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Fermion bound states in the Aharonov-Bohm field in \(2+1\) dimensions. (English. Russian original) Zbl 1196.81122
Theor. Math. Phys. 163, No. 1, 511-516 (2010); translation from Teor. Mat. Fiz. 163, No. 1, 132-139 (2010).
Summary: We find exact solutions of the Dirac equation that describe fermion bound states in the Aharonov-Bohm potential in \(2+1\) dimensions with the particle spin taken into account. For this, we construct self-adjoint extensions of the Hamiltonian of the Dirac equation in the Aharonov-Bohm potential in \(2+1\) dimensions. The self-adjoint extensions depend on a single parameter. We select the range of this parameter in which quantum fermion states are bound. We demonstrate that the energy levels of particles and antiparticles intersect. Because solutions of the Dirac equation in the Aharonov-Bohm potential in \(2+1\) dimensions describe the behavior of relativistic fermions in the field of the cosmic string in \(3+1\) dimensions, our results can presumably be used to describe fermions in the cosmic string field.

MSC:
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
81U15 Exactly and quasi-solvable systems arising in quantum theory
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