SCATTERING IN NETWORKS AND DESIGN OF THE RESONANCE TRIADIC QUANTUM SWITCH

A B MIKHAYLOVA

Department of Mathematical Physics
Institute for Physics
St Petersburg State University
Ulyanovskaya 1
St Petersburg 198904, Russia

B S PAVLOV

pavlov@aitken.scitec.auckland.ac.nz

Department of Mathematical Physics
Institute for Physics
St Petersburg State University
Ulyanovskaya 1
St Petersburg 198904, Russia

Department of Mathematics
University of Auckland
Private Bag 92019
Auckland, New Zealand

Based on connection between the Dirichlet-to-Neumann Map and the scattering matrix for scattering problem on the networks the resonance transmission through the splitting is considered.

The mathematical design of a realistic three-position quantum switch controlled by the constant electric field is suggested in form of a circular quantum well - a unit disc on a plane- with four straight channels attached to it. The magnitude of the constant electric field directed parallel to the disc may be defined such that rotation of this field in the plane of the device permits manipulation of the electron current through the triple splitting. Explicit expression for transmission coefficient from one channel to another is obtained via reduction of the analysis of the corresponding infinitely-dimensional spectral problem to the analysis of a relevant finite-dimensional analytic matrix function. Our techiques is based on methods developed in [7,6,8] and the results are published in other papers quoted below.

Our main practical result is the calculation of the working point of the switch in the multi-dimensional space of the numerical parameters of the switch , to enable the resonance manipulation of the current across the quantum well from one wire to another.

Bibliography

1

V. Bogevolnov, A. Mikhailova, B. Pavlov, A. Yafyasov.About Scattering on the ring, in: Operator Theory: Advances and Application, Vol. 124 (The Israel Gohberg Anniversary Volume : International Workshop in Gronningen, June 1988 ) ed. A. Dijksma, A.M. Kaashoek, A.C.M. Ran ; Birkhauser, Basel, 2001, pp 155-187.

2

A. Mikhaylova, B. Pavlov. Quantum domain as a triadic relay, in: Unconventional Models of Computations UMC'2K (Proceedings of the UMC'2K Conference, Brussels, Dec 2000) eds.I. Antoniou, C. Calude, M.J. Dinneen, Springer Verlag Series for Discrete Mathematics and Theoretical Computer Science (2001), pp 167-186.

3

B.Pavlov, I. Popov, V. Geyler, O. Pershenko, Possible construction of a quantum multiplexer Europhyics Letters, 52,(2),2000,pp 196-202

4

A.Mikhailova, B.Pavlov,I.Popov,T.Rudakova,A.M.Yafyasov Scattering on a compact domain with few semiinfinite wires attached: resonance case. Mathematishe Nachrichten, 235 (2002), p.101-128

5

A. Mikhailova, B. Pavlov.Quantum Domain as a triadic relay. April 2000, Department of Mathematics Report series No.420,ISSN 1173-0889, the University of Auckland, Auckland, NZ,16p. New technologies for narrow-gap semiconductors. Esprit project N 28890 ESPRIT NTCONGS Progress Reports (July 1, 1999 - December 31,1999)

6

B.Pavlov. Splitting of acoustic resonances in domains connected by a thin channel. New Zealand Journal of Mathematics, Vol. 25,(1996), p 199-216.

7

B. Pavlov. The theory of extensions and explicitly solvable models, Uspekhi Mat. Nauk,42(1987) p 99-131.

8

S. Albeverio, P. Kurasov Singular Perturbations of Differential Operators London Math. Society Lecture Notes Series 271, Cambridge Univ. Press (2000), 429 pp

9

A. Mikhailova, B. Pavlov Resonance triadic Quantum Switch Accepted for publication in Operator Theory Adv. Appl. Birkhäuser, Basel (Proceedings of Sonja Kovalevski Conference)