# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Y. Aharonov and D. Bohm, Phys. Rev., 115, 485–491 (1959). · Zbl 0099.43102 [2] M. Peshkin and A. Tonomura, The Aharonov-Bohm Effect (Lect. Notes Phys., Vol. 340), Springer, Berlin (1989). [3] K. Huang, Quarks, Leptons, and Gauge Fields, World Scientific, Singapore (1982). [4] P. de Sousa Gerbert and R. Jackiw, Comm. Math. Phys., 124, 229–260 (1989). · Zbl 0685.35093 [5] P. de Sousa Gerbert, Phys. Rev. D, 40, 1346–1349 (1989). [6] M. G. Alford, J. March-Pussel, and F. Wilczek, Nucl. Phys. B, 328, 140–158 (1989). [7] M. G. Alford and F. Wilczek, Phys. Rev. Lett., 62, 1071–1074 (1989). [8] I. V. Tyutin, ”Electron scattering by a solenoid [in Russian],” Preprint FIAN No. 27, Lebedev Inst. Phys., Moscow (1974); arXiv:0801.2167v2 [quant-ph] (2008). [9] V. R. Khalilov and I. V. Mamsurov, Theor. Math. Phys., 161, 1503–1512 (2009). · Zbl 1183.81117 [10] Y. Aharonov and A. Casher, Phys. Rev. Lett., 53, 319–321 (1984). [11] S. P. Gavrilov, D. M. Gitman, and A. A. Smirnov, Eur. Phys. J. C, 32, No. S1, 119–142 (2004); arXiv:hepth/0210312v3 (2002). [12] S. P. Gavrilov, D. M. Gitman, A. A. Smirnov, and B. L. Voronov, ”Dirac fermions in a magnetic solenoid field,” in: Focus on Mathematical Physics Research (C. V. Benton, ed.), Nova Science, Hauppauge, N. Y. (2004), pp. 131–168; arXiv:hep-th/0308093v2 (2003). [13] C. R. Hagen, Phys. Rev. Lett., 64, 503–506 (1990). · Zbl 1050.81707 [14] V. R. Khalilov and C.-L. Ho, Ann. Phys., 323, 1280–1293 (2008); arXiv:0708.3131v2 [hep-th] (2007). [15] V. R. Khalilov, Modern Phys. Lett. A, 21, 1647–1656 (2006). · Zbl 1099.81051 [16] M. A. Naimark, Linear Differential Operators, Frederick Ungar, New York (1967). · Zbl 0219.34001 [17] S. G. Krein, ed., Functional Analysis [in Russian], Nauka, Moscow (1972). [18] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, Self-Adjointness, Acad. Press, New York (1975). · Zbl 0308.47002 [19] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics [in Russian], Nauka, Moscow (1980); English transl. (Vol. 4 of Course of Theoretical Physics by L. D. Landau and E. M. Lifshitz), Pergamon, Oxford (1982).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.