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

Dolbeault, Jean; Esteban, Maria J.; Loss, Michael (2008), Characterization of the critical magnetic field in the Dirac-Coulomb equation, Journal of Physics A: Mathematical and Theoretical, 41, 18, p. 185-303. http://dx.doi.org/10.1088/1751-8113/41/18/185303

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00201095/en/
Date
2008
Journal name
Journal of Physics A: Mathematical and Theoretical
Volume
41
Number
18
Pages
185-303
Publication identifier
http://dx.doi.org/10.1088/1751-8113/41/18/185303
Metadata
Show full item record
Author(s)
Dolbeault, Jean cc
Esteban, Maria J. cc
Loss, Michael
Abstract (EN)
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.
Subjects / Keywords
Landau levels; pair creation; relativistic hydrogen atom; Dirac-Coulomb Hamiltonian; Dirac equation; magnetic field; min-max levels; ground state; Relativistic quantum mechanics

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