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 DirichlettoNeumann 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 threeposition 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 infinitelydimensional spectral problem to the
analysis of a relevant
finitedimensional 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
multidimensional 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.

 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 155187.
 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 167186.
 3
 B.Pavlov, I. Popov, V. Geyler, O. Pershenko,
Possible construction of a quantum multiplexer Europhyics
Letters, 52,(2),2000,pp 196202
 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.101128
 5
 A. Mikhailova, B. Pavlov.Quantum Domain as
a triadic relay. April 2000, Department of Mathematics Report
series No.420,ISSN 11730889, the University of Auckland,
Auckland, NZ,16p. New technologies for narrowgap 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 199216.
 7
 B. Pavlov. The theory of extensions and
explicitly solvable models,
Uspekhi Mat. Nauk,42(1987) p 99131.
 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)
EPSRC Gregynog Workshop, 2126 July 2002Gregynog Abstracts