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TMF, 2007, Volume 150, Number 1, Pages 41–84 (Mi tmf5965)  

This article is cited in 43 scientific papers (total in 43 papers)

The Dirac Hamiltonian with a superstrong Coulomb field

B. L. Voronova, D. M. Gitmanb, I. V. Tyutina

a P. N. Lebedev Physical Institute, Russian Academy of Sciences
b Universidade de São Paulo

Abstract: 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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English version:
Theoretical and Mathematical Physics, 2007, 150:1, 34–72

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Received: 08.08.2006

Citation: B. L. Voronov, D. M. Gitman, I. V. Tyutin, “The Dirac Hamiltonian with a superstrong Coulomb field”, TMF, 150:1 (2007), 41–84; Theoret. and Math. Phys., 150:1 (2007), 34–72

Citation in format AMSBIB
\by B.~L.~Voronov, D.~M.~Gitman, I.~V.~Tyutin
\paper The~Dirac Hamiltonian with a~superstrong Coulomb field
\jour TMF
\yr 2007
\vol 150
\issue 1
\pages 41--84
\jour Theoret. and Math. Phys.
\yr 2007
\vol 150
\issue 1
\pages 34--72

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