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\begin{document}
{ \Large
\begin{center}

 RESONANCE QUANTUM  SWITCH\\
 AND QUANTUM  GATE\\
\vskip1cm \large N.T.\,Bagraev $^1$, A.B\,.Mikhailova $^2$,\\
B.S.\,Pavlov $^{2,3}$, L.V. Prokhorov $^4$ .\vskip10pt
\end{center}
}
%\normalsize\em
\noindent $^1$ A.F. Ioffe  Physico-Technical Institute, St.
Petersburg, 194021, Russia.\\ $^2$ Laboratory of the Theory of
Complex Systems, V.A. Fock Institute of Physics, St.Petersburg
State University, Ulianovskaya 1, Petrodvorets, St.Petersburg,
198504, Russia\\ $^3$Department of Mathematics , University of
Auckland, Private Bag 92019,\\ Auckland, New Zealand;\\
$^4$Department of Theoretical Physics, V.A. Fock Institute of
Physics, St.Petersburg State University, Ulianovskaya 1,
Petrodvorets, St.Petersburg, 198504, Russia\\

%\newtheorem{th}{Theorem}[section]
\newtheorem{lem}{Lemma}[section]
\newtheorem{cor}{Corollary}[section]

\centerline{\large\bf Abstract}

Solvable models for two- and  three-terminal Quantum Switches and
Quantum  Gates are suggested in form of a quantum ring witha  few
one-dimensional quantum wires  attached to it. In resonance case
when the Fermi level in the wires coincides with the  resonance
energy level on  the  ring , the magnitude of the governing
 electric field may  be specified such that the quantum
current through the switch from up-leading  wire  to the outgoing
wires may be controlled via rotation of the orthogonal  projection
of the field onto the plane of the device.The working parameters
of the switches  and gates  are defined in dependence of the
desired working temperature, the Fermi level \footnote{See
\cite{Madelung}}  and the effective mass of the electron  in the
wires. \vskip5pt

\section{Resonance Quantum Switch}
Interference of the  wave function may  serve a  base  for design
of  the  quantum  electronic  devices, see  \cite{Exner}. Though
the the  basic problems  of  the  mathematical  design of quantum
electronic  devices  were already formulated  in  term  of quantum
scattering by  the  beginning  of  nineties, see \cite{Adamjan},
still the  design of  most of modern resonance quantum devices,
beginning from classical Esaki diode up to  modern devices, see
for  instance \cite{Samuelson},\,\,\cite{Compano} are based on the
resonance of energy levels rather than on resonance properties of
the corresponding wave functions. At the same time modern
experimental technique already permits to observe resonance
effects caused by  details of  the shape of the resonance wave
functions, see \cite{B1}, \cite{B2}, \cite{B3}. In  frames  of the
EC-project "New technologies for narrow-gap semiconductors" (
ESPRIT-28890 NTCONGS,\,\,1998 - 1999) the problem on  mathematical
design of a  four-terminal  Quantum  Switch for triadic logic was
formulated by Professor  G. Metakides  and  Doctor  R.Compano from
the Industrial  Department of  the  European  Comission. Results
of the theoretical part of the project were published in  papers
\cite{Ring}, \cite{Domain}, \cite{Solvay}, \cite{P00}, \cite{MP00}
where a  new design of  Resonance Quantum Switches (RQS) was
suggested in  form of  a quantum domain or quantum ring with a few
quantum wires (terminals) attached to it. The idea of the new
design, as presented in \cite{MP00} for the  device designed in
form of a quantum domain (quantum well), is based on properties of
the {\it shape} of the {\it resonance eigenfunction} which
corresponds to the {\it resonance eigenvalue} \footnote{Resonance
eigenvalue is equal to the Fermi level in the up-leading wires} of
the Schr\"{o}dinger equation in the quantum well. These resonance
properties were observed first, see \cite{Opening}, in the
scattering problem  for acoustic scattering on a resonator with a
small  opening: an additional term in  the scattering amplitude
caused  by  the opening appeared to  be proportional to the value
of the resonance eigenfunction at  the opening (for the Neumann
boundary condition on  the  walls of the resonator) or to the
value of it's normal derivative (for the Dirichlet boundary
conditions).
 \par
We consider a  single  act of computation as  a scattering
process.  In the corresponding scattering problem for RQS formed
as a ring of radius $R$ a few (straight) quantum wires weakly
connected to the ring via tunneling of electrons of  mass $m$
across the potential barrier power of width $l$, hight $H$ with
electrons  charge  already  included into  it  in  proper  units.
The potential  on  the  wires  is  assumed to  be  equal  $V_2 <
E_f$, so  that  the  resonance  eigenvalue  $E_2 = E_f$  on  the
ring is  embedded  into  continuous  spectrum   of  the
Schr\"{o}dinger  operator on  the  whole  graph.  If  the
potential barriers  separating  the  wires  from  the  ring  are
  strong  enough then  the  connection between the  wires  and
  the  ring  may  be  reduced  to  the  boundary  condition  with
  the  small  parameter
 $$ \beta = \left(\cosh\frac{\sqrt{2m H}}{\hbar}l\right)^{-1} $$
at the contact points  $a_s$, see  below  Sections  3  and  4. For
weak  connection between the  ring  and  the  wires  the
transmission coefficient from one wire to another in the resonance
case appears to  be proportional to the product of the values of
the normalized resonance eigenfunctions $ \varphi(a_s)$ at  the
contact points see \cite{Ring}. If the renormalized energy
$\lambda = k^2 = (E - V_2)\frac{\hbar^2 R^2}{2m}$ proportional to
the depth of the quantum  wires $E - V_2$  at infinity  is close
to the renormalized  Fermi  level $\lambda_f = (E_f -
V_2)\frac{\hbar^2 R^2}{2m}$ and $\beta $  is small,then the
following approximate expression in  scaled  variables, see below
Section 4, is true for the transmission coefficient from the wire
attached to $a_s$ to the wire attached to $ a_t$ :
\[
S_{s,t} (\lambda) =
 \frac{2k|\beta|^2}{k |\beta|^2
|\vec\varphi|^2 - i(\lambda_f - \lambda) } \varphi(a_s) \varphi
(a_t) + O (\beta^2), \,\, s\neq t.
\]
where $|\vec\varphi|^2$ is the length  of  the  {\it
channel-vector} $\left(\varphi a_1,\,\varphi a_2,\,\dots \varphi
a_4 \right)$ the second term is uniformly small when $\beta\to 0$,
but the first one exhibits a nonuniform behaviour in dependence on
ratio $(\lambda_f - \lambda)/\beta^2$. The  last formula  being
applied formally  to the case $\lambda = \lambda_f$
 shows, that the  transmission  coefficient is approximately equal to
 \[
S_{s,t} (\lambda) = \frac{2}{|\vec{\varphi}|^2}\varphi(a_s)
\varphi (a_t) + O(\beta^2).
\]
This looks rather surprising  for  $\beta \approx 0$  since  it
would supply a nonzero transmission coefficient  for almost  zero
connection. Actually this  means that  the transmission
coefficients are not continuous  with  respect  to the
renormalized  energy $\lambda$ uniformly in $\beta$. The
physically significant  values of  the transmission coefficient
may be obtained in limit  case via
 averaging  over  intervals $ |E - E_f|< \kappa T$   for {\it
relatively} small and  {\it relatively} large  temperature. In
the first  case we still have:
\[
\overline{|S_{ij} (T)|^2} \approx
\frac{2\vert\varphi(a_s)\varphi(a_t)\vert^2} {|\vec{\varphi}|^4}
\]
but  in the second  case, we  have :
\[
\overline{|S_{ij} (T)|^2} \approx
4\frac{\vert\varphi(a_s)\varphi(a_t)\vert^2}{|\vec{\varphi}|^4}
\frac{1}{1 + \frac{\kappa^2 T^2}{\lambda_0|\beta|^4
|\vec{\varphi}|^4 }}.
\]
Hence  for  small  $\beta$ and non-zero temperatures  the averaged
transmission  coefficient  is small, according to natural physical
expectations.
\par
Nevertheless the  above  formulae show  that  in certain  range of
temperatures  the  transmission is  proportional to  the product
of  values of  the resonance eigenfunctions at  the  contact
points. Similar observation  takes  place for  switches  based on
the  quantum  well  with  Neumann boundary  conditions ,see
\cite{MP00} and analog of  it with  normal derivatives of  the
resonance eigenfunction remains true  for the Dirichlet boundary
conditions, see \cite{MP01}.
 \par
 One may obviously construct the
{\it dyadic} RQS basing on this observation. But even {\it
triadic} (four-terminal) RQS may be constructed with minimal
alteration of the geometrical construction. For instance, on a
circular quantum well $\Omega_0 : |\vec x|\leq R$ for a {\it
special choice} of the constant electric field ${\cal E} \vec
\nu,\,\, |\vec \nu|= 1,$ and  the shift potential $V_0$, see
\cite{MP00}, the corresponding Schr\"{o}dinger equation
 \begin{equation}
 \label{Schred}
 -\frac{\hbar^2}{2m}\bigtriangleup \psi +
 \left( {\cal E}e \langle \vec x,\, \vec \nu \rangle + V_0\right) \psi = E
\psi
\end{equation}
may  have  a resonance eigenfunction with  an  eigenvalue equal to
the Fermi level  $E = E_{_f}$ in the wires such that it  has only
one smooth  line of zeroes which  crosses the  circle  dividing it
in ratio $ 1:2 $. Then, attaching the wires $\Gamma$ to the domain
at the points $a_1,a_2,a_3,a_4$ characterized by  the central
angles $ 0,\, \pm \pi/3,\, \pi $ we obtain the Resonance Quantum
Switch manipulated by rotating $\vec \nu \to \vec \nu'$ of the
constant electric field  ${\cal E} \vec \nu$  in the plane
parallel to the plane of the device. In particular, if for some
direction $\vec\nu$ the line of zeroes arrives to  the  boundary
exactly at the contact points : $a_2,a_3$  (or $a_3,a_4$, or else
$a_4,a_2$ ) , then  the corresponding wires $\Gamma_2,\, \Gamma_3$
 ( or respectively $\Gamma_3,\, \Gamma_4$ , or else  $\Gamma_4,\,
\Gamma_2$)  are  blocked. With two outgoing  wires blocked, one
up-leading wire $\Gamma_1$ and  one outgoing  wire (respectively
$\Gamma_4$, or  $\Gamma_2$, or $\Gamma_3$) remain open. Hence the
electron current may go across the  well  from the up-leading wire
to the outgoing wire. The corresponding transmission and
reflection coefficients may be calculated  in course  of solution
of the corresponding scattering  problem, \cite{MP00}.
 \par
For fixed contact points the working regime of the
 RQS is defined by the position of the  working point $R,\,
{\cal E}, \, V_0 $ in the three-dimensional space of the
parameters. This position is uniquely  defined, see Section $2$,
by the desired working temperature $T$ and by the Fermi level
$E_{_f}$ in the up-leading quantum wires. Note, that {\it the
position of the working point can't be defined experimentally }
just  by naive scanning on one of parameters for other parameters
fixed at random, since the probability of proper choice of the
remaining parameters is zero (proportional to the zero-measure of
a point on  a $2-d$ plane).
\par
Discussing the  RQS based on  a  quantum  well in \cite{MP00} we
assumed  that the connection between the wires  and  the  quantum
well  was defined by some small parameter $\beta$ in  the
corresponding boundary conditions (Section $2$). This  boundary
parameter was interpreted in \cite{MP00} as an ``exponential
power" of the potential barrier separating the  wires from the
well. At the moment  of submitting of  the paper  the authors  did
not  have any idea how  the power of  the potential  barrier  may
be controlled practically. It appeared that actually  not the {\it
hight of the barrier} but the {\it width of the the part of the
up-leading channel} connecting the wire and the well may be
controlled by a special nano-electronic construction -- the {\it
split-gate}. The power  of the potential barrier inside the
split-gate may  be manipulated by the classical electric field
applied to the brush of boron's dipoles sitting on  the shores of
the channel. This construction was suggested in experimental
papers \cite{B1}, \cite{B2}, \cite{B3}, \cite{PBar}. The hight of
the barrier, and hence the power of it, is defined by the position
of the  lowest energy  level  in  the cross-section  of  the
channel of variable width.
\par
 The plan of our paper is the
following. In  the  second Section  we  suggest a procedure of
selection of  size  of  the device in form of a circular quantum
well in dependence on  desired  working temperature.  In the third
Section we show, that  for  weak connection between  the wires and
the ring the scattering matrix near  the
 resonance eigenvalue  may  be  approximated  by an analytic
 matrix function  with two  poles  only. Then  we calculate
 the  life time of  resonances and  suggest
 conditions  of elimination  of  the  Coulomb  blockade. In
the  fourth Section we discuss  the  realistic  boundary
conditions  at  the contact points defined  by  the  adjacent
split-gates and  estimate the position of the working point of the
RQS modeled as a Quantum Ring, assuming that the working
temperature and the Fermi level are fixed. We omit essential part
of reasoning concerning  the spectral and the scattering problem
for the Schr\"{o}dinger operator on the Quantum Ring, see
\cite{Ring}, but derive the boundary conditions for the scattered
waves at the Fermi level ``from the first principles" and define
the life time of the  corresponding
 resonance for the  selected  working  point in  dependence  of
 the  power of  the potential  barrier. The life time  defines both  the
 speed  of switching  and  the  minimal  current  through  the  device.
 In  the last  Section we  calculate the working  parameters of  the
ring-based three-terminal Quantum Gate, manipulated by  the
single-hole charging of  electrodes situated inside the  ring. We
show, that the  problem  of  calculating of  the  working
parameters  of the Gate  may  be  reduced to  the  situation
discussed  in Section 4.
 \par
     Note  that  the  derivation of  the  boundary condition
 at  the  contact  points  "from  the  first  principles" is
actually  a  part  of  the  general  program  of replacement  of
the partial Schr\"{o}dinger equation on the quasi-one-dimensional
structures  with properly  chosen extensions on a  one-dimensional
Schr\"{o}dinger  operators, see also \cite{GP}, \cite{Albeverio},
\cite{Extensions}, \cite{Novikov}, \cite{Schrader}.  Using  of
this  approach in mathematical design of  nano-electronic devices
permits  predict qualitative features of  devices  and   gives
preliminary estimation  of their working parameters.
 \par
  Note  that  the next two important  problems of  the
  mathematical design  of  the  Resonance Quantum  Switches and
  Gates are : calculation of the Voltage-Current characteristics
  and  estimation of an affordable  precision of the geometrical
  details of it. These  problems  are not discussed yet here: we just
  estimate  the minimal  current eliminating  the  Coulomb  Blockade
  and  assume  that both the  ring  and  the  wires have the ideal
  geometry. Since the technologically affordable deviation of geometrical
  parameters is now circa $2$ nm, it is clear that the  device should
  work  at the  nitrogen  temperature  77 K , but it is  not clear
  yet  if  this precision  is  sufficient  to  guarantee  the
  stability of  the  working regime of the switches and  gates
  at  room  temperature.
\vskip5pt

\section{High-temperature triadic  RQS }
\vskip5pt
 Consider a  RQS  constructed  in  form of  a quantum  domain -
 a  circular  quantum well - with  four  terminals - quantum  wires -
 attached  to  it  at  the contact points $a_1,\, a_2,\, a_3,\, a_4 ,$
 selected as  suggested  above. To  choose  the  working  point of
 the  switch  in  dependence of  desired  temperature we consider first the
 {\it dimensionless}   Schr\"{o}dinger equation
 \begin{equation}
 \label{dimless}
 -\bigtriangleup u  + \epsilon \langle \vec \xi ,\, \vec \nu \rangle u =
 \lambda u
 \end{equation}
in  the  unit  disc $|\vec \xi| < 1$  with  Neumann  or  Dirichlet
boundary  conditions  at  the  boundary. The  dimensionless
  Schr\"{o}dinger  equation  may  be  obtained  from  the
  original  equation  by  scaling  $\vec x = R \vec \xi$ :
  \begin{equation}
  \label{connect}
 -\bigtriangleup_{\xi} u  +
 \frac{2 m e {\cal E}R^3}{\hbar^2}\langle \vec \xi ,\, \vec \nu\rangle u =
 \frac{2m R^2 (E - V_0)}{\hbar^2}u.
  \end{equation}
Here ${\cal E}$ is  the  magnitude of  the  selected  electric
field and  the  unit  vector $\vec \nu$ defines  it's  direction,
$e$ is the  absolute  value of the  electric charge of the
electron  and $R$ is the radius of the circular  well. Selecting $
\displaystyle \epsilon = \frac{2me {\cal E}R^3}{\hbar^2} =3.558$
for Neumann boundary conditions one may  see, \cite{MP00}, that
the eigenfunction corresponding to the  second lowest eigenvalue
$\mu_2 = 3.79$ of the dimensionless  equation (\ref{dimless}) has
only one smooth  zero line in the  unit  disc which  crosses  the
unit circle  at  the points  situated  on  the  ends  of  radii
forming  the angles  $\pm \frac{\pi}{3}$  with  the electric field
$\epsilon \vec \nu$. The minimal distance $\delta_0$ of  $\mu_2$
to the nearest  eigenvalues (the  spacing of eigenvalues  at
$\mu_2$), depending  on  boundary  condition  on  the border of
the well, may be between $2$ and $10 $.
 For  Dirichlet  or  Neumann  boundary  conditions  the
 eigenfunctions of  the  spectral  problem  for  the  above
 Schr\"{o}dinger equation (\ref{dimless})  are  even  or  odd
 with  respect  to  reflection  in  the  normal  plane containing
 the  electric  field  $\epsilon \vec{\nu}$. In  particular for
 the Neumann boundary  conditions  the nearest  eigenvalues
corresponding  to {\it even } eigenfunctions are equal $ \mu_1 = -
0.79$ and $\mu_3 = 9.39$, that it the spacing  between  $\mu_2$
and  other  eigenvalues of the  {\it even} series  may be
estimated as $\delta_0 := $ min$\{|\mu_2 - \mu_1|,|\mu_2 -
\mu_3|\} \approx 4$. Generally for the  circular domain  the
spacing between  the second  lowest eigenvalue $\mu_2$ and  other
eigenvalues ( of both  even  and odd  series )  may  be estimated
from below as $\delta_0 \geq 2$. The  working  regime  of  the
switch  will  be  stable if  the bound  states  corresponding  to
the  neighboring eigenvalues will not be excited at the
temperature $T$ :
\begin{equation}
\label{Temper}
 \displaystyle \kappa T \,\,\, \frac{2m R^2}{\hbar^2} \leq
\frac{\delta_0}{2}.
\end{equation}
This  condition  may  be  formulated  in  terms of the  {\it
scaled  temperature } $\displaystyle \Theta = \frac{2mR^2
T}{\hbar^2}$ as
\begin{equation}
\label{reduced} \kappa\Theta < \frac{\delta_0}{2}.
\end{equation}
The  temperature which  fulfils  the above  condition we  call
{\it low} temperature  for  the  given  device. If the  radius $R$
of  the  corresponding  quantum  well  is  small enough , then  it
may  work at  the (absolutely) high  temperature, which correspond
to  the  {\it low} scaled temperature. It  may  take  place if
 the radius of the well is  sufficiently  small. Importance
of developing technologies of producing  devices of small size
with rather high potential barriers is systematically underlined
when discussing  the  prospects  of nano-electronics, see  for
instance \cite{Compano}.
\par
  We  assume that the effective depth $V_f$ of the bottom
value $V_2$ of the potential on the wires from the Fermi-level
$E_f$ in the wires is  positive $V_f = E_f - V_2 > 0 $, and the
De-Broghlie wavelength on Fermi  level is defined as $$
\displaystyle \Lambda_f =\frac{h}{\sqrt{2m V_f}}.$$  Then we
obtain the estimate of the  radius $R$ of  the  domain from
(\ref{Temper}) as:
\begin{equation}
\label{R} \frac{R}{\Lambda_f}\leq  \sqrt{\frac{V_f}{\kappa T}}
\sqrt{\frac{\delta_0}{8 \pi^2}}.
\end{equation}
For fixed radius $R$, the shift potential $V_0$ may be defined
from the condition
\[
\frac{2m R^2 [E_f - V_0]}{\hbar^2} = \mu_2.
\]
 For  instance, if  we  choose the  radius $R$ of  the  domain
as  $\displaystyle R^2 = \frac{\delta_0 \hbar^2}{4 m \kappa T}$,
we  obtain :
\[
V_0 = E_f - \frac{\hbar^2 \mu_2 }{2 m R^2} = E_f - 2\kappa
T\frac{\mu_2}{\delta_0}.
\]
 Finally, the electric field ${\cal E}$ may be found from
 the  condition
  \[
\epsilon = 3.8 = e {\cal E}\frac{2m R^3}{\hbar^2},
  \]
where  $e$  is  the absolute value of the electron charge. Hence
for the value of $R$ selected above  we have :
\[
e{\cal E} R = \epsilon  \frac{\hbar^2 }{2 m R^2} = \frac{2
\epsilon \kappa T}{\delta_0}.
\]
Hence  the  switch  may  work even  at  room  temperature if the
radius $R$ of  the  quantum  well  is  small enough  and  the
geometrical details are  exact.
\par
Similar  calculations  may  be  done  for the  circular quantum
well with  Dirichlet boundary conditions. It  appeared  that for
the dimensionless equation with  the  potential  factor $\epsilon
= 18.86$ the eigenfunction with a single zero-line dividing the
unit circle into  ratio $1:2$ corresponds to  the second smallest
eigenvalue $\mu_2 = 14.62 $. The lowest eigenvalue which
corresponds to the {\it even} eigenfunction is $\mu_1 = 2.09$ and
the  spacing between  $\mu_2$ and  other  eigenvalues of  both
even  and  odd  series  is  estimated  as  before, $\delta_0 \geq
2 $. This gives proper base for calculation of the radius of the
quantum well,the intensity of the electric field and the shift
potential subject to given temperature and the Fermi level.
Similar calculation  may be done  for RQS based  on  quantum ring,
see  below Section $4$.
 \vskip0.5cm
\section{ Scattering  matrix: two-poles  approximation, resonances and
estimation  of  the  speed  of  switching }

Consider RQS based  on  quantum  ring. It  is  constructed  in
form of a graph constituted of a {\it circular quantum ring} and
four quantum wires  attached to it at the contact points selected
as suggested above. We  choose the boundary condition connecting
the boundary values at  the end-point of the wire $\Gamma_s$
 similarly to the  choice made in \cite{Ring},that
is the jump of  the derivative $[\psi'_s] $ and the value of  the
wave-function $\psi_s$ at the corresponding contact point $a_s$ on
the ring :
\begin{equation}
\label{bcond} \left(
\begin{array}{c}
[\psi ']\\ \psi_s
\end{array}
\right)=
 \left(
\begin{array}{cc}
0& \beta\\ \beta & 0
\end{array}
\right) \left(
\begin{array}{c}
\psi \\ -\psi'_s
\end{array}
\right).
\end{equation}
Later  in  Section 4  we shall suggest arguments approving the
choice.
\par We find the scattered waves as solutions of a system of
Schr\"{o}dinger equations on the graph:
\[
-\frac{\hbar^2}{2 m}\frac{d^2 \psi}{d x^2} + \left( {\cal E} e
\langle x,\nu \rangle  + V_0 \right)\psi = E \psi,
\]
\begin{equation}
\label{Schred} - \frac{\hbar^2}{2 m}\frac{d^2 \psi_s}{d x^2} + V_2
\psi_s = E \psi_s
\end{equation}
with  the  above  boundary  conditions (\ref{bcond}). It  is
convenient  to  reduce  the  system (\ref{Schred}) to  the
dimensionless form  with  respect  to  the  new  coordinate $\xi =
x/R $ and the  new  spectral  parameter $\displaystyle k^2 = R^2
\frac{2 m (E - V_2) }{\hbar^2}$ :
\[
- \psi_{\xi \xi} + R^3 \frac{ 2m {\cal E}e}{\hbar^2} \psi +
R^2\frac{2m (V_0 - V_2)}{\hbar^2} \psi = k^2 \psi,
\]
\begin{equation}
\label{schred} -\frac{d^2 \psi_s}{d \xi^2} \psi_s = k^2 \psi_s, \,
s = 1,2,3,4.
\end{equation}
The  scattered  wave iniciated  by the  input from  the  wire
$\Gamma_1$ may be found as smooth solutions of the system
(\ref{schred}) on  the graph which have the standard  form  on the
wires ( with positive values of the spectral parameter $k$ )
\[
\psi_s = e^{ik\xi} \delta_{s1} + S_{s1} e^{-ik\xi} :=\left\{
\left( e^{ik\xi} I +  e^{-ik\xi} S \right) \vec{e}_1\right\}_s.
\]
 Here $\vec{e}_1$ is the vector $(1,0,0,0)$ in the $4$-dimensional
" channel-space" of the system (\ref{schred}) - the  space, where
the  results of the  scattering processes  are  observed . The
component of the scattered wave on the ring is presented as a
linear combination of Green  functions $G (x,\, a_s, k^2) $ of the
equation  on  the ring :
\[
\psi (\xi) = \sum_{s = 1}^{4} u_s G (\xi,\, a_s/R, k^2) := G
\vec{u}
\]
which  satisfies the  boundary  condition $[\psi'] (a_s/R) = -
u_s$. The  Ansatz  $\{\psi,\,\psi_1,\,\psi_2,\,\psi_3,\,\psi_4
\}$, which obviously  satisfies the  equation,  may  be inserted
into boundary  conditions and  used for calculation of the
scattering matrix :
\begin{equation}
\label{scatm} \left(
\begin{array}{c}
-\vec{u}\\ ( I + S ) e_1
\end{array}
\right)=
 \left(
\begin{array}{cc}
0& \beta\\ \beta & 0
\end{array}
\right) \left(
\begin{array}{c}
G \vec{u} \\ - ik (I - S)\vec{e}_1
\end{array}
\right).
\end{equation}
Using  the  notation $\left\{G (a_s/R,\,a_t/R,\,k^2)\right\} =
Q_{st} (\lambda) ,\,\, \lambda = k^2 $, we  obtain the explicit
formula for the scattering matrix $S$ of the dimensionless
Schr\"{o}dinger equation, see also \cite{Ring}, \cite{MP00},
\begin{equation}
\label{Smatrix}
 \displaystyle S = - \frac{\frac{I}{ik\beta^2} - Q}{\frac{I}{ik\beta^2} +
 Q}.
\end{equation}
In  the  case of  quantum  domain, see \cite{MP00}, the  role of
$4\times 4$ matrix $Q$ is  played  by  the  matrix combined of the
{\it regularized} values of the Green function at the contact
points $a_s$. In the  case  of  the  quantum  ring  the  matrix
$Q$ admits  the  spectral  representation in  terms  of  the
orthogonal and  normalized  eigenfunctions $\phi_l$ of  the
dimensionless  Schr\"{o}dinger equation on the  ring. This  matrix
is  an analytic function of  the spectral parameter $\lambda =
k^2$, and has a positive imaginary part in  upper  half-plane $\Im
\lambda > 0$ and simple poles  on the  real axis at  the
eigenvalues $\mu_l,\, l=1,2\dots$  of the Schr\"{o}dinger equation
on the ring:
  $$Q_{st}(\lambda) = \left\{G (a_s/R,\, a_t/R,\, \lambda)\right\}_{s,t} = $$
 $$ \left\{ \sum_{l =1}^{\infty}
  \frac{\varphi_l (a_s/R) \, \varphi_l (a_t/R)}
 {\mu_l - \lambda} \right\}_{s,t}.$$
 It  is  easy  to  see  that  the  zeroes  of  the  numerator of
 the  scattering  matrix (\ref{Smatrix})
 sit  in  the  lower  half-plane of  the  spectral parameter $k =
\sqrt{\lambda}$
 symmetrically   with respect  to  the  reflection  in the
 imaginary  axis  of  $k$
  and  are  complex conjugate to  the poles (zeroes of  the  denominator) in
  the upper  half-plane.
   \par
Assuming  that the  normalized  eigenfunctions  are uniformly
bounded \footnote{This  fact may  be  derived  for the
Schr\"{o}dinger operator on the  Quantum Ring with  smooth
potential.} one may derive the approximate expression for the
matrix $Q (\lambda)$ when $\lambda = k^2$ sits on the real axis
close to the resonance eigenvalue $\mu_2$ , $|\mu_2 - \lambda|<
\delta_0 /2$. Using  the notation $\vec{\varphi}_l = \{ \varphi_l
(a_1),\, \varphi_l (a_2),\dots \varphi_l (a_4) \}$ for the vector
of the  channel space  we obtain :
\[
Q (\lambda) = \frac{|\vec{\varphi}_2|^2 P_2 }{\mu_2 - \lambda} +
\sum_{l \neq 2 }^{\infty}\frac{|\vec{\varphi}_l|^2 P_l }{\mu^l -
\lambda} :=
\]
\[
\frac{|\vec{\varphi}_2|^2 P_2 }{\mu_2 - \lambda} + K,
\]
where $P_l$ is the orthogonal projection in $R_N$ onto
$|\vec{\varphi}_l|^{-1} \vec{\varphi}_l$ and $K$ admits an
estimate outside  the  spectrum  $\{\mu_l\},\, l\neq 2$ as  an
operator in $4$-dimensional complex euclidean space with the
square of the Hilbert-Schmidt  norm  estimated  as
\[
||  K  ||^2  \leq \sup_l |\vec{\varphi}_l|^2 \sum_{l\neq 2}
\frac{1}{|\mu_l - k^2 |}.
\]
This suggests considering the additional  term   $K$ as  a weak
perturbation. A  convenient  technique  for estimation  of errors
appearing  from  neglecting  additional terms  was developed in
series of papers  by  R. Mennicken  concerning spectral problems
for matrix operators see, for instance \cite{Mennicken3}. In
particular one may derive with use  of this  technique an
approximation for the scattering matrix  in the  small
neighborhood  of  the resonance eigenvalue $\mu_2$ {\it embedded}
into  the  absolutely continuous spectrum of  the Schr\"{odinger}
operator on the graph:
\[
 \displaystyle S = - \frac{\frac{I}{ik\beta^2} -
  \frac{|\vec{\varphi}_2|^2 P_2}
 {\mu_2 - \lambda} - K}
 {\frac{I}{ik\beta^2} + \frac{|\vec{\varphi}_2|^2 P_2}
 {\mu_2 - \lambda} + K}.
\]
\par
 {\bf Assertion.}
{\it The  subordinate  terms  $\pm K$  in the numerator and in the
denominator in the above expression for  the  scattering  matrix
are {\it dominated} by the leading terms
 $$ M (k) = \displaystyle \frac{I}{ik\beta^2} \pm
  \frac{|\vec{\varphi}_2|^2 P_2}
 {\mu_2 - \lambda}$$
 on  the  boundary  of  a ``small"  $\frac{|\vec{\varphi}_2|^2\beta^2}{4} $ -
 neighborhoods $\omega_{\pm}$ of the  zeroes $k_{\pm}$
  of the group  $M (k)$  the leading terms. In  particular
  \begin{equation}
 sup_{k \in \partial \omega_s} ||M^{-1} K (k)|| < 3 \alpha \beta^2
 ||K|| < 1,
 \end{equation}
 if   $\frac{|\vec{\varphi}_2|^2\beta^2}{4}  < 1$. }
 \par
 Rather technical verification of the  Assertion may be found in the  Appendix 1.
 \par
 If  the  functions $M{\pm}$ have  zeroes $k_{\pm}$ with
 orthogonal projections $P_{\pm}$  onto  the  corresponding zero-
  subspaces, then  zeroes  and  projections  onto  the
  zero-subspaces  of  the   perturbed  operator-functions
  $M_{\mp}$ may  be characterized with a  help  of the
  matrix-valued  version of  Rouche  theorem, see
\cite{Gohberg}. The analytic  operator  function  $M(k) + K (k)$
which fulfills the  domination  condition on  the  boundary
$\partial \omega_0$ of a neighborhood  $\omega_0$ of  the (vector)
zero $k_0$ of the analytic function $M : M (k_0) P_0 = 0$ with
some maximal orthogonal projection $P_0$,
\[
\sup_{k\in \partial \omega}|| M_{\mp}^{-1} K (k)|| < 1,
\]
has  inside the  neighborhood  vector zeroes $\tilde{k}_l$ with
the total multiplicity dim$P_0 $
\[
\left[M(\tilde{k}_l) + K (\tilde{k}_l)\right] P_l = 0
\]
 with  the  sum  of  multiplicities equal  to  the  multiplicity
 of  the  zero of  the  leading  term:
 \[
\sum_l \mbox{dim}\tilde{P}(\tilde{k}_l) = \mbox{dim}P_0.
\]
If the vector-zeroes  of  the  leading terms  are  simple, then
the root - vectors of  the  scattering  matrix are simple as  well
and the  corresponding root-vectors are close to the corresponding
root - vectors of the approximate scattering matrix
\begin{equation}
\label{approx}
 \displaystyle S_{approx} = - \frac{\frac{I}{ik\beta^2} -
  \frac{|\vec{\varphi}_2|^2 P_2}
 {\mu_2 - \lambda} - K}
 {\frac{I}{ik\beta^2} + \frac{|\vec{\varphi}_2|^2 P_2}
 {\mu_2 - \lambda} + K}.
\end{equation}.
\par
 In  case of the  weak connection $\beta << 1$ the
numerator of the scattering matrix has zeroes $\tilde {k}_{\pm}$
in ``small" neighborhoods $\omega_{\pm}$ of zeroes of the
corresponding leading  terms. Due to the symmetry principle for
the unitary operator functions the zeroes of the denominator are
situated in the  complex-conjugated points $\bar{\tilde
{k}}_{\pm}$. These zeroes are usually interpreted as resonances,
see below. These resonances  are caused by the embedded eigenvalue
$\mu_2 = \alpha^2> 0$ of the Schr\"{o}dinger operator on  the
ring. Using the root-decomposition of the polynomial $k^2 + i k
\beta^2 |\vec{\varphi}_2|^2 - \mu_2 = (k - k_+)(k - k_-)$ one may
derive the ``two-pole approximation" which corresponds  to the two
poles $\bar{k}_{\pm}$ and  two  zeroes $k_{\pm}$ of  the
approximate scattering  matrix in  $k$-plane :
\[
S_{approx} (k) = - \frac{(k - k_+)(k - k_-)}{(k - \bar{k}_+)(k
-\bar{k}_-)} P_2 - (I - P_2)
\]
In this case under condition  of  domination of the matrix $K$  by
the leading terms one  may derive from the  Rouche  theorem
 quoted  above, that there exist zeroes
$\tilde{k}_{\pm}$ of the original Scattering matrix which sit in a
small neighborhood of zeroes $k_{\pm}$ of it's two-poles
approximation $S_{approx}$ and have  root-vectors  $\tilde{e},\,
S(\tilde {k}) \tilde{e}= 0 $ which are close to the  corresponding
root-vectors $\vec{\varphi}_2$ of the approximate $S$-matrix. One
may derive from it that the original scattering matrix may be
presented near  the  point $\tilde{k}_+$ in form
\[
S(k) =  S^+_0 (k) \left[ \frac{k - \tilde{k}_+}{k -
\bar{\tilde{k}}_+}\tilde{P}^+_2  - (I - \tilde{P}^+_2)\right],
\]
and  near  $\tilde{k}_-$ in  form 
\[
S(k) =  S^-_0 (k) \left[ \frac{k - \tilde{k}_-}{k -
\bar{\tilde{k}}_-}\tilde{P}^-_2  - (I - \tilde{P}^-_2)\right],
\]
where the  projections $\tilde{P}^{\pm}_2$  onto  root-vectors of
the  scattering  matrix  at  the   vector-zeroes
$\tilde{k}_{\pm}$,are  close  to  $P^{\pm}_2$ and 
 the factors $S^{\pm}_0$ are  analytic invertible functions near
 the zeroes  $\tilde{k}^{\pm}$ respectively. The  operators 
$P^{\pm}$ are  commuting  and the  operators
$\tilde{P}^{\pm}_2$ are almost commuting, we  may use 
on  the  real  axis $k$ near  the  points  $\pm\alpha$  matrix any  of
three representations  for  the  scattering -  with  small errors :
 \[
S(k) =  S^{\pm}_0 (k) \left[ \frac{k - \tilde{k}_-}{k -
\bar{\tilde{k}}_-}\tilde{P}^-_2  - (I - \tilde{P}^-_2)\right]
 \left[ \frac{k - \tilde{k}_-}{k -
\bar{\tilde{k}}_-}\tilde{P}^-_2  - (I - \tilde{P}^-_2)\right] +
o(1) =
\]
$$
 S^{\mp}_0 (k) \left[ \frac{k - \tilde{k}_-}{k -
\bar{\tilde{k}}_-}\tilde{P}^-_2  - (I - \tilde{P}^-_2)\right]
\left[ \frac{k - \tilde{k}_+}{k -
\bar{\tilde{k}}_+}\tilde{P}^+_2  - (I - \tilde{P}^+_2)\right] + o(1) = 
$$
\[
 S_{00} (k) \left[ \frac{k - {k}_-}{k -
\bar{{k}}_-}{P}^-_2  - (I - {P}^-_2)\right]
\left[ \frac{k - {k}_+}{k -
\bar{{k}}_+}\tilde{P}^+_2  - (I - {P}^+_2)\right] + o(1) 
\]
with analytic invertible  near  $k_{\pm}$ factors $S^{\pm}_0,
\,S^{\mp}_0,\,S_{00}$. These  representations play  the  role  of 
an asymptotical two-poles  representation for the  S-matrix near
the  resonance eigenvalue. 
\par
The  imaginary  part  $\displaystyle -\frac{\beta^2
|\vec{\varphi}_2|^2 }{2}$ of resonances in  the  plane of  the
spectral  parameter $k$ is usually  interpreted  as  a  inverse
life time of the resonance state (with respect to the  scaled time
variable in our  case, see  below). Being correct for acoustic
equation, see \cite{Lax}, which contains the second derivative on
time, this interpretation should be reconsidered for  the
non-stationary Schr\"{o}dinger equation. We follow here the
 classical analysis  of  the  problem  via  contour deformation
 described in \cite{Zeld}, but  add  few  details  concerning
specific  two-poles  approximation  of  the
  Scattering  matrix  on  the  graph  for  the  Schr\"{o}dinger
  equation with proper boundary condition at the contact points:
\begin{equation}
\label{nonstat}
 \frac{1}{i} \frac{\partial \psi}{\partial t} =
-\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + q
(x)\psi,
\end{equation}
where $q (x) = \left( {\cal E}e \langle \vec x,\,\vec \nu \rangle
+ V_0\right)$ on the  ring  and  $q(x) = V_2$ on  the  wires.
Introducing the scaled coordinate $\xi = x/R$ and scaled time
variable $\displaystyle \tau = \frac{\hbar^2 t}{ 2 m R^2}$ we may
rewrite the non-stationary equation in form
\[
\frac{1}{i} \frac{\partial \psi}{\partial \tau} =
-\frac{\partial^2 \psi}{\partial \xi^2} +\frac{ 2 m R^2}{\hbar^2}
q (\xi R)\psi,
\]
and  represent the  solution of Cauchy problem with  given initial
data $u_0$ in spectral form via decomposition in scattered waves.
In particular the diffracted wave  may be represented using the
above two-pole approximation and substituting  $k_{\pm},\, P_2$
for $\tilde{k}(\pm),\, \tilde{P}^{\pm}_2$ respectively:
\[
-\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ik^2 \tau}  S_0 (k)
\frac{(k-k_+)(k - k_- )}{(k - \bar{k}_+)(k - \bar{k}_- )} \langle
u_0,\psi (k) \rangle e^{-ik\xi} 2 k dk .
\]
If the initial data have a compact support, then  all  terms of
the  integrand  are  analytic  functions  near  the  real  axis
and  the  integral  may  be  deformed  to  the  new  path passing
the  origin  along  the  direction  arg$k = \pi/4 $ (the direction
of the  most rapid  descent of  the  function $e^{ik^2 \tau}$ for
positive $\tau$ ) and  approaching  infinity  along  the line $
\Im k = h, \,\, k \to + \infty + ih,\, h > \Im \bar{k}_{\pm}$.
Then the residue calculated  at  the  pole $\bar{k}_{+}$ gives the
exponentially decreasing  term
\[
i e^{i \bar{k}^2_+ \tau}\frac{(\bar{k}_+ - k_+)(\bar{k}_+ - k_-
)}{\bar{k}_- - \bar{k}_+} \langle u_0,\psi (\bar{k}_+) \rangle 2
\bar{k}_+ e^{-i\bar{k}_+ \xi},
\]
which is  essential for  relatively  small  time before the
non-exponential terms  begin  play  a  role. The real part of the
coefficient in front of the  physical time $t = \tau \frac{2 m R^2
}{\hbar^2}$ in the exponent
\[
 \sqrt{\alpha^2 - \frac{\beta^4 |\vec{\varphi}_2|^4}{4}}\,\,\,\,\,
\frac{\beta^2 |\vec{\varphi}_2|^2  \hbar^2}{ m R^2} \approx
 \frac{\alpha \beta^2 |\vec{\varphi}_2|^2  \hbar^2}{2 m R^2}
   \]
plays  the  role  of  the  {\it  inverse  life time } of  the
resonance state for  the  non-stationary  Schr\"{o}dimger
equation. It  differs  from  the  imaginary  part  of  the
resonance  by  the  presence  of  the  factor $\sqrt{\alpha^2 -
\frac{\beta^4 |\vec{\varphi}_2|^4}{4}} \approx \sqrt{\mu_2} $ if
$\beta |\vec{\varphi}_2|<<1$. More  accurate  estimate of  the
inverse life-time  taking  into  account  realistic  boundary
conditions at  the  contact  points  see  in the  next Section,
(\ref{lifetime}).
 The life time shows how long an  electron may stay  in the
resonance  state and  hence  defines how fast is the switching
process. On the other hand, if one electron already sits in  this
resonance state , another electron in the same spin state can't
appear in the same position. This  situation is interpreted as
Coulomb blockade: to pass the quantum well, the next electron has
to tunnel under the higher potential barrier defined by  the
spacing  between  $\mu_2$ and the  nearest bigger eigenvalue. The
product of the charge of electron by  the  inverse  life-time
gives an estimate for the minimal current through the device. It
is essentially defined by the value of ``small " parameter $\beta$
in the boundary conditions, by the size $R$ of the ring, by  the
value of the effective  mass of the electron $m$, and by  the
``geometrical" parameter $|\vec{\varphi}_2|^2 $.
 \vskip5pt
\section{ Boundary conditions at the  contact points.}
\vskip5pt
 In  this  Section  we obtain ``from  the  first principles"
  the  parameter $\beta$
in the boundary  conditions for the  RQS based  on the quantum
ring  with  few  terminals. We assume that the  potential  barrier
separating  the  ring  from  the quantum  wire at  the  contact
point  may be controlled by  the  split-gate described above,
Section $1$, see  also \cite{B2}, \cite{B3}.
 \par
 Consider a  quantum switch constructed in form of a
 circular  ring  of  quasi-one-dimensional  quantum  wire
 $\Gamma_0$ with a few  straight  radial  up-leading  wires
  $\Gamma_s = \Gamma_{s1} \cup \Gamma_{s2}$ attached  to  it
 orthogonally  at  the  contact  points $a_s,\, s = 1,2,\dots 4$.
 The  Schr\"{o}dinger  equation  on the  ring $\Gamma_0$
is  defined  by  some  smooth  potential  $ q(x)+ V_0$, and the
Schr\"{o}dinger equations  on  the  wires $\Gamma_s = \Gamma_{s1}
\cup \Gamma_{s2}$ have piecewise  constant  potentials
\[
V_s (x) = \left\{
\begin{array}{cc}
V_1 ,\,\, \mbox{if}& x \in \Gamma_1 :-l < x < 0,\\
 V_2, \,\,\mbox{if}& x \in \Gamma_2 : 0<x<\infty
\end{array}
\right.
\]
The  hight $H = V_1 - E_f$ of  the  potential  over the  Fermi
level on the initial part of the quantum wire (within the
split-gate $x \in (-l, 0)$) is controlled  by the  electric field
orthogonal to the wire , which  may  change  the  width  of  the
the channel via turning  the  boron's  dipoles  sitting  on  the
shore  of it. We assume, that the Fermi level in the wires lies
between $V_1, V_2$: $ H> 0> V_2 $ , the boundary  condition at the
point of contact is chosen in Kirchhoff form \footnote{In fact one
may show that  the  boundary  condition  connecting  the solutions
of the  differential  equations  on  the  wires  and  on the  ring
at the  contact  points  depends  on  local  geometry of the
joining. We  consider the  Kirchhoff  condition as  a zero-order
approximation  for the  realistic boundary conditions.}:
\begin{equation}
\label{bc} [u'_0](a_s) + u'_{s}(a_s) = 0 ,
\end{equation}
and the solutions $u_{s} = \{ u_{s1},\, u_{s2}\} $ of  the
Schr\"{o}dinger equations on  $\Gamma_s = \{ \Gamma_{s1},\,
\Gamma_{s 2} \}$  are smooth  functions on the joint interval
$(-l,0)\cup (0,\infty) = \Gamma_1 \cup \Gamma_2$ for  which the
matching  conditions  are  fulfilled
 $$u_{s1}(-0) = u_{s2}(0),\,u'_{s1}(-0) = u'_{s2}(0).$$
\par
For ``low"  temperature  $\kappa \Theta < \frac{\delta_0}{2}$ one
may  assume that  the  dynamics  of electrons is described as the
restriction of the evolution defined by the non-stationary
Schr\"{o}dinger equation (\ref{nonstat}) onto  the spectral
interval length $\kappa T$ near the Fermi level (that is near the
corresponding resonance eigenvalue on  the  ring). Practically we
should calculate the scattering matrix on the graph for  values of
energy inside  this interval. We may do  it in different ways:
beginning with the Green function $G_0 (x,y,\lambda)$ on the ring
with the smooth potential $V (x)$
  \[
Lu = -\frac{\hbar^2}{2 m}\frac{d^2 u}{d x^2} + V (x)u = E u +
\delta (x-y),
  \]
defined  by  the  macroscopic  electric  field, or  beginning with
the Green function $\tilde {G}_0 (x,y,\lambda)$  of  the perturbed
problem on the ring which already  takes  into  account the wires
attached  to the ring at  the  contact  points. We  choose the
second  option of the  two equivalent possibilities, which  gives
better  approximation  for  solutions in  case of weak connection
  between  the  terminals  and  the  quantum  ring.
The Kirchhoff conditions for the solutions $\psi, \psi_s$ of the
Schr\"{o}dinger equation on the ring and the Schr\"{o}dinger
equation on the wires at the contact points
\[
[\psi '] + \psi'_s \big|_{a_s}  = 0,\, \psi\big|_{a_s} = \psi_s
\big|_{a_s} ,
\]
may  be  simplified at  the  resonance $E = E_f$ due  to the
 assumption  that  the potential barrier on the  initial
part  of the  wire  is  strong  enough, so  that  the solution of
the  Schr\"{o}dinger  equation on  the  initial  interval of the
quantum  wire  may  be  presented just  as a solution  of the
equation  with  the  constant
 potential  $V_1$ :
\[
\psi_s = C e^{-\frac{\sqrt{2m(V_1 - E_f)}}{\hbar} (x + l)}.
\]
Then eliminating  the Cauchy  data of  the  decreasing  solution
on the wire we obtain from the above  Kirchhoff condition the  the
jumping boundary condition for the wave-function on the ring at
the contact points :
 \[
[\psi '] - \frac{\sqrt{2 m (V_1 - E_f)}}{\hbar} \psi
 \,\, \big|_{a_s} =  0.
\]
 This boundary  condition  may be also  presented  in  form of an
additional singular  potential :
 $$V(x) \longrightarrow  V (x) +
   \sum_{s = 1}^4 \delta (x - a_s) \frac{\hbar\sqrt{2m(V_1 -
   E_f)}}{2m} := \tilde V (x).$$
The  formally   introduced ``perturbed"  Schr\"{o}dinger  operator
$$\tilde{L} = -\frac{\hbar^2}{2m} \frac{d^2 \psi}{d x^2} + V(x)
\psi + \sum_{s =0}^{4} \delta (x - a_s) \frac{\hbar\sqrt{2m(V_1 -
   E_f)}}{2m} \psi $$
   on the ring with the new potential
$\tilde V$ serves as a convenient step to construct the scattered
waves on the whole graph. It  is exactly the operator which
substitutes asymptotically  the operator $L$ on the  ring for
energy  close to  Fermi-level  when the power of the  potential
barrier is growing, $l \to \infty$.
 Using of the  Green function $\tilde {G} (x,y,\lambda)$
 of  the  ``perturbed" operator $\tilde{L}$ instead of the  Green function
 $ {G} (x,y,\lambda)$  of  the  operator $L$  with the smooth
  potential $V (x)$  is  more practical, because
it suggests  a  convenient  Ansatz for the  component of  the
scattered  wave  on  the  ring with resonance  energy and proper
asymptotic behaviour for strong barriers.
\par
 Note that the
Green function $\tilde{G}$ of the perturbed Schr\"{o}dinger
operator $\tilde{L}$ may be easily calculated as  a linear
combination of the Green  functions of the Schr\"{o}dinger
operator $L$ with the smooth potential:
\[
\tilde {G} (x,y,\lambda) =  {G} (x,y,\lambda) + \sum_{s = 1}^4 g_s
(y) {G} (x,a_s,\lambda),\,\, [\frac{d\tilde {G}
(x,y,\lambda)}{dx}]_{x=y} = - 1.
\]
where the coefficients $g_s$ may be found from the linear system
defined by the boundary conditions at the contact points :
\[
-\frac{\hbar^2}{2m}[\tilde {G}' (a_t,y,\lambda)]= g_t (y)=-\left[
{G} (a_t,y,\lambda) + \sum_{s = 1}^4 g_s (y) {G}
(a_t,a_s,\lambda)\right] \frac{\hbar \sqrt{2m(V_1 - E_f)}}{2m}.
\]
Denoting  by  ${\bf G}$ the  matrix $ {G} (a_t,a_s,\lambda)$
combined  of  the  values of the non- perturbed Green-function
 and by $\vec{G}(y)$ the vector $({G} (a_1,y,\lambda),{G}
(a_2,y,\lambda),{G} (a_3,y,\lambda),{G} (a_4,y,\lambda))$ in  thje
channel- space, one may represent  the solution $\vec{g}: =
(g_1,g_2,g_3,g_4) $ of the above linear equation  in form
 $$
\vec{g} = - \frac{\hbar \sqrt{2m(V_1 - E_f)}}{2m} ({\bf I} +
\frac{\hbar \sqrt{2m(V_1 - E_f)}}{2m} {\bf G })^{-1} \vec{G}(y).
$$
 We  use, further, the perturbed  Green function  and the corresponding
 matrix when  combining  the  Ansatz for  the  scattered  waves  of
 the  Schr\"{o}dinger  equation at  the  Fermi level.
  \par
 Note  that  the  Kirchhoff boundary condition imposed at the point
 of  contact $a$, which has  the coordinate $x = -l$ may  be
 transformed  to  the  point  $x = 0$ of  the  wire  with  use
 of the corresponding transfer-matrix for the Schr\"{o}dinger equation
 on the wire $\Gamma_s$ for  the corresponding component of the scattered
 wave:
 \[
-\frac{\hbar^2}{2 m}\frac{d^2 \psi_s}{d x^2} + ( V_1 - V_2) \psi_s
= (E_f - V_2)\psi_s.
 \]
We  shall  use  the scaled  equation on  the  scaled  wire:
\[
\tilde{l}_s  \tilde{\psi}_s = -\frac{d^2 \tilde{\psi}_s}{d\xi^2} +
\frac{2 m R^2}{\hbar^2}( V_1 - V_2)  \tilde{\psi}_s =  \frac{2 m
R^2}{\hbar^2}(E_f - V_2) \tilde{\psi}_s := k^2 \tilde{\psi}_s,
\]
and  on  the  scaled quantum  ring with $V (\xi R) = \epsilon R
\cos (\xi,\nu) + V_0 - V-2 $:
\[
\tilde{l}  \tilde{\psi}_0 = -\frac{d^2 \tilde{\psi}_0}{d\xi^2} +
\frac{2 m R^2}{\hbar^2} V (\xi R) \tilde{\psi}_0 +
 \sum_{s =0}^{4} \delta (\xi - \frac{a_s}{R}) R \frac{\sqrt{2m(V_1 -
   E_f)}}{\hbar} \tilde{\psi}_0
 = k^2 \tilde{\psi}_0,
\]
 Then Cauchy  data of  the solution
$\tilde{\psi}_s (\xi) = \cosh Rr\xi\,\, \tilde{\psi}_s (0)  +
\frac{\sinh Rr\xi}{Rr}\,\,\tilde{\psi}'_s (0),
$
 at the points $\xi =-l/R$ and $\xi = 0 $ are connected  by  the
transformation which leaves the
 boundary  form  $\psi' \bar{\varphi} - \psi \bar{\varphi}'$
 invariant :
\begin{equation}
\label{monod}
 \left(
\begin{array}{c}
\tilde{\psi}'_s (-l/R)\\ \frac{d\tilde{\psi}_s}{d\xi}(-l/R)
\end{array}
\right)
=
 \left(
\begin{array}{cc}
\cosh rl & -\frac{\sinh rl}{Rr}\\ - Rr \sinh rl\,\, & \cosh rl
\end{array}
\right)
 \left(
\begin{array}{c}
\tilde{\psi}'_s (0)\\ \frac{d\tilde{\psi}'_s}{d\xi}(0)
\end{array}
\right).
\end{equation}
Here the  notation  $r = \frac{\sqrt{2 m (V_1 - E_f)}}{\hbar},\,\,
$ is  used. We  shall  transmit the Kirchhoff  boundary condition,
via transform-matrix on  the  wire for the scaled equation, from
the point $\xi = -l/R$ to the point $\xi = 0$. This way we connect
the boundary values $\,\,\left\{[\frac{d\tilde{\psi}_0}{d \xi}]
(a_s),\,\tilde{\psi}_0 (a_s)\right\}\,\,$ of the solution  of the
scaled Schr\"{o}dinger  equation  on  the ring with  the boundary
values $\left\{\psi_s (0), - \frac{d\psi_s}{d\xi}(0)\right\}$ of
the solution of the scaled Schr\"{o}dinger equation  on  the wire:
\[
[\frac{d\tilde{\psi}_0}{d\xi}](a_s/R) = Rr \sinh rl\,\,
\tilde{\psi}_s (0) - \cosh rR\,\, \frac{d\tilde{\psi}_s}{d\xi}
(0),
\]
\[
\tilde{\psi}_0 (a_s/R) = \cosh rl\,\, \tilde{\psi}_s (0) -
\frac{\sinh rl }{Rr} \frac{d\tilde{\psi}_s}{d\xi} (0).
\]
Eliminating the function $\tilde{\psi}_s (0)$ from the first
equation we obtain the  connection  between  symplectic variables
on  the  ring  at  the  contact  point  and  on  the  wire at
 the point  $\xi = 0$ in  hermitian  form :
 \begin{equation}
 \label{BoundCond}
 \left(
 \begin{array}{c}
[\frac{d\tilde{\psi}_0}{d\xi}] (a_s/R)\\
 \tilde{\psi}_s (0)
 \end{array}
 \right)
==
 \left(
 \begin{array}{cc}
Rr \tanh rl & \frac{1}{\cosh rl}\\
 \frac{1}{\cosh rl} & \frac{-1}{Rr}  \tanh rl
 \end{array}
 \right)
 \left(
 \begin{array}{c}
\tilde{\psi}_0 (a_s /R)\\
 - \frac{d\tilde{\psi}_s}{d\xi} (0)
 \end{array}
 \right).
\end{equation}
 Denote  by  $\tilde{g} (\xi,\, \eta)$ the  Green function of  the
 scaled  perturbed  Schr\"{o}dinger  equation on  the  ring. Then
 the jump of  the  derivative  of  it  is equal  to  $-1$ at  all
 points on the  unit  circle, except contact  points,
 where  the  jump  is  calculated  as:
 \[
[\tilde{g}']_{\xi = a_s/R} - rR \tilde{g}(a_s/R,\, a_s/R) = - 1.
 \]
 We choose  an Ansatz for  the component   $\tilde{\psi}_0 (\xi)$
 of  the  scattered  wave on  the  ring  in  form
 \[
\tilde{\psi}_0 (\xi) = \sum_{s = 1}^4 \tilde{g}(\xi,\, a_s/R)
u_0^s = \tilde{\bf g}(\xi) \vec{u}_0.
 \]
Inserting  this  Ansatz into the  boundary  condition
(\ref{BoundCond}) we  obtain  the  equation
\[
-u_0^s = Rr ( \tanh rl -1 ) \tilde {\psi}_0(a_s) + \frac{1}{\cosh
rl } (-\tilde {\psi}'_s (0)),
\]
\[
\tilde {\psi}_s (0) =  \frac{1}{\cosh rl }\,\,\tilde {\psi}_0(a_s)
- \frac{1}{Rr}\,\,\tanh rl \,  (-\tilde {\psi}'_s (0)).
\]
We  may  substitute  now into  the  first  equation the
corresponding component of the  Ansatz  $\tilde{\bf g}({a_s}/R)
\vec{u}_0 := \left\{\tilde{\bf g} \vec{u}_0\right\}_s $ instead of
$\tilde {\psi}_0 (a_s)$, and  the $s$-component of the standard
Ansatz  $ \left[ e^{ik\xi} + S(k) e^{-ik\xi} \right] \vec{e}$  of
the scattered  wave  for $\tilde {\psi}_s$:
   \[
\tilde {\psi}_s (0) = ( [I + S]\vec{e})_s := u_s (0),\,\,\,\,
\tilde {\psi}'_s (0) = ik( [I - S]\vec{e})_s := u'_s (0),
   \]
then the equations  may  be  rewritten  in  vector form  as:
\begin{equation}
\label{BCcompl} \left(
\begin{array}{c}
-\vec{u}_0 \\
 \vec{u} (0) -\frac{1}{Rr} \vec{u}' (0)
\end{array}
\right) = \left(
\begin{array}{cc}
Rr\frac{e^{- rl}}{\cosh rl} & \frac{1}{\cosh rl}\\
 \frac{1}{\cosh rl} & - \frac{1}{Rr} \frac{e^{- rl}}{\cosh rl}
\end{array}
 \right)
\left(
\begin{array}{c}
\tilde{\bf g} \vec{u}_0 \\ - ik [I - S]\vec{e}
\end{array}
\right),
\end{equation}
where  $\tilde{\bf g}$ is  a  matrix combined  of  values
 of  the  perturbed  scaled  Green function $\tilde{g}(a_s/R,\,a_t/R)$
 at  the  contact  points.
 The scattering matrix may be found from these equations via
eliminating of the variables $\vec{u}_0$ on the ring.
\par
 We shall assume now, that the ratio $\displaystyle 1/\cosh rl$
plays a role of the  small parameter. Then the diagonal elements
of  the  matrix  in the right-hand side of  the last equation have
the second exponential order, meanwhile the anti-diagonal elements
are  of the  first  exponential  order.
 \par
 Seems natural that for  $\cosh^2 rl>>1$ one may cancel the
diagonal elements  thus obtaining in the above boundary  condition
a matrix  similar to  the matrix used in (\ref{scatm}) for
calculation of  the scattered waves. But now we already have  an
explicit expression for the  parameter $ \beta = (\cosh rl)^{-1}$.
Notice, that the role of the non-perturbed operator is played by
the Schr\"{o}dinger operator with a special boundary conditions
(or singular potentials)  at the contact points. These potentials
take into  account the  presence of  the potential barrier hight
$V_1 - E_f$ over  the Fermi level  on the  initial interval
$(-l,0)$ of the quantum wire. If  we  assume  $\cosh rl>>1$ and
use the
 corresponding  simplified  version  of  the  boundary
 conditions
\begin{equation}
\displaystyle \label{SMBC}
 \left(
\begin{array}{c}
-\vec{u}_0 \\\vec{u}(0)- \frac{1}{Rr} \frac{d \vec{u}}{d\xi}(0)
\end{array}
\right) = \left(
\begin{array}{cc}
0 & \frac{1}{\cosh rl}\\
 \frac{1}{\cosh rl} &0
\end{array}
 \right)
\left(
\begin{array}{c}
\tilde{\bf g} \vec{u}_0 \\ - \frac{d \vec{u}}{d\xi}(0)
\end{array}
\right),
\end{equation}
 then  the  calculation of  the  scattering  matrix  may
 follow  the pattern  suggested  in the  previous Section 3.
Taking into account that $\vec{u}(0) = \frac{ik(I - S)}{\cosh rl}
\vec{e} $ we may solve the equation (\ref{SMBC}) with respect to
the $S \vec{e}$ for any $4$-vector $\vec{e}$ and then obtain an
expression for the scattering matrix  in  form
\begin{equation}
\label{SM} \displaystyle
 S(k) = \frac {\frac{\tilde{\bf g}(\lambda)}{\cosh^2 rl}
+ \frac{1}{Rr} - \frac{1}{ik}}
 {\frac{\tilde{\bf g}(\lambda)}{\cosh^2 rl} + \frac{1}{Rr} + \frac{1}{ik}}.
\end{equation}
Here  $\tilde{\bf g}(\lambda)$ is a  matrix  combined  of values
at the contact  points of the Green- function of  the scaled
perturbed equation on the ring: $\left\{\tilde{\bf g} (\lambda)
\right\}_{st} = \tilde{g}(a_s/R,\, a_t/R, \, \lambda),\, \lambda =
k^2 $.
 Now, similarly to analysis done in Section $3$, we
may select the leading terms in the numerator and denominator of
the matrix function $\tilde{\bf g} (\lambda)$ near the resonance
eigenvalue $\mu_2$ of the perturbed operator :
\[
\tilde{\bf g} = \frac{|\vec{\varphi}_2|^2 P_2}{\mu_2 - \lambda} +
K.
\]
Here $P_2$ is the orthogonal projection  in  $4$-dimensional
euclidean space onto the vector  $\vec{\varphi}_2$ formed of the
values of the resonance eigenfunction ${\varphi}_2$ at the contact
points; the  non-singular  addend  $K$  may  be estimated
similarly  as in Section $2$. If the  condition of domination of
the  non-singular  term  $K$ by  the group  of the
 leading  terms,
 \[
\displaystyle \cosh^2 rl \left (\frac{1}{ik}- \frac{1}{Rr} \right)
\left( I - P_2 \right) + \left[ \cosh^2 rl \left(\frac{1}{ik}-
\frac{1}{Rr}\right) - \frac{|\vec{\varphi}_2|^2}{\mu_2 - \lambda}
\right] P_2 := M (k),
\]
is  fulfilled,
\[
|| K  M^{-1}(k)|| << 1,
\]
 in a  small  real neighborhood  of  the resonance
eigenvalue $\mu_2$ then calculating  of an  approximate expression
for the scattering matrix in this neighborhood one may   pertain
the leading  terms only:
\begin{equation}
\label{simple} \displaystyle S_{approx}(k) = \frac{
\frac{|\vec{\varphi}_2|^2}{\mu_2 - \lambda} P_2 + \cosh^2 rl
(\frac{1}{R r} - \frac{1}{ik})} {\frac{|\vec{\varphi}_2|^2}{\mu_2
- \lambda}  P_2 + \cosh^2 rl (\frac{1}{R r} + \frac{1}{ik})}.
\end{equation}
To  obtain  the  corresponding  two-poles  approximation  for the
scattering matrix and  estimate  the  decay  of  the  resonance
 terms  in solution of  the  non-stationary  Schr\"{o}dinger
 equation it  suffice to calculate  zeroes  of  the  leading  term
 in the  numerator  assuming that  $\cosh rl >> 1$. Using the
 notation $\alpha = \pm \sqrt{\mu_2}$,
 we  obtain  two  zeroes in  lower  half-plane :
 \begin{equation}
 \label{appzero}
k \approx  \alpha + \frac{irR }{ i\alpha  - rR}\,\,\,
\frac{|\vec{\varphi}_2|^2}{2 \cosh^2 rl}.
 \end{equation}
One  may  derive  from  it  an  analog  of the  two-poles
approximation, see the  previous  section, and  define  the
inverse life time $\tilde{\gamma}$ of the resonances. The  scaled
time $\tau $ corresponding  to the scaled  equation  and  the real
time $t$  are connected  by the formula  $k^2 \tau = (E - V_2)t$,
or $t = \frac{2mR^2}{\hbar^2}\tau $. The  exponential  decay  of
the resonance states of both scaled and the  non-scaled equations
is defined by  the behaviour of he exponential  factor  $e^{i k^2
\tau} = e^{i \Re k^2 \tau} e^{-\Im k^2 \tau}$. The  decreasing
exponential factor may be rewritten with  respect  to  real  time
as $e^{\Im k^2 \frac{\hbar^2}{m R^2} t}$. Hence  the  role  of the
real  inverse lifetime  is played  by  $\Im k^2 \frac{\hbar^2}{m
R^2} $ and may be  calculated  approximately  for $\cosh rl >>1$
as
\begin{equation}
\label{lifetime}
 \frac{\alpha r^2 \hbar^2 |\vec{\varphi}_2|^2}{2 m
(\alpha^2 + r^2 R^2) \cosh^2 rl}.
\end{equation}

 In  fact even substituting the  above  Ansatz  for  scattered  waves
 into the non-simplified boundary  conditions (\ref{BCcompl})
 we  actually  may  obtain a slightly  inconvenient, but
 still  exact  expression
 for  the  scattering  matrix  in form
 \begin{equation}
 \label{Sfull}
S (k) = \frac {\left(1 - \frac{rR}{ik} + \frac{e^{-rl}}{\cosh rl}
\right) e^{-rl}\cosh rl + 1}
 {\left(1 + \frac{rR}{ik} +
\frac{e^{-rl}}{\cosh rl} \right) e^{-rl}\cosh rl + 1} \,\,
 \,\, \frac{\cosh^2 rl  \frac{1 - \frac{rR}{ik} +
\frac{e^{-rl}}{\cosh rl}} {\left(1 - \frac{rR}{ik} +
\frac{e^{-rl}}{\cosh rl} \right) e^{-rl}\cosh rl + 1} + Rr \tilde
{g}} {\cosh^2 rl \frac{1 + \frac{rR}{ik} + \frac{e^{-rl}}{\cosh
rl}}{\left(1 + \frac{rR}{ik} + \frac{e^{-rl}}{\cosh rl} \right)
e^{-rl}\cosh rl + 1} + Rr \tilde {g}}.
\end{equation}
For intermediate  values of $\cosh^2 rl >> \frac{1}{ \sqrt{1 +
\frac{r^2 R^2 }{\mu_2}}}$ one may  simplify the expressions in
both  terms of  the  previous formula for the scattering matrix
neglecting $e^{-2rl}$ compared with $ \sqrt{1 + \frac{r^2 R^2
}{\mu_2}}$. Then  we obtain more convenient  approximate
expression for the {\it approximate scattering matrix} near the
resonance  eigenvalue $\mu_2$:
\[
S_{approx} (k) = \frac{3ik - Rr}{3ik + Rr}\,\,\,\,\frac{2\cosh^2
rl \frac{Rr - ik}{Rr - 3ik} + Rr \tilde {g}}{2\cosh^2 rl \frac{Rr
+ ik}{Rr + 3ik} + Rr \tilde {g}}.
\]
There  two zeroes which  coincide with the approximate zeroes
(\ref{appzero}) calculated above. $S_{approx}$ gives more accurate
two-poles approximation for the scattering matrix of  the switch,
than the  expression (\ref{simple}) derived from the simplified
boundary  condition. The comparison of the  explicit expression
(\ref{Sfull}) for the scattering  matrix and  the approximate
scattering  matrix shows that under some natural domination
conditions the  scattering matrix  has also three zeroes  in small
neighborhoods of zeroes  of  the approximate scattering matrix.
Complete  analysis of  this alterantive will  be  done  in
another publication.
\par
 We shall discuss  now  a  version  of  RQS
 based  on  a  circular {\it quantum  ring} $\Gamma_{_0}$ of radius
 $R$ with three {\it outgoing} straight radial  quantum wires
$\Gamma_{_s},\, s= 1,2,3 $ attached  to  it  at  the points
$\varphi = \pm \pi/3,\, \pi$ via tunneling  across  the potential
barriers controlled  by  the  split-gates. We  assume,  that the
up-leading quantum  wire  is  supplied  with  so  high potential
barrier  that  the  jump   of  the  derivative  of  the
wave-function on  the  ring  at   this  point  may  be  neglected
when calculating the  eigenvalues  and  eigenfunctions  of  the
perturbed operator $\tilde{L}$. Still  we  pertain  the  jumps  at
the  contact points  of  the  outgoing  channels $\varphi = \pm
\pi/3,\, \pi$, characterized  by  the  potential  barriers  width
$l$ and hight $H = V_1 - E_f$ over  the Fermi level  $E_{_{f}}$ in
the radial quantum  wires $\Gamma_{_s}$. In  this  Section we
assume  that RQS  is  manipulated by  the  constant macroscopic
electric field ${\cal E} \vec \nu $ which  generates  the
potential ${\cal E} e R \langle \vec \nu,\,\vec x \rangle  +  V_0
$ in  the Schr\"{dinger} equation on  the ring $\vec x = R \vec
\xi , |\vec \xi| = 1 $.  We  assume as before that the influence
of the field  on  the  quantum  wires  is  eliminated by  some
additional construction, so  that  the  potential on the wires
produced  by  the macroscopic field  is  equal to zero.  It  means
that the Schr\"{o}dinger equation on the network combined  of  the
up-leading  wire, the ring $\Gamma_{_0} $  and  outgoing  wires $
\Gamma_{_1},\, \Gamma_{_2},\,\Gamma_{_3}\,$  may  be written as a
system  of Mathieu equation on  the  ring  $\Gamma_0$
   \[
  - \frac{\hbar^2}{2m}\frac{d^2 u}{d x^2} +
   \left({\cal E} e \langle \nu,\,x \rangle  +
     V_0 \right) u = E u,\,\,\, u = u_0,
   \]
 and the Schr\"{o}dinger  equation with  the  step-wise  potential
\[
 V_{_s} (x) =
 \left\{
 \begin{array}{cc}
H + E_{_f}, & \mbox{\,\, if } - l < x < 0,\\ E_{_f} + V ,
&\mbox{\,\, if \,\,} 0 \leq x < \infty,\,\, V  <0
 \end{array}
 \right. :
\]
  \[
  - \frac{\hbar^2}{2m}\frac{d^2 u}{d x^2} +
    V_{_s} (x)  u = E u,\,\,\, u = u_s,
   \]
on the outgoing  wires $\Gamma_{_s}, \, s=1,2,3$ and on the
incoming  wire $\Gamma_{_4}$. We assume also, that the incoming
wire $\Gamma_{_4}$ is attached to the quantum ring at some point
$a$ different from the above points $a_s$. The connection of the
outgoing  wires with the  quantum ring is characterized by the
``small" parameter $\beta =(\cosh \frac{\sqrt{2m (V_1 -
E_f)}}{\hbar} l )^{-1}$.
\par
 An  important  engineering  problem  is actually {\it the  proper
choice of the macroscopic electric field } ${\cal E}$  such  that
the  corresponding differential operator  on  the  ring  has  an
eigenfunction  with  special distribution  of  zeroes: the zeroes
of the eigenfunction corresponding to the second smallest
eigenvalue  should  divide the  ring  in  ratio $1:2$. We  assume
that  the  potential barrier at the  contact  point $a_{_4}$ with
the incoming  wire  is  so  high that we  may  neglect  the jump
of the  derivative of the perturbed  operator
 $\tilde{L}$ eigenfunction at  this  point. Then the   whole  potential  of
 $\tilde{L}$
on  the  ring  is  combined  of the  smooth  potential  defined by
the macroscopic  field ${\cal E}$ and  an
 additional  singular  potential appearing  from the Kirchhoff's
 conditions  of  smooth matching of the  solution $\psi$ at
  the  contact  points  $a_s,\, s= 1,2,3$ on  the  ring
  with  proper  solutions  of  the
 equations  on  the  wires when the energy is fixed  on  the
 Fermi level $ E = E_f$ :
\[
[\psi'_0] - \frac{\sqrt{2m (V_1 - E_f)}}{\hbar} \psi\,\,
\bigg|_{a_{s}} = 0,
\]
\[
-\frac{\hbar^2}{2m} \frac{d^2 \psi}{d x^2} +
  {\cal E} e \langle \vec x,\vec \nu \rangle +
   \sum_{s = 1}^N \delta (x - a_s) \frac{\hbar\sqrt{2m(V_1 -
   E_f)}}{2m}\psi  + \left( V_0 - V_2 \right) \psi=  E \psi.
\]
We  select  the  field  ${\cal E}$  such that the resonance
eigenfunction for  $E = E_f$  would  have, for certain direction
of  the  unit  vector $\vec \nu $, two  zeroes on  the ring
sitting  at  the  points $\varphi = \pm\pi/3$ . When
  using  the  standard  form  of the scaled  Mathieu  equation
  with  properly  renormalized  coefficients
   $q = \frac{4m {\cal E}e R^3}{\hbar^2},\,
   a = frac{8m R^2 (E-V_0 + V_2)}{\hbar^2} $,
 \begin{equation}
 \label{standard}
  y'' + (a - 2  q \,\,\, \cos(2 z)) y = 0,
 \end{equation}
  we  should pass from  the  angular (scaled) variable
$\xi =x / R $  to  the  new  variable $z = \frac{x}{2R} $ which is
changing on the interval $-\pi/2,\, \pi/2$.  We  have found that
if the vector $\vec \nu$ is  directed   toward  $ z = 0$, and  the
solution $\psi$, we are looking for, is  an even  (cosine-type )
solution of the Mathieu equation on the scaled  ring  $-\pi/2 < z
< \pi/2$ with a positive value  at the point $ z = 0$, and zeroes
at  $z = \pm \pi/6 $, then  it  is smooth  at  the contact points
with $z = = \pm \pi/6$   and has a jump  of  the  derivative
$[\tilde {\psi}']_{\pi} = \frac{R\sqrt{2m (V_1 - E_f)}}{\hbar}
\tilde{\psi} (\pi) $ at the point $\xi = \pm\pi/2$. Hence, $y =
\tilde{\psi}(2z) $ satisfies the Mathieu equation on the interval
$(0,\pi /2)$ and the boundary conditions
\begin{equation}
\label{resbc}
 y' (0) =
0, $$ $$
 \frac{d y}{dz} (\pi/2) + \frac{R\sqrt{2m (V_1 -
E_f)}}{\hbar} y (\pi/2) = 0.
\end{equation}
 The dimensionless Mathieu equation in  standard  form
 (\ref{standard}) with properly  scaled variable $z$ ,
  $-\pi/2 < z <\pi/2 $,  was  analyzed with  Mathematica
 in dependence  of  the re-normalized  electric field
 and  the parameter  $ \displaystyle \gamma = \frac{R\sqrt{2m (V_1
- E_f)}}{\hbar}$ in  the  boundary condition
   $[\frac{d\tilde{\psi}}{d\xi}] - \gamma \tilde{\psi} = 0 $ at  the  contact
   points. It  was  found  that for  the  following  values of  the
parameters $q,\, \gamma $ the  resonance eigenfunction with two
zeroes  at  $ z = \pm \pi/6 $ exists, for  instance :
\[
\gamma = 10,\,\, q = -1.98,\,\, a = 5.24.
\]
For the  parameter $q$  selected as shown  above, there  exist  an
eigenfunction $u$ of the Mathieu  equation  perturbed  by  the
$\delta$-potentials attached to  the  points $a_s$ with  weight
$\gamma$ such that the zeroes of $u$ divide  the unit  circle in
ratio  $1 : 2$. These eigenfunctions  may  play a role  of the
resonance eigenfunctions for  the  corresponding triadic Resonance
Quantum Switch. Being normalized by the  condition $\varphi(0) = 1
$ this function has square $L_2$-norm
 $3.5234 $. The  spacing between the resonance
  eigenvalue $\mu_2 = \frac{a}{4} = 1.30$
 on  the unit  ring  and the  nearest  eigenvalue
   $\mu' =\frac{a'}{4}= 1.49 $ (from
 the  odd series of the eigenfunctions) is  estimated  as $0.19$.
 Now  the  working  temperature of  the  switch  may  be estimated
 as  in  Section 3: $\kappa T \leq \frac{0.19\,\,\hbar^2}{2m
 R^2}$. For quantum  rings with  radius  $10$ nm  the  switching
 time estimated  from  the life time of the corresponding
 resonance  may  be  circa $10^{-17}$ sec.
  \vskip1cm

\section{Resonance Quantum Gate}

The Resonance Quantum  Switch  manipulated  by  the  macroscopic
electric field is actually a  {\it classical device for
manipulating  the quantum  current}. It can't be used as a detail
of a quantum network since the  macroscopic  electric  field can't
be used , generally , in the quantum network, since  it  would
affect simultaneously all elements of  the  network neighboring
with the switch. In this Section we consider a completely quantum
device manipulated by a single electron or hole. Mathematical
modeling of this device requires solution of a two-electron
problem  on a network, similar to  one solved in the simplest case
in \cite{MP95}. In this note  we consider a one-body version of
the problem, assuming that a single hole is sitting inside the
circular ring at the hight $h$ over some point $b_s$ on the
continuation of  one of the radii corresponding to the contact
point $a_s $ with outgoing wires, at the distance $|b_s| = b$ from
the center of the ring. We may have
 three   electrodes   inside  the  ring, and hence  three  possible
potentials to be used for redirecting  of  the
 electron  current  to  different outgoing  wires. The
Coulomb potential  on the ring, produced   by  the  single  charge
sitting  on  one  of  electrodes, is  equal to
\[
 \frac{e}{\sqrt{R^2 + h^2 + b^2 - 2 b R
\cos (\theta - \theta_s) }}.
\]
If the  condition  $\frac{2bR}{R^2 + h^2 + b^2} << 1$ is
fulfilled, then using the Taylor expansion   we  may find the
approximate expression for the  potential energy of the
Schr\"{o}dinger equation on  the  ring in form :
\[
\label{Coulomb} V_s^C (x) \approx - \frac{e^2}{\sqrt{R^2 + h^2 +
b^2 }} -
\]
\begin{equation}
 \frac{e^2 bR}{(R^2 + h^2 + b^2)^{3/2} } \cos(\theta - \theta_s) +
  O(\frac{e^2 b^2 R^2}{(R^2 + h^2 + b^2)^{5/2}}).
\end{equation}
Neglecting the comparatively  small second  addend  we  obtain the
renormalized harmonic potential in  form
\[
V_s (x) = - Q \cos (\theta - \theta_s)  - A ,
\]
and  thus  we  arrive  again  to  the  Mathieu  equation  of  the
same  type  as  described  in  the  previous  Section.  The  only
difference  is  that after introducing  the  new variable
  $z = 1/2 \,\,\,(\theta - \theta_s)$ the coefficients of the
 Mathiew equation  in  standard  form  (\ref{standard})
 are calculated  as
 $$ a  = \frac{8mR^2}{\hbar^2}\,
 (E + \frac{e^2}{\sqrt{R^2 + h^2 + b^2 }}),$$
 $$
 q = -  \frac{8 m e^2 bR^3}{\hbar^2 (R^2 + h^2 + b^2)^{3/2} }.
 $$
Calculation of the working parameters may  be accomplished
similarly  to one in  previous Sections 3,4.
\par
The above  one-body  approximation  may  be  used  if  the life
time of  the  single  hole  on  the  electrode  is  greater than
the life time of  the resonance, but  still  small enough to
provide  necessary  speed  of  switching.  If  this condition  is
not  fulfilled, then  the  corresponding scattering problem should
be analysed in two- body approximation similarly  to \cite{MP95}.
\vskip1cm
\section{Appendix 1: Verification of the Assertion.}
Denoting  $\mu_2 $ by $\alpha^2$ and assuming that
$\frac{|\vec{\varphi}_2|^2\beta^2}{\alpha}\leq 1$,
  we  may  calculate  the
 vector  zeroes of  the  leading  term  $M_{+}:= M$,
 \[
M (k)= \frac{1}{ik\beta^2} (I - P_2) + \left( \frac{1}{ik\beta^2}
-\frac{|\vec{\varphi}_2|}{\alpha^2 - k^2} \right)P_2
 \]
 \[
M (k) e = 0.
 \]
 Due  to the  orthogonality  of  the  projections $P_2,\, I-P_2$ we
 see that  the  root- vectors  may  lie  in the subspace $P_2 {\cal H}$
 of  the  channel-space ${\cal H}$ only  if
  the  values of  the spectral parameter $k$  fulfil the
  equation  $ \frac{1}{ik\beta^2}
- \frac{|\vec{\varphi}_2|}{\alpha^2 - k^2} = 0$. Then we may
 find  this  values approximately  as
\[
k_{\pm} \approx \pm \alpha - i
\frac{|\vec{\varphi}_2|^2\beta^2}{2}.
\]
 This implies some useful estimates for  the  zeroes :
\[
|k_{\pm} \mp \alpha|\approx \frac{|\vec{\varphi}_2|^2\beta^2}{2},
\]
\[
|k_{-} - k_{+}|\approx 2\alpha,
\]
\[
 2\alpha - \frac{|\vec{\varphi}_2|^2\beta^2}{2}\leq |k_{\pm}
\pm \alpha| \leq 2\alpha + \frac{|\vec{\varphi}_2|^2\beta^2}{2}
\]
and  the  estimates of  distances  of  points  $k'_{pm}$  on the
boundaries  $\partial \omega_{pm}$ of  the  neighborhoods
 $ \omega_{pm}$  of  the zeroes :
\[
| k - k_{pm}| = \frac{|\vec{\varphi}_2|^2\beta^2}{4}
\]
 from $\pm \alpha$:
\[
 \frac{|\vec{\varphi}_2|^2\beta^2}{4} \leq | k'_{pm} \mp \alpha
|\leq \frac{3|\vec{\varphi}_2|^2\beta^2}{4},
\]
 from $k_{\mp}$
\[
2 \alpha -  \frac{3}{4} |\vec{\varphi}_2|^2\beta^2 \leq  |k'_{\pm}
- k_{\mp}| \leq  2 \alpha + \frac{3}{4}
|\vec{\varphi}_2|^2\beta^2,
\]
and  from $k_{\mp}$ :
\[
2\alpha - \frac{3|\vec{\varphi}_2|^2\beta^2}{4} \leq |k'_{\pm}-
k_{\mp}|\leq 2\alpha +  \frac{3|\vec{\varphi}_2|^2\beta^2}{4}.
\]
The  above  estimates imply  the  estimate of the ratio
\[
\frac{(k - k_+)(k - k_-)}{(k - \alpha)(k + \alpha)}
\]
on  the  boundaries of  the  $\frac{|\vec{\varphi}_2|^2\beta^2}{4}
$ - neighborhoods $\omega_{\pm}$ of the zeroes $k_{\pm}$:
\[
\frac{2 -  \frac{|\vec{\varphi}_2|^2\beta^2}{4\alpha}}{2 +
\frac{|\vec{\varphi}_2|^2\beta^2}{4\alpha}} \leq \bigg|\frac{(k' -
k_+)(k' - k_-)}{(k' - \alpha)(k' + \alpha)}\bigg| \leq \frac{2 +
\frac{|\vec{\varphi}_2|^2\beta^2}{4\alpha}}{2 -
\frac{|\vec{\varphi}_2|^2\beta^2}{4\alpha}}
\]
Hence the norm of the inverse of  the operator function  $M$
\[
M^{-1} = i k \beta^2 \left[ (I-P_2) + \frac{(k - \alpha)(k +
\alpha)}{(k - k_+)(k - k_-)} P_2 \right]
\]
for  $\frac{|\vec{\varphi}_2|^2\beta^2}{\alpha} < 1$  may  be
estimated on  the boundaries $\partial\omega_{\pm}$ of the
neighborhoods $\omega_{\pm}$ as
\[
|| M^{-1}||\leq  \alpha \beta^2 \frac{2 +
\frac{|\vec{\varphi}_2|^2\beta^2}{4\alpha}}{2 -
\frac{|\vec{\varphi}_2|^2\beta^2}{4\alpha}} < 3 \alpha \beta^2
\]
and hence  the domination condition is  fulfilled on  the
boundaries $\partial\omega_{\pm}$ :
 \[
  || M^{-1} K|| \leq 3 \alpha \beta^2
  \sup_{k \in \omega_{\pm} }||K (k)|| < 1.
\]
for $\beta$ small  enough.



\section{Acknowledgements}
The  authors (A.M and B.P) acknowledge the  support  from the
Russian Academy of Sciences, Grant  RFFI 97 - 01 - 01149.

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