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The Dirac Hamiltonian with a superstrong Coulomb field. (English. Russian original) Zbl 1118.81027
Theor. Math. Phys. 150, No. 1, 34-72 (2007); translation from Teor. Mat. Fiz. 150, No. 1, 41-84 (2007).
Summary: 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.

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81V45 Atomic physics
70H40 Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics
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