Geodesic
The Dirac Hamiltonian with a superstrong Coulomb field
Teoretičeskaâ i matematičeskaâ fizika, Tome 150 (2007) no. 1, pp. 41-84 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the quantum mechanical problem of a relativistic Dirac particle moving in the Coulomb field of a point charge $Ze$. It is often declared in the literature that a quantum mechanical description of such a system does not exist for charge values exceeding the so-called critical charge with $Z=\alpha^{-1}=137$ because the standard expression for the lower bound-state energy yields complex values at overcritical charges. We show that from the mathematical standpoint, there is no problem in defining a self-adjoint Hamiltonian for any charge value. Furthermore, the transition through the critical charge does not lead to any qualitative changes in the mathematical description of the system. A specific feature of overcritical charges is a nonuniqueness of the self-adjoint Hamiltonian, but this nonuniqueness is also characteristic for charge values less than critical $($and larger than the subcritical charge with $Z=(\sqrt{3}/2)\alpha^{-1}=118)$. We present the spectra and $($generalized$)$ eigenfunctions for all self-adjoint Hamiltonians. We use the methods of the theory of self-adjoint extensions of symmetric operators and the Krein method of guiding functionals. The relation of the constructed one-particle quantum mechanics to the real physics of electrons in superstrong Coulomb fields where multiparticle effects may be crucially important is an open question.
Keywords: Dirac Hamiltonian, Coulomb field, self-adjoint extensions, spectral analysis. 
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B. L. Voronov; D. M. Gitman; I. V. Tyutin. The Dirac Hamiltonian with a superstrong Coulomb field. Teoretičeskaâ i matematičeskaâ fizika, Tome 150 (2007) no. 1, pp. 41-84. http://geodesic.mathdoc.fr/item/TMF_2007_150_1_a2/

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