|
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 Universitya
|
|
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 tight-binding approximation widely used in the physics literature for studying such nanostructures. We have proved the existence and uniqueness of the solution of the corresponding Lippmann-Schwinger equation and obtained the asymptotic formula for it. The non-stationary 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 Lippmann-Schwinger 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. 74-84.
|
|
References
|
- Tinyukova T.S. Quasi-levels of the discrete Schrödinger operator for a quantum waveguide, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 2, pp. 88-97.
- Tinyukova T.S. The Lippmann-Schwinger equation for quantum wires, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 1, pp. 99-104.
- 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.
- Miroshnichenko A.E., Kivshar Y.S. Engineering Fano resonances in discrete arrays, Phys. Rev. E, 2005, vol. 72, 056611 (7 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., Izhevsk, 2004, no. 1 (29), pp. 85-94.
- Chuburin Y. P. A discrete Schrödinger operator on a graph, Theor. Math. Phys., 2010, vol. 165, no. 1, 1335-1347.
- Reed М., Simon B. Metody sovremennoi matematicheskoi fiziki. I. Funktsionalnyi analiz (Methods of Mathematical Physics. I. Functional analysis), Moscow: Mir, 1977, 357 p.
- Reed М., Simon B. Metody sovremennoi matematicheskoi fiziki. III. Teoriya rasseyaniya (Methods of Mathematical Physics. III. Scattering Theory), Moscow: Mir, 1982, 443 p.
|
|
Full text
|
|
+7 (3412) 91 60 92
Russian