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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.

MSC:
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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