phone +7 (3412) 91 60 92

Archive of Issues

Russia Izhevsk
Section Mathematics
Title The scattering problem for a discrete Schrödinger operator with the “resonant” potential on the graph
Author(-s) Morozova L.E.a
Affiliations Izhevsk State Technical Universitya
Abstract We consider a discrete Schrödinger operator on the graph, which is the Hamiltonian in the tight-binding approach of an electron in the system consisting of a quantum wire, and two embedded quantum dots. This operator describes the double-barrier resonant nanostructure, in which one of the barriers is a non-local potential. The essential and absolutely continuous spectra of this operator are described. We study the scattering problem in the stationary approach for two possible directions of particles propagation. The conditions of total reflection and total transmission are found.
Keywords discrete Schrödinger operator, spectrum, the Lippmann-Schwinger scattering problem, quantum dot
UDC 517.958, 530.145.6
MSC 81Q10, 81Q15
DOI 10.20537/vm130104
Received 10 December 2012
Language Russian
Citation Morozova L.E. The scattering problem for a discrete Schrödinger operator with the “resonant” potential on the graph, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 1, pp. 29-34.
  1. Orellana P.A., Domingulz-Adame F., Gomes I., Ladron de Guevara M.L. Transport through a quantum wire a side quantum-dot array, Phys. Rev. B., 2004, vol. 67, 085321 (5 p).
  2. Gores J., Goldhaber-Gordon D., Heemeyer S., Kastner M.A. Fano resonances in electronic transport through a single-electron transistor, Phys. Rev. B., 2000, vol. 62, no. 3, pp. 2188–2195.
  3. Fuhrer A., Brusheim P., Ihn T., Sigrist M., Ensslin K., Wegscheider W., Bichler M. Fano effect in a quantum-ring-quantum-dot system with tunable coupling, Phys. Rev. B., 2006, vol. 73, 205326 (9 p).
  4. Chakrabarti A. Fano resonance in discrete lattice models: Controlling lineshapes with impurities, Phys. Letters A., 2007, vol. 336, issues 4–5, pp. 507–512.
  5. Reed M., Simon B. Metody sovremennoi matematicheskoi fiziki. I. Funktsional’nyi analiz (Methods of Mathematical Physics. I. Functional analysis), Moscow: Mir, 1977, 357 p.
  6. Baranova L.Y., Chuburin Y.P. Quasi-levels of the two-particle discrete Schrödinger operator with a perturbed periodic potential, J. Phys. A.: Math. Theor., 2008, vol. 41, no. 435205 (11 p).
  7. Tinyukova T.S., Chuburin Y.P. Quasi-levels of the discrete Schrödinger equation with a decreasing potential on a graph, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2009, no. 3, pp. 104–113.
  8. Baranova L.E., Chuburin Yu.P. On quasi-levels of the discrete two-particle Schrödinger operator with a decreasing small potential, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2008, no. 1, pp. 35–46.
  9. Chuburin Yu.P. A discrete Schrödinger operator on a graph, Theor. Math. Phys., 2010, vol. 165, no. 1, pp. 1335–1347.
  10. Tinyukova T.S. The Lippmann–Schwinger equation for quantum wires, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 1, pp. 99–104.
  11. Zakhar’ev B.N. Novaya azbuka kvantovoi mekhaniki (A new alphabet of quantum mechanics), Izhevsk: Udmurt State University, 1997, 160 p.
Full text
<< Previous article
Next article >>