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Russia Izhevsk
Section  Mathematics 
Title  Scattering in the case of the discrete Schrödinger operator for intersected quantum wires 
Author(s)  Tinyukova T.S.^{a} 
Affiliations  Udmurt State University^{a} 
Abstract  The paper considers the discrete Schrödinger operator on a graph with vertices on two intersecting lines, which is perturbed by a decreasing potential. This operator is the Hamiltonian of an electron near a structure formed by a quantum dot and four outgoing quantum wires in the tightbinding approximation widely used in the physics literature for studying such nanostructures. We have proved the existence and uniqueness of the solution of the corresponding LippmannSchwinger equation and obtained the asymptotic formula for it. The nonstationary scattering picture has been studied. The scattering problem for the above operator in the case of a small potential, and also in the case of both a small potential and small velocity of a quantum particle, is investigated. Asymptotic formulas for the probabilities of the particle propagation in all possible directions have been obtained. 
Keywords  discrete LippmannSchwinger equation, reflection and transmission amplitudes 
UDC  517.958, 530.145.6 
MSC  81Q10, 81Q15 
DOI  10.20537/vm120308 
Received  7 April 2012 
Language  Russian 
Citation  Tinyukova T.S. Scattering in the case of the discrete Schrödinger operator for intersected quantum wires, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 3, pp. 7484. 
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