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Archive of Issues

Russia Izhevsk
Year
2016
Volume
26
Issue
4
Pages
535-542
 Section Mathematics Title The quasi-levels of the Dirac two-dimensional difference operator in a strip Author(-s) Tinyukova T.S.a Affiliations Udmurt State Universitya Abstract In the last decade, topological insulators have been actively studied in the physics literature. Topological insulator is a special type of material that is within the scope of an insulator and conducts electricity on the surface. Topological insulators have interesting physical properties, for example, the topological properties of this material can be stably maintained up to high temperatures. Topological insulators can be used in a wide variety of microelectronic devices ranging from very fast and efficient processors to topological quantum computers. The electron in topological insulators is described by the massless Dirac operator. Such operators in quasi-one-dimensional structures (for example, strips with different boundary conditions) are very interesting not only from a physical, but also from a mathematical point of view, but they are still poorly understood by mathematicians. In this article, we have found the eigenvalues of the Dirac difference operator for a potential of the form $V_0 \delta_{n0}.$ We have studied the quasi-levels (eigenvalues and resonances) of the operator in the case of small potentials. Keywords Dirac difference operator, resolution, spectrum, quasi-level, eigenvalues, resonance UDC 517.958, 530.145.6 MSC 81Q10, 81Q15 DOI 10.20537/vm160408 Received 14 October 2016 Language Russian Citation Tinyukova T.S. The quasi-levels of the Dirac two-dimensional difference operator in a strip, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 4, pp. 535-542. References Hasan M.Z., Kane C.L. Colloquium: topological insulators, Rev. Mod. Phys., 2010, vol. 82, pp. 3045-3067. DOI: 10.1103/RevModPhys.82.3045 Bardarson J.H., Moore J.E. Quantum interference and Aharonov-Bohm oscillations in topological insulators, Reports on Progress in Physics, 2013, vol. 76, 056501. DOI: 10.1088/0034-4885/76/5/056501 Blokhintsev D.I. Osnovy kvantovoi mekhaniki (Foundations of quantum mechanics), Moscow: Vysshaya shkola, 1963, 619 p. Yokoyama T., Tanaka Y., Nagaosa N. Anomalous magnetoresistance of a two-dimensional ferromagnet / ferromagnet junction on the surface of a topological insulator, Physical Review B, 2010, vol. 81, 121401, 4 p. DOI: 10.1103/PhysRevB.81.121401 Chuburin Y.P. Electron scattering on the surface of a topological insulator, Journal of Physics A: Mathematical and Theoretical, 2014, vol. 47, 255203, 13 p. DOI: 10.1088/1751-8113/47/25/255203 Tinyukova Т.S. Two-dimensional difference Dirac operator in the strip, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2015, vol. 25, issue 1, pp. 93-100 (in Russian). DOI: 10.20537/vm150110 Shabat B.V. Vvedenie v kompleksnyi analiz. Chast' II. Funktsii neskol'kikh peremennykh (Introduction to complex analysis. Part II. Functions of several variables), Moscow: Nauka, 1976, 400 p. Reed М., Simon B. Metody sovremennoi matematicheskoi fiziki. IV. Analiz operatorov (Methods of modern mathematical physics. IV. Analysis of operators), Moscow: Mir, 1982, 430 p. Morozova L.I., Chuburin Y.P. On levels of the one-dimensional discrete Schrödinger operator with a decreasing small potential, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2004, issue 1 (29), pp. 85-94 (in Russian). Albeverio S. Reshaemye modeli v kvantovoi mekhanike (Solvable models in quantum mechanics), Moscow: Mir, 1991, 568 p. Full text