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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2018, Volume 28, Issue 2, Pages 222–230 (Mi vuu633)

MATHEMATICS

Majorana states in a $p$-wave superconducting nanowire

T. S. Tinyukova

Department of Mathematical Analysis, Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia

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.

DOI: https://doi.org/10.20537/vm180208

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Bibliographic databases:

UDC: 517.958, 530.145.6
MSC: 81Q10, 81Q15

Citation: T. S. Tinyukova, “Majorana states in a $p$-wave superconducting nanowire”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 28:2 (2018), 222–230

Citation in format AMSBIB
\Bibitem{Tin18} \by T.~S.~Tinyukova \paper Majorana states in a $p$-wave superconducting nanowire \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2018 \vol 28 \issue 2 \pages 222--230 \mathnet{http://mi.mathnet.ru/vuu633} \crossref{https://doi.org/10.20537/vm180208} \elib{http://elibrary.ru/item.asp?id=35258689} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. T. S. Tinyukova, Yu. P. Chuburin, “Otrazhenie Andreeva v kontakte $p$-volnovoi sverkhprovodnik–normalnyi metall”, Izv. IMI UdGU, 54 (2019), 55–62
2. T. S. Tinyukova, Yu. P. Chuburin, “Cuschestvovanie maioranovskikh lokalizovannykh sostoyanii v prostoi modeli perekhoda Dzhozefsona”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 29:3 (2019), 351–362
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