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Russia Izhevsk
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.
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