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## Archive of Issues

Russia Izhevsk
Year
2017
Volume
27
Issue
2
Pages
257-266
 Section Mathematics Title Scattering and quasilevels in the SSH model Author(-s) Tinyukova T.S.a Affiliations Udmurt State Universitya Abstract Topological insulator is a special type of material that represents an insulator in the interior (“in bulk”) and conducts electricity on the surface. The simplest topological insulator is a finite chain of atoms in polyacetylene. In the last decade topological insulators are actively studied in the physics literature. A great interest to topological insulators (and also to topologically similar superconducting systems) is due to the presence of a link between “volume” and “boundary”. In this article, we have studied the discrete model SSH (Su-Schrieffer-Heeger) for polyacetylene. This model describes an electron in a one-dimensional chain of atoms with two alternating amplitudes of the transition to a neighboring atom. We have found the spectrum and resolution of this operator. The quasilevels (eigenvalues and resonances) in the case of a small potential have been investigated. In addition, we obtained a solution of the Lippmann-Schwinger equation and asymptotic formulas for the probability of transmission and reflection in case of small perturbation. Keywords resolution, spectrum, eigenvalue, resonance, Lippmann-Schwinger equation, probability of reflection UDC 517.958, 530.145.6 MSC 81Q10, 81Q15 DOI 10.20537/vm170209 Received 1 February 2017 Language Russian Citation Tinyukova T.S. Scattering and quasilevels in the SSH model, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 2, pp. 257-266. References Hasan M.Z., Kane C.L. Colloquium: topological insulators, Reviews of Modern Physics, 2010, vol. 82, issue 4, pp. 3045-3067. DOI: 10.1103/RevModPhys.82.3045 Bardarson J.H., Moore J.E. Quantum interference and Aharonov-Bohm oscillations in topological insulators, Rep. Progr. Phys., 2013, vol. 76, no. 5, 056501. DOI: 10.1088/0034-4885/76/5/056501 Asbóth J.K., Oroszlány L., Pályi A. A short course on topological insulators: band-structure topology and edge states in one and two dimensions, Lecture Notes in Physics, 2016, vol. 919. DOI: 10.1007/978-3-319-25607-8 Ruzicka F. Hilbert space inner products for $\mathcal {PT}$-symmetric Su-Schrieffer-Heeger models, International Journal of Theoretical Physics, 2015, vol. 54, issue 11, pp. 4154-4163. DOI: 10.1007/s10773-015-2531-4 Leijnse M., Flensberg K. Introduction to topological superconductivity and Majorana fermions, Semiconductor Science and Technology, 2012, vol. 27, no. 12, 124003. DOI: 10.1088/0268-1242/27/12/124003 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 Tinyukova Т.S. Scattering in the case of the discrete Schr\"odinger operator for intersected quantum wires, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2012, issue 3, pp. 74-84 (in Russian). DOI: 10.20537/vm120308 Full text