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{ \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 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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