Z. The
computed value is however significantly too big for a large Z, and
the wave function is not well approximated. Hence, accurate numerical
computations involving the Dirac equation cannot systematically rely
on the Landau level ansatz.
Our approach is based on a scaling property. The critical magnetic
field is characterized in terms of an equivalent eigenvalue problem.
This is our main analytical result, and also the starting point of
our numerical scheme.
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