We consider the quantum-mechanical problem of a relativistic Dirac particle moving in the Coulomb field of a point charge Ze. In the literature, it is often declared that a quantum-mechanical description of such a system does not exist for charge values exceeding the so-called critical charge with Z = α −1 = 137 based on the fact that 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 value of charge. What is more, 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 non uniqueness of the self-adjoint Hamiltonian, but this non uniqueness is also characteristic for charge values less than the critical one (and larger than the subcritical charge with Z = ( √ 3/2)α −1 = 118). We present the spectra and (generalized) eigenfunctions for all self-adjoint Hamiltonians. The methods used are 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 of crucial importance is an open question.
In this paper, we present a mathematically rigorous quantum-mechanical treatment of a one-dimensional motion of a particle in the Calogero potential αx −2 . Although the problem is quite old and well-studied, we believe that our consideration, based on a uniform approach to constructing a correct quantum-mechanical description for systems with singular potentials and/or boundaries, proposed in our previous works, adds some new points to its solution. To demonstrate that a consideration of the Calogero problem requires mathematical accuracy, we discuss some "paradoxes" inherent in the "naive " quantum-mechanical treatment. We study all possible self-adjoint operators (self-adjoint Hamiltonians) associated with a formal differential expression for the Calogero Hamiltonian. In addition, we discuss a spontaneous scale-symmetry breaking associated with self-adjoint extensions. A complete spectral analysis of all self-adjoint Hamiltonians is presented. 4 oscillator representation for the Calogero Hamiltonians.2 A "naive" treatment of the problem and related paradoxesAs mentioned above, the consideration of this section is on the so-called "physical level of rigor", or, in other words, "naive", so we actually repeat here a negative experience of the first researches. We start with the formal differential expression, or differential operation (d x = d/dx),for the Calogero Hamiltonian, and consider it as an s.a. operatorĤ in the Hilbert space H = L 2 (R + ) of quantum states for any α, conventionally without any reservations about its domain. We say in advance that the latter is precisely the reason for paradoxes. In QM, the time evolution governed by an s.a. HamiltonianĤ is unitary and is defined for all moments of time, although, as we have mentioned in Introduction, an analogue of a "fall to the center" is well-known from textbooks in the case of α < −1/4: in this case, the spectrum ofĤ is unbounded from below. This is argued [15] by considering the singular Calogero potential as a limit of bounded regularized potentials,
We consider the Dirac equation in the magnetic-solenoid field (the field of a solenoid and a collinear uniform magnetic field). For the case of Aharonov-Bohm solenoid, we construct self-adjoint extensions of the Dirac Hamiltonian using von Neumann's theory of deficiency indices. We find self-adjoint extensions of the Dirac Hamiltonian and boundary conditions at the AB solenoid. Besides, for the first time, solutions of the Dirac equation in the magnetic-solenoid field with a finite radius solenoid were found. We study the structure of these solutions and their dependence on the behavior of the magnetic field inside the solenoid. Then we exploit the latter solutions to specify boundary conditions for the magnetic-solenoid field with Aharonov-Bohm solenoid.
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