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## Archive of Issues

Russia Izhevsk
Year
2018
Volume
28
Issue
2
Pages
222-230
 Section Mathematics Title Majorana states in a $p$-wave superconducting nanowire Author(-s) Tinyukova T.S.a Affiliations Udmurt State Universitya Abstract At the present time, the Majorana bounded states (MBSs) and associated phenomena, such as the variation of the conductance, are being actively studied in the physical literature because of the highly probable use of MBSs in quantum computations. In spite of the urgency, a rigorous mathematical study of the spectral properties and scattering for the one-particle Bogolyubov-de Gennes operator $H,$ commonly used for investigation of MBS's, has almost never been carried out. The methods proposed in the article allow one to obtain mathematically and physically interesting results. In this paper, we study the problem of the existence of MBSs (that is, the existence of a zero eigenvalue) for the Bogolyubov-de Gennes Hamiltonian in the case of an infinite one-dimensional superconducting structure in the presence of a potential. Conditions for the existence of MBSs are obtained. The scattering problem for the Bogolyubov-de Gennes operator with a potential is studied. The Green's function of the operator $H$ used in solving these problems is also found. Keywords Bogolyubov-de Gennes Hamiltonian, Green's function, spectrum, eigenvalue, scattering problem, transmission probability, Majorana bounded states UDC 517.958, 530.145.6 MSC 81Q10, 81Q15 DOI 10.20537/vm180208 Received 10 May 2018 Language Russian Citation Tinyukova T.S. Majorana states in a $p$-wave superconducting nanowire, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 2, pp. 222-230. References Elliot S.R., Franz M. Colloquium: Majorana fermions in nuclear, particle, and solid-state physics, Rev. Mod. Phys., 2015, vol. 87, issue 1, pp. 137-163. DOI: 10.1103/RevModPhys.87.137 Alicea J. New directions in the pursuit of Majorana fermions in solid state systems, Rep. Prog. Phys., 2012, vol. 75, issue 7, 076501. DOI: 10.1088/0034-4885/75/7/076501 Sato M., Fujimoto S. Majorana fermions and topology in superconductors, Journal of the Physical Society of Japan, 2016, vol. 85, no. 7, 072001. DOI: 10.7566/JPSJ.85.072001 Sarma S.D., Nag A., Sau J.D. How to infer non-Abelian statistics and topological visibility from tunneling conductance properties of realistic Majorana nanowires, Phys. Rev. B., 2016, vol. 94, issue 3, 035143. DOI: 10.1103/PhysRevB.94.035143 Chuburin Yu.P. Existence of Majorana bound states near impurities in the case of a small superconducting gap, Physica E: Low-dimensional Systems and Nanostructures, 2017, vol. 89, pp. 130-133. DOI: 10.1016/j.physe.2017.02.017 Kitaev A.Yu. Unpaired Majorana fermions in quantum wires, Phys.-Usp., 2001, vol. 44, pp. 131-136. DOI: 10.1070/1063-7869/44/10S/S29 Karzig T., Refael G., von Oppen F. Boosting Majorana zero modes, Phys. Rev. X., 2013, vol. 3, issue 4, 041017. DOI: 10.1103/PhysRevX.3.041017 Sarma S.D., Sau J.D., Stanescu T.D. Splitting of the zero-bias conductance peak as smoking gun evidence for the existence of the Majorana mode in a superconductor-semiconductor nanowire, Phys. Rev. B., 2012, vol. 86, issue 22, 220506. DOI: 10.1103/PhysRevB.86.220506 Full text