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## Archive of Issues

Russia Izhevsk
Year
2019
Volume
29
Issue
3
Pages
351-362
 Section Mathematics Title Existence of Majorana bounded states in a simple Josephson transition model Author(-s) Tinyukova T.S.a, Chuburin Yu.P.b Affiliations Udmurt State Universitya, Udmurt Federal Research Center, Ural Branch of the Russian Academy of Sciencesb Abstract For the last 15 years, Majorana bounded states (MBSs) and associated phenomena, such as variation of conductance and the Josephson effect, have been actively studied in the physical literature. Research in this direction is motivated by a highly probable use of MBSs in quantum computing. The article studies the eigenfunctions of the one-dimensional Bogolyubov-de Gennes operator with a delta-shaped potential at zero, describing localized states with energy in the spectral gap (superconducting gap). The transmission probabilities are found in the scattering problem for this operator, when the energies are close to the boundary of the superconducting gap. These problems are studied both for a superconducting order that is the only one on the whole straight line and is defined by the real constant $\Delta,$ and for a superconducting order defined by the function $\Delta\theta(-x)+\Delta e^{i\varphi}\theta(x)$ for $\varphi=0,\pi$ (i.e., for zero superconducting current and for current close to critical). The Hamiltonian used can be considered as the simplest model of the Josephson junction. It is proved that in both cases there are two MBSs, but with certain values of the parameters, i.e., MBSs are unstable. Moreover, the probability of passage is zero in both cases. 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/vm190306 Received 12 June 2019 Language Russian Citation Tinyukova T.S., Chuburin Yu.P. 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