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\begin{center}
{\Large  Modelling of quantum networks}\\
\vskip1cm A.B.\,Mikhailova $^1$,
B.S.\,Pavlov $^{1,2}$, L.V.\,Prokhorov $^1$.
\vskip10pt
\end{center}

\noindent $^1$ V.A. Fock Institute of Physics, St.Petersburg
State University, Ulianovskaya 1, Petrodvorets, St.Petersburg,
198504, Russia\newline
$^2$Department of Mathematics , University of Auckland, Private Bag 92019,%
\newline
Auckland, New Zealand;
\vskip1cm
\begin{center}
Dedicated to the memory of  Ilya  Prigogine.
\end{center}
\vskip1cm
%\newtheorem{th}{Theorem}[section]
\newtheorem{lem}{Lemma}[section] \newtheorem{cor}{Corollary}[section]
\vskip1cm
\centerline{\large\bf Abstract}

\noindent
{\small A mathematical design of a quantum network is equivalent to
the Inverse Scattering  Problem for the Schr\"{o}dinger equation on a composite
domain consisting of quantum wells and finite or semi-infinite quantum wires
attached to them. We  suggest an alternative approach, based on using
a solvable
model, to this difficult problem and to the relevant problems of choice and
optimization of the construction and working parameters of the
quantum network.
In this paper we suggest a general principle of construction of
quantitatively consistent solvable models for one-particle scattering processes
in the network assuming that the  transmission of an  electron across
the wells from
one quantum wire to the other happens due to  the excitation of
oscillatory modes in the well.
This approach permits us to obtain an approximate formula for the
transmission coefficients
based on numerical values for the discrete spectrum of the
Schr\"{o}dinger operator of  the quantum wells.}
    \vskip5pt
\section{Introduction}

The modern  interest to  quantum networks  is  due to the
engineering  of  quantum electronic  devices. Basic problems of
quantum conductance were related to Scattering
Processes long ago, see \cite{Landauer70,Buttiker85,PalmThil92}, and
the role of scattering in mathematical design of quantum
electronic devices was  clearly understood  by the beginning  of the
nineties, see \cite{Adamyan,BH91,Buttiker93}. Still the practical
design of devices beginning from the classical Esaki diode up to
modern types, see for instance \cite{Compano}, was based on the
resonance of energy levels rather than on resonance properties of
the corresponding wave functions. Importance of  interference
in the mathematical design of  devices was noticed in \cite{Exner88,Alamo}
and intensely  studied in \cite{PalmThil92,Exner96,Interf1,Interf2},
see also recent papers \cite{Safi99,Aver_Xu01,ScattXu02,Kouw2002}.
\par
    Modern experimental techniques already permit us to observe
resonance effects caused by details of the shape of the resonance
wave functions, see \cite{B1,B2,Carbon01}. In  our previous papers, see
\cite {Helsinki2,PRB} and other references therein, we proposed using
these effects as a tool for manipulation of the quantum current
across the resonance quantum switch based on using the  shape
of  the  resonance eigenfunction. In particular we  consider the
process of resonance transmission of an electron  across the quantum well
from one quantum wire to another based on excitation of resonance
modes inside the domain---a phenomenon first  noticed in \cite{Opening}.
In this paper we  apply this basic idea to description of  one-electron
transmission in general quantum networks which can be formed on the
surface of a
semiconductor with quantum wires attached to quantum wells.
The depth of the wires and the wells is assumed large enough
to replace the matching boundary conditions on the boundary of the
network with Dirichlet boundary conditions---at least for
electrons with energy close to the Fermi level  in the wires. We
do not assume here that the wires are thin or the connection of the wires to
the quantum wells is weak but we assume that the dynamics of the
electrons in the wires is single-mode and ballistic on large
intervals of the wires compared  with the size of geometric
details of the construction (the width of the wire or the size of
the contacts). We use  analysis of the one-body scattering problem
for  the relevant Schr\"{o}dinger equation on the network.
\par
The quantum network we study this paper is a composite domain of a
sophisticated form, a sort of a ``fattened graph", see for  instance
\cite{KZ01}. Analytical solution of
the Schr\"{o}dinger equation on this domain is impossible and
direct computation is not efficient for optimization of  the
construction because of the large number of essential parameters and
the resonance transmission effect observed  only at a  triple point,
or generally,
a multiple point of the  space of    parameters. Use of certain solvable
models, see for instance \cite{Kurasov}, may simplify the problem of
search of  optimal parameters of  the network. The program of preliminary
search of the working point based on a solvable model  was developed
for  the quantum switch, see \cite{MPP02}.
\par
Replacement of  the network by
the one-dimensional graph with proper boundary  conditions at
the  nodes looks like an  attractive alternative, see
\cite{Gerasimenko,Novikov,Schrader}. Analysis  of the
one-dimensional Schr\"{o}dinger equation on the graph  is a
comparatively  simple mathematical problem but estimation of
the error appearing  from the substitution of the network by a
corresponding graph is difficult. In
\cite{KZ01,RS01} the authors develop a comprehensive  technique
for studying  of the spectrum of the Schr\"{o}dinger operator on a
``fattened graph". Based on a variational approach developed in
\cite{Schat96} they noticed that the (discrete) spectrum of the
Laplacian on a system of finite length shrinking wave-guides, width
$\varepsilon$, attached to the shrinking vertex domain diameter
$\varepsilon^{\alpha},\, 0< \alpha < 1 $ tends to the spectrum of the
Laplacian  on the corresponding one-dimensional graph but with
different boundary conditions at the vertices depending on the speed
of shrinking. In the case of ``large protrusion" $1/2 < \alpha < 1 $
the spectrum in the corresponding composite domain tends to the
spectrum on a graph with Kirchhoff boundary conditions at the
vertices; for the intermediate case $\alpha = 1/2$ the boundary
condition becomes energy-dependent, and  for ``small
protrusion", $0 < \alpha < 1/2 $, the spectrum on a system of
wave-guides, when shrinking, tends to  the spectrum on the graph
with  zero boundary conditions at the  points of contact so the
vertices play the role of a ``black hole" according to
\cite{KZ01}.
\par
In distinction from the papers \cite{KZ01,RS01} quoted above, our
theoretical analysis of the  Schr\"{o}dinger  equation on the ``fattened
graph'' \cite{P02} was based on a {\it single-mode  one-body
scattering problem}
in the corresponding  composite domain. The aim of this paper is
the construction
of a ``solvable  model''  for  the relevant Resonance Scattering  problem  on
the  network.  We do not consider the shrinking process when the width of the
wires tends to zero, but still our case is similar to one of the
``large protrusion", with $\alpha = 1$, see a footnote in
section 3. The  role of the Scattering matrix of the  solvable model is  played
by the proper resonance factor of the  complete Scattering matrix.
\par
There also exists another aspect of our mathematical  model. It  is  connected
with  general  analytic perturbation techniques, see for  instance
\cite{Kato} where
the  case of additive perturbations $A + \varepsilon B$  for
operators with  discrete
spectrum is  studied. Probably  I. Prigogine  was the  first who
attracted the attention of
specialists to  the  fact  that the  standard technique does  not
work for operators
with  continuous  spectrum. Trying to  reconsider  analytical perturbation
techniques  for  continuous  spectrum,
I. Prigogine  suggested the use of resonances which form a discrete set
in the complex plane  of the
spectral parameter and may be  treated with analytical perturbation
techniques. Unfortunately,
the  price of replanting  the  whole  perturbation problem into the
theory of resonances is enormous, the  set  of  resonance states is 
normally complete
only  in a  tiny  subspace
of the original space. I. Prigogine  assumed  that  this  cost
may  be  reduced  if  we  leave
the  cosy  Hilbert  environment  and  pass  to  the  larger  class
of  Banach  or  topological
spaces say via rigging  the  original Hilbert space,
\cite{Prig95,Reichl}. The  elegant construction
developed  from this idea  remained  essentially  mathematical  until
now since  the  choice of  triples of  the  rigged  spaces is not 
uniquely defined by the
physical content of the problem,
   it is rather in hands of the researcher.
Even for  classical  model dynamical  systems the choice  may  affect
the  whole  construction
and the final results essentially thereby violating the basic
requirement of uniqueness.
    \par
According to evidence from long-time Prigogine's collaborator
Professor I. Antoniou,
see  footnote  on the  page  $7$  below, I.Prigogine alos attempted
to use the idea of
an {\it intermediate  operator}  as a  base  for development of
analytical perturbation theory for operators with continuous 
spectrum. Later he
abandoned this idea for unclear
reasons. We will return to this idea again now in a  special case of
quantum networks where the  candidates for the role of an
intermediate operator are  almost obvious.
It  appears  that practically  the idea of an intermediate
   operator  is  automatically associated  with  resonances. The  essential,
and even  decisive difference, between  our version of  the  intermediate
operator  approach
and the  version which  I. Prigogine probably  envisaged,
is  the natural  principle of  {\it spectral locality}  which we  impose:
we  attempt  to  construct  the
approximate scattered waves via the a special perturbation
procedure not globally---but  locally---on a  selected  spectral 
interval. It is easy to define this interval
mathematically: it
is the  maximal interval containing the  Fermi  level,\cite{Madelung},
where the  spectral  multiplicity $n$ is constant and  the
scattering  matrix is
   a square $n \times n$ matrix.
The  structure of  the  corresponding  scattered  waves within this  spectral
interval  is given  by  the  structure of  the  scattering
matrix and  corresponds  to  an
n-dimensional input-output. The structure of  scattered waves on
neighboring spectral
intervals  is different. We call this maximal interval the {\it
spectral terrace  height $n$} and
   denote it by $\Delta_n$. We suggest this term here  to  distinguish 
our situation
from the  basic case of a single cylindrical (or ``asymptotically
cylindrical'') single wire $\omega$
where the terraces coincide  with spectral steps
$(\lambda^{^{\omega}}_{_{\bot,\,k}} ,\,
\lambda^{^{\omega}}_{_{\bot,\,k+1}}) $  defined  by  the eigenvalues
$\lambda^{^{\omega}}_{_{\bot,\,k}}$ of the  part $l^{^{\bot}}$ of the
Schr\"{o}dinger operator
on the cross-section of the  wire with multiplicity $n_{_{\Lambda}} $
coinciding with
the  sum  of  multiplicities
$\sum_{_{\lambda^{^{\omega}}_{_{\bot,\,r}}\,\leq\,
\lambda^{^{\omega}}_{_{\bot,\,k}}}}\,\,n_r = n_k$
of all preceding eigenvalues  the  operator $l^{^{\bot}}$.
In the case of terraces the summation is spread over all preceding eigenvalues
$\lambda^{^{\omega^{^ m}}}_{_{\bot,\,k}}$ of operators
$l^{^{\bot}}_{\omega^{^ m}}$ on the
cross-section of all wires $\omega ^{^ m}$ involved.
\par
For quantum networks constructed  of  several quantum wells
$\Omega_t$ and a finite number
of straight finite or  semi-infinite quantum wires  $\omega^m$ the
spectral steps and terraces are  well defined. They are  separated by
proper thresholds $T_n$
which coincide with the eigenvalues of the  Schr\"{o}dinger operator on
the cross-section
of  the semi-infinite  wires  and  form a  decomposition of
the  real axis  of  the spectral variable $ R = \sum_{n = 0}^{\infty}
\Delta_n \cup T_{n+1}$. Hence for
the construction of the perturbation procedure we need generally {\it
not one, but a countable number of
intermediate operators}. In the case  of quantum networks on the surface
of a semiconductor the set of
intermediate operators has a straightforward physical meaning and
generically only one
or  two of them
correspond  to  terraces which are  close to the Fermi-level
\cite{Madelung} and  thus can play a role in
Scattering processes. Nevertheless we consider here the whole family
of intermediate operators.
Quantum networks are the simplest object for which the construction
of a family of intermediate operators
with finite-dimensional spectral steps can be realized. One may guess
  that I. Prigogine
abandoned his fruitful  idea of the intermediate operator because the
problem of modelling quantum networks was  not  a  problem of quantum
mechanics at that  time (in the seventies--eighties). Probably due to 
this Prigogine
   missed the necessity of considering intermediate operators
locally for  proper  spectral
terraces. We try to fill this gap now,
see the next section, and  calculate  the  Scattering matrix  based
on  intermediate  operators.
It is important that the  eigenvalues of the intermediate operators
give rise to the resonances, see
the formula (\ref{Scatt}) below. Positions  of  the  resonances
(the non-spectral poles of  the  Scattering matrix on the
terrace $\Delta_n$ ) are  defined  essentially  by  the eigenvalues
of the intermediate operator below
the  nearest
upper  threshold $T_{n+1}$, see  (\ref{Sapprox}). In simplest  case
when just one  quantum well
with few  wires  attached
   is  involved  and  only one simple eigenvalue is  sitting on a  terrace,
the  corresponding  Scattering matrix  is  presented  approximately
by  the  one-dimensional factor
containing  the  resonance eigenvalue  $\lambda^{\Lambda}_0$ of  the
corresponding intermediate operator
$L_{_{\Lambda}}$ and the  normal derivative of the corresponding
eigenfunction $\varphi_0$ restricted
onto the  bottom-sections of  the  adjacent  semi-infinite  wires and
projected  onto  the  corresponding
entrance  subspaces  $E^{^{\Lambda}}_{_{+}} = E_{_{+}}$ on the bottom
cross-sections,
see below:
\[
S (\lambda) = -\frac{{\cal D}{\cal N}^T_{\Lambda} + i K_+}
{{\cal D}{\cal N}^T_{\Lambda} - i K_+} \approx
\]
\begin{equation}
\label{Sapprox1}
\displaystyle
- \frac{\frac{P_{+}\frac{ \partial \varphi_0 (x)}{\partial n_x}\rangle
\,\,\langle P_{+}\frac{ \partial \varphi_0 (s)}{\partial n_s}}
{\lambda - \lambda^{^{\Lambda}}_0} + i(2\mu)^2 K_{+} (\lambda^{^{\Lambda}}_0)}
{\frac{P_{+}\frac{ \partial \varphi_0 (x)}{\partial n_x}\rangle
\,\,\langle P_{+}\frac{ \partial \varphi_0 (s)}{\partial n_s}}
{\lambda - \lambda^{\Lambda}_0}-i(2\mu)^2 K_{+} (\lambda^{^{\Lambda}}_0)}.
\end{equation}
Here $K_{_{+}}$ is the exponent obtained from  oscillating
exponential solutions in semi-infinite quantum
wires, $\mu$ is the effective  mass in the quantum well
and $P_{+}\frac{ \partial \varphi_r (x)}{\partial n_x}\rangle
\,\,\langle P_{+}\frac{ \partial \varphi_r (s)}{\partial n_s}$ is the
resonance  term  associated
with the resonance eigenfunction $\varphi_0$,
$\lambda_0^{^{\Lambda}}$ being  the  closest eigenvalue to the
   Fermi level. It can be  presented in
a  standard  form  $\parallel P_{+}\frac{ \partial \varphi_0
}{\partial n} \parallel^2 P_0$
with  orthogonal projection  $P_0$ onto the  one-dimensional
subspace $E^{^0}_{+}$ in  $E_+$ spanned
   by $P_{+}\frac{ \partial \varphi_0 }{\partial n} $. If  $P_0$ commutes  with
$K_{+} (\lambda^{^{\Lambda}}_0)$
   then  the  factor  has a standard  Blashke-Potapov form, see
\cite{Potapov,Ginzburg}. In the general
   case  the  interplay between the  eigenvalues of the  intermediate
   operator and  the  resonances  is  more  sophisticated.
   \par
The  plan  of  our  paper  is  the  following: after the  formal
description  of
our network in the  next  section  and the introduction of  the
intermediate operator
we prove in section 3 that the intermediate operator $l_{\Lambda}$ is
self-adjoint, hence possess a
resolvent kernel. We then introduce the  corresponding
Dirichlet-to-Neumann map and calculate it
in terms of  the Schr\"{o}dinger  operator  on the  Quantum
well. The  following
   section, 4,   is  dedicated  to the ecalculation of the scattering matrix
and derivation
of the ``one-pole  approximation'' based  on the use of the
Dirichlet-to-Neumann map of the
intermediate operator. In Appendix  1 we suggest  a  solvable  model
of the  simplest
Quantum Network (Resonance Quantum Switch). In  Appendix 2  we develop
the  technique of the intermediate
operator for the Friedrichs model. It  gives  the  ``one-pole approximation"
for the corresponding  Scattering  matrix which  contains the
non-analytic part
of  the  Scattering  matrix near  the  resonance. The  remaining
factor of the  Scattering
   matrix  proves to be  analytic with respect to the  perturbation parameter.
This fact may be  interpreted as the simplest Hilbert-space  counterpart of the
idea  of  I. Prigogine  of  the resonance   variant  of
analytical  perturbation
techniques  for  the  continuous  spectrum.
\vskip1cm
\section{Intermediate Operators}
\noindent
   Consider  a  network  on a  plane  combined  of  several
non-overlapping quantum wells
(vertex domains)  $\Omega_t,\,\,t=1,2,\dots T$ and a few straight
finite or semi-infinite
wires $\omega^m,\,\, l= 1,2,\dots M$ of  constant  width  $\delta_m$ and
length $d_m \leq \infty$
attached  to the  domains
$\Omega_{t_{1}},\,\,\Omega_{t_2},\dots $ orthogonally such that  the
orthogonal bottom
sections $\gamma^m_{t_1},\,\gamma^m_{t_2}$ of  the  wire $\omega^m$
are parts of
the piece-wise smooth boundaries  $\partial \Omega_{t_1},\,\,
\partial \Omega_{t_2},\dots $
of  the  domains  $\Omega_{t_1},\,\, \Omega_{t_2},\dots $
respectively. Each  wire
is attached to one (if  the  wire is semi-infinite ) or two  domains
(if it is  finite),
so that the  function $m\to t$ or $m \to t_1,t_2$ is  defined although
the  inverse
function may not be since there may be a few  wires   $\omega^m$ 
connecting two
given domains $\Omega_{t_1},\,
\Omega_{t_2}$. We consider the spectral problem for the
Schr\"{o}dinger operator
\begin{equation}
\label{Schredinger}
{\cal L} u = -\bigtriangleup_{\mu} u + V(x)u = \lambda u
\end{equation}
on the  network $\Omega = \cup_{t}\Omega_t \cup_m \omega^m$ with zero
boundary condition
on  $\partial \Omega$. The kinetic term $-\bigtriangleup_{\mu}$
containing the tensor  $\mu$ of  effective mass is  defined as
\[
\displaystyle
-\bigtriangleup_{\mu} = \left\{
\begin{array}{cc}
\label{mass}
-(2 \mu_t)^{-1}\bigtriangleup,\,&\mbox{if} \,\in \Omega_t\\
-\frac{1}{2\mu^{^{\parallel}}_{_{m}}}\frac{\partial^2}{\partial x^2}
-\frac{1}{2\mu^{^{\bot}}_{_{m}}}\frac{\partial^2}{\partial y^2}
&\mbox{if} \,\in \omega^m
\end{array}
\right.,
\]
where $x,y$ are the coordinates along and  across  the  wire
$\omega^m$, and $\mu_t,\,
\mu^{^{\bot}}_{_{m}},\,\mu^{^{\parallel}}_{_{m}}$ are  positive 
numbers playing the
roles of the  average effective  mass  in the well $\Omega_t$ and
effective masses across
and  along the wire  $\omega^m$ respectively.
We  impose the  Meixner condition at the inner corners at the boundary
of the domain
in form $D({\cal L})\subset W_2^1 (\Omega)$. We assume that the
potential $V$  is  constant
on the wires $V \bigg|_{\omega_m} = V_m$ and is a real bounded
measurable function
   $V_t$ on each vertex  domain  $\Omega_t$. Without
loss of generality we may assume  that the overlapping wires do not
interact, otherwise
we  may treat the overlapping as an additional quantum well.
The absolutely-continuous spectrum of the  operator  $L$ coincides
with the absolutely-
continuous spectrum of the restriction $\sum_{m = m(t)}\oplus L_m$ of
the operator $L$
onto the  sum of  all semi-infinite  wires  $\omega^m,\,\, m = m(t)$
with  zero boundary
conditions on  the boundary $\cup_m \partial \omega^{m}$. The  spectrum of
the  restriction  $L_{m}$ consists of a countable number of branches
$\cup_{m=1}^{\infty}\left[\left. V_{_m} + \frac{l^2 \pi^2}
{2 \mu^{^{\bot}}_{_m}\delta^2_m},\,\infty \right.\right)$
   which correspond to the
eigen-modes   $\displaystyle e^m_l(y)\,\,\sin\sqrt{\lambda - \left[
V_{_m} + \frac{l^2 \pi^2}
{2 \mu^{^{\bot}}_{_m}\delta^2_m}\right]}\,\,\,x,\,\,\,
   V_{_m} + \frac{l^2 \pi^2}
{2 \mu^{^{\bot}}_{_m}\delta^2_m}\,\,\, < \,\lambda\, $ spanned by
the eigenfunctions of the cross-section $\displaystyle e^m_l(y) =
\sqrt{\frac{2}{\sqrt{2 \mu^{^{\bot}}_{_m}}\delta_m}}
\sin \frac{l \pi y}{\sqrt{2 \mu^{^{\bot}}_{_m}}\delta_m},\,\, l= 1,2,\dots $.
For any  value  $\Lambda$ of the spectral parameter $\Lambda$
different  from the  thresholds,
$\Lambda \neq  V_{_m} + \frac{l^2 \pi^2}
{2 \mu^{^{\bot}}_{_m}\delta^2_m} ,\,\, l = 1,2,\dots$
the linear hulls of eigenfunctions of  cross-sections  may be considered as
entrance subspaces  of ``open  channels''
   $E^m_{+}(\Lambda) = \bigvee_{l\pi \leq \sqrt{2
\mu^{^{\bot}}_{_m}(\Lambda-V_{_m})}\,\delta_m} e^m_l $.
These  correspond  to
to the lower thresholds $\frac{l^2 \pi^2}{2
\mu^{^{\bot}}_{_m}\delta^2_m} + V_{_m} < \Lambda $.
The entrance subspaces of ``closed  channels'' $E^m_{-} (\Lambda) =
\bigvee_{l \pi > \sqrt{2 \mu^{^{\bot}}_{_m} (\Lambda - V_{_m})}\,
\delta_m} e^m_l
= L_2 (\gamma^m) \ominus E^m_{+}(\Lambda)$,
correspond to the  upper thresholds:
$\frac{l^2 \pi^2}{2 \mu^{^{\bot}}_{_m}\delta^2_m} + V_{_m} > \Lambda $.
   Denote by  $P^m_{\pm} (\Lambda)$ the  orthogonal projections onto
the  subspaces
$E^m_{\pm} (\Lambda) \subset L_2 (\gamma^m_t)$ on the  bottom
sections of  the  finite and/or
infinite  wires $\omega^m$ lying on  the  border of the  well $\Omega_t$.
\par
An important feature of the  quantum network  described  above  is  the
``step-wise'' structure of  the
continuous  spectrum  of  the  operator  ${\cal L}$ with
terraces  of growing spectral multiplicities
separated  by  thresholds. Presence  of  this  structure permits us to
introduce the  structure
of intermediate operators which play  an  important  role  in the
calculation of the  Scattering matrix.
\par
For a given value of the  spectral parameter $\Lambda > 0$ consider the
{\it intermediate  operator}
\footnote{This construction is an extension
of  the construction suggested  in our  previous  papers
\cite{P02,PRB} for the  star-shaped
Resonance  Quantum Switch.}
${\cal L}_{\Lambda}$  in the  space $L_2 (\Omega)$ of  all
square-integrable functions
defined on the components $\left\{\omega^m,\,\Omega_t\right\}$ of the network
with special  boundary  conditions  at the  bottom sections
$\gamma^m \in \partial
\Omega_t,\, t= 1,2, \dots $ of  the  wires $\omega^m$. We  submit
components $ u_t,\,\,u^{m} $
from $W_2^2(\Omega_t)\oplus W_2^2 (\omega^m)$ to the following
boundary conditions on elements
$\gamma^m_t$ of the common boundary $\partial \omega^m \cap \partial \Omega_t$:
\[
P^m_{+}(\Lambda) u_t \bigg|_{\gamma^m_t} = 0,\,\,\,\,\,
P^m_{+}(\Lambda) u^m \bigg|_{\gamma^m_t} = 0
\]
\begin{equation}
\label{boundcond}
P^m_{-}(\Lambda)\left[ u^m \bigg|_{\gamma^m_t} -
   u_t \bigg|_{\gamma^m_t}\right]= 0,\,\,\,\,\,
P^m_{-}(\Lambda)\left[ \frac{1}{2\mu_m^{^{\parallel}}}\frac{\partial
u^m}{\partial n} \bigg|_{\gamma^m_t}
       - \frac{1}{2 \mu_{_t}}\frac{\partial u_t }{\partial n}
\bigg|_{\gamma^m_t} \right]= 0.
\end{equation}
To  present the  above boundary  conditions in a more  compact form we
can introduce  orthogonal sums  of  upper and  lower entrance spaces
$\sum^{\pm}_{m}\oplus E^m_t := E_{t,\pm}$ on the wires attached to
the  domain $\Omega_t$ and the corresponding projections
$\sum^{\pm}_{m} u^m := P_{t,\pm} {\bf u}^{\omega}$ of the vectors
$\sum_{m} u^m := {\bf u}_t^{\omega}$ obtained via  restriction of
the  functions
defined  on the wires attached to the given domain $\Omega_t$ onto
the bottom sections
$\gamma_t^m \in \partial \Omega_t\cap\partial \omega^m$. Similarly
we can introduce the
restrictions of the  functions $u_t$ defined  in $\Omega_t$ onto  the sum
   $\Gamma_t = \cup_m\gamma_t^m$ of all bottom
sections $\gamma_t^m$ of the wires $\omega^m$ attached to the domain $\Omega_t$
and the associated projections:
\[
{\bf u}_{_{_{\Omega_t}}} = u_t \bigg|_{_{_{\Gamma_t}}},
\]
and  then form an orthogonal sum:
\[
{\bf u}_{_{_{\Omega}}}= \sum_t \oplus u_{_{\Omega_t}},\,\,P_{\pm
}{\bf u}_{_{_{\Omega}}} =
\sum_{tm}\oplus  P_{_{m,\pm }} {\bf u}_{_{\Omega_t}}.
\]
The  above boundary  conditions may be  re-written  in a  compact form  as
\begin{equation}
\label{bcond}
P_{+}{\bf u}_{_{_{\Omega}}}= P_{+}{\bf u}^{\omega} = 0,\,\,
P_{-}\left[{\bf u}_{_{_{\Omega}}}-{\bf u}^{\omega}\right] = 0,\,\,
P_{-}\left[\frac{1}{2{\bf \mu}_{_{\Omega}}}\frac{\partial{\bf
u}_{_{_{\Omega}}}}{\partial n} -
\frac{1}{2{\bf \mu}^{^{\parallel}}}\frac{\partial{\bf
u}^{\omega}}{\partial n}\right] = 0.
\end{equation}
Here $\mu_{\Omega},\, \mu^{\parallel}$  are diagonal tensors composed
of the values of
the average effective masses in domains  $\Omega_t$ and along the
wires $\omega^{m}$ respectively.
The boundary condition \ref{bcond}  may be  interpreted  as a
``chop-off'' condition  in  all
open (lower)  channels  $\frac{\l^2 \, \pi^2}{2
\mu^{\bot}_{_{\omega}}\delta_m^2} +  V_{_m} < \Lambda $
and  a ``matching'' condition in all closed (upper)  channels
$\frac{\l^2 \, \pi^2}{2 \mu^{\bot}_{_{\omega}} \delta_m^2} +
V_{_m} > \Lambda $.
   The restrictions $\hat{l}^m (\Lambda)$ of  the  operator
${\cal L}_{\Lambda}$ onto the invariant subspaces
${\cal H}^m_+ = \bigvee_{\frac{\l^2 \, \pi^2}{2
\mu^{\bot}_{_{\omega}}\delta_m^2} + V_{_m} < \Lambda} e^m_l
   \times L_2(0,\, d_m)
= E^m_{+} \times L_2(0,\, d_m)$
   which correspond to the (open) channels
   $\frac{l^2 \,\pi^2}{2 \mu^{\bot}_{_{\omega}}\delta^2_m} + V_{_m}< \Lambda$,
are self-adjoint operators and
admit a spectral analysis in explicit form. We denote by
$l_{_{\Lambda}}$ the  orthogonal sum of
all  operators $\hat{L}^m_l(\Lambda)$ in all  open channels. We  will
prove in the next  section that  the
restriction $\hat{L}_{\Lambda}$ of the  ${\cal L}_{\Lambda}$ onto
the complementary invariant
   subspace is  also  a  self-adjoint  operator:
\[
\hat{L}_{\Lambda} = {\cal L}_{\Lambda}\,\ominus\,
\sum_{\frac{l^2 \,\pi^2}{2 \mu^{\bot}_{_{\omega}}\delta^2_m} + V_{_m}
< \Lambda}
{\oplus}\hat{L}^m_l(\Lambda) :=
{\cal L}_{\Lambda}\,\ominus\, \hat{l}_{_{\Lambda}},
\]
here the orthogonal sum is  extended over all lower channels.
\par
   The above chopping-of construction may be applied  either to all
open channels,
finite and infinite, resulting in the  above operators
$\hat{L}_{_{\Lambda}}$ and $\hat{l}_{_{\Lambda}}$;
   or it may be  applied to the semi-infinite open channels only,
resulting in the  operators
   $L_{_{\Lambda}},\,\,\, l_{_{\Lambda}}$ respectively.
      One of the intermediate operators
   $\hat{L}_{\Lambda},\,\,L_{_{\Lambda}} $
   may  be  more  convenient  for  calculating the  Scattering matrix 
depending on  the
  architecture of  the  network.
   In this  paper we  choose  $L_{_{\Lambda}}$ as an
   intermediate operator, chopping off  only  the  open channels in the
semi-infinite  wires.
\vskip1cm
\section{Dirichlet-to-Neumann map}
\noindent
Analytic perturbation theory for discrete spectrum, see \cite{Kato}, 
permits us,
in the generic case of simple spectrum, to construct a branch of
eigenvalues and corresponding  eigenfunctions  of  the  perturbed problem
$A \to A + \varepsilon V = A^{\varepsilon}$ which is  analytic
in  the
perturbation parameter $\varepsilon$ on the  complex  plane. A similar
construction is
impossible for eigenfunctions of the continuous spectrum but may be  replaced
by  the  corresponding construction for  resonances. Although the
resonances are analytic
functions  of  the perturbation parameter $\varepsilon$ the set  of
resonances and  resonance states depends  on the rigging of the
space by  the observer with a proper motivation necessary for each system,
see for instance the discussion in \cite{Prig95}. Hence, due  to the 
non-uniqueness involved in
it, the  resonance-based construction of the  analytic perturbation 
theory is not perfect either.
\par
Developing  the perturbation procedure for
Quantum Networks we  follow another suggestion  by I. Prigogine which was
almost completely forgotten during  the last decade\footnote{
This early idea of  Prigogine, together with his
latest criticism on it directed mainly against
the Hilbert Space formalism, was  communicated  by  Prof.I. Antoniou
in his  talk on the conference on Random  Processes in  Nagoya, 9-11
January 2003
organized by  professor T. Hida.}: we introduce an intermediate
operator $A^{\varepsilon}_0$---a sort of a ``canonic model'' of  the 
perturbed operator
$A_{\varepsilon}$. This  model
should be simple enough to obtain explicit formulae  for the
eigenfunctions and/or evolution
but still able to  reflect typical features of the  spectral
structure of  the  perturbed  operator  $A^{\varepsilon}$, at  least
for a given interval of
energies. We obtain the Scattered waves of the intermediate operator
on the  network via matching the
solutions of  the  Schr\"{o}dinger equation on the  wells with
elementary exponentials
in the  wires. Thus matching  solutions is  an  initial step of the
suggested perturbation procedure.
\par
Now  we focus on the  multi-dimensional techniques of matching based
on a special version of the Dirichlet-to-Neumann map \cite{SU2,DN01} modified
for the Quantum Network. We need, first of all, the   self-adjointness
of  the  intermediate operator introduced  above, the corresponding
Green  function
and the Poisson map. Then, based on this, we suggest an  algorithm for the
construction of  the
   Dirichlet-to-Neumann map of  the  intermediate operator which gives
the  corresponding
   Scattering  matrix.

   \begin{theorem}\label{intermed} {\it The  operators
$l_{_{\Lambda}}\,$,$\, L_{_{\Lambda}}$ defined
by the differential expression (\ref{Schredinger})  and the above
boundary  conditions
(\ref{bcond}) on the sum of all open semi-infinite channels
${\cal H}_{+} (\Lambda) = \sum_{m}{\cal H}^m_{+}(\Lambda)$  and in
the  orthogonal complement
${\cal H}_{-} (\Lambda) = L_2 (\Omega)\ominus \sum_{m} {\cal H}^m_{+}
(\Lambda)$
respectively  are  self-adjoint. The
absolutely-continuous spectrum  $\sigma (l_{_{\lambda}}) $ of the operator
   $l_{_{\lambda}}$ is  a  joining of all
lower branches in semi-infinite  wires:
\[
\sigma (l_{_{\lambda}}) = \cup_{_{\left\{\frac{l^2 \,\pi^2}{2
\mu^{\bot}_{_m} \delta^2_m} <
\Lambda- V_{_m} \right\}}}
\left[\left.\frac{l^2 \,\pi^2}{2 \mu^{\bot}_{_m}\delta^2_m} +
V_{_m},\,\infty \right.
\right).
\]
The original  operator ${\cal L}$ on the  whole network may be
obtained  from the  operator
$L_{\Lambda}$ via replacement of the first of  the  boundary  conditions
(\ref{bcond}) by  the corresponding
   matching boundary condition in the lower channels.}
\end{theorem}
{\it Proof}   It is  sufficient to  prove that the  operator ${\cal
L}_{_{\Lambda}}$ is symmetric and
it's  adjoint is  symmetric  too. Then the  operator
$L_{_{\Lambda}}$  is self-adjoint
as  a  difference of  self-adjoint  operators ${\cal L}\ominus l_{_{\Lambda}}$.
\par We introduce the  notation
   $\cup_{m,s} \gamma^m_s:= \Gamma$ and consider the
decomposition of the  network $\Omega= \cup_{m}\omega^m
\,\cup\,\cup_s \Omega_s $. Then integrating
by parts with  functions  continuously  differentiable on each
component of the above  decomposition
we  may present the  boundary  form  of the  Schr\"{o}dinger  operator $ L$
without
boundary conditions in terms  of the  jumps
$\displaystyle \left[ {\bf u} \right]= u^{\omega}- u^{\Omega},\,
\,\left[ \frac{\partial {\bf u}}{\partial n} \right] =
\frac{1}{2\mu^{^{\parallel}}_{_m}}\frac{\partial {u}^{\omega}}{\partial n}-
\frac{1}{2\mu_{_t}}\frac{\partial { u}_{\Omega}}{\partial n}$
of the  functions,
normal derivatives  on the  bottom sections of the  wires
and  the  corresponding mean values $\left\{ {\bf u} \right\}\,=
\frac{1}{2}\left\{ u^{\omega} + u_{_{_\Omega}}\,\right\},
   \left\{ \frac{\partial {\bf u}}{\partial n} \right\} =
   \frac{1}{2}\left\{\frac{1}{2\mu^{^{\parallel}}_{_m}}\frac{\partial
{u}^{\omega}}{\partial n} +
\frac{1}{2\mu_{_t}}\frac{\partial { u}_{_{_\Omega}}}{\partial n}\right\}$:
   \[
\langle {\cal L} {\bf u},\, {\bf v} \rangle - \langle u,\,{\cal L}
{\bf v} \rangle  =
\]
\[
\int_{\Gamma} \left(\left[\frac{\partial {\bf u}}{\partial
n}\right]\left\{ \bar{{\bf v}}\right\}
- \left\{ {\bf u}\right\} \left[\frac{\partial \bar{{\bf
v}}}{\partial n}\right] \right)
   d\Gamma +
\]
\[
\int_{\Gamma} \left(\left\{\frac{\partial {\bf u}}{\partial
n}\right\}\left[ \bar{{\bf v}}\right]
- \left[ {\bf u}\right] \left\{\frac{\partial \bar{{\bf v}}}{\partial
n}\right\} \right) d\Gamma .
\]
Denote  by  $P^{^{\Lambda}}_+$ the  orthogonal sum of orthogonal projections
onto the entrance subspaces of the lower (open) channels
in $L_2 (\Gamma)$ and by  $P^{^{\Lambda}}_{_{+}}$ the complementary projection,
   $I = P^{^{\Lambda}}_+ + P^{^{\Lambda}}_{-}$. Inserting
this decomposition of unity into the integration over $\Gamma =
\cup_{m,t} \gamma^m_t$ we  can
see  that  the  above  boundary  form is presented as
\[
\int_{\Gamma} \left(P^{^{\Lambda}}_{+}\left[\frac{\partial {\bf
u}}{\partial n}\right]\,
   ^{^{\Lambda}}_{+}\left\{\bar{{\bf v}}\right\}
-  P^{^{\Lambda}}_{+}\left\{{\bf u}\right\}
P^{^{\Lambda}}_{+}\left[\frac{\partial \bar{{\bf v}}}{\partial
n}\right] \right)
   d\Gamma +
\]
\[
\int_{\Gamma} \left(P^{^{\Lambda}}_{-}\left[\frac{\partial {\bf
u}}{\partial n}\right] \,
P^{^{\Lambda}}_{-}\left\{\bar{{\bf v}}\right\}
- P^{^{\Lambda}}_{-} \left\{{\bf u}\right\} P^{^{\Lambda}}_{-}
\left[\frac{\partial \bar{{\bf v}}}{\partial n}\right] \right)
   d\Gamma+
\]
\[
\int_{\Gamma} \left( P^{^{\Lambda}}_{+}\left\{\frac{\partial {\bf
u}}{\partial n}\right\}\,
    P^{^{\Lambda}}_{+}\left[\bar{{\bf v}}\right]
-  P^{^{\Lambda}}_{+}\left[ {\bf u}\right]
   P^{^{\Lambda}}_{+}\left\{ \frac{\partial \bar{{\bf v}}}{\partial
n}\right\} \right)
d\Gamma +
\]
\[
\int_{\Gamma} \left(P^{^{\Lambda}}_{-}\left\{\frac{\partial {\bf
u}}{\partial n}\right\},
   P^{^{\Lambda}}_{-}\left[ \bar{{\bf v}}\right]
- P^{^{\Lambda}}_{-} \left[ {\bf u} \right]\,
   P^{^{\Lambda}}_{-} \left\{\frac{\partial \bar{{\bf v}}}{\partial n}
\right\} \right)
d\Gamma.
\]
One can see from (\ref{boundcond}) that the operator $L$ supplied
with the above
   boundary conditions is  symmetric. Vice versa, if  $L^+$ is  the
adjoint operator
   then the boundary form with ${\bf u} \in D(L),\, {\bf v} \in D(L^+)$
vanishes. Then
   $P^{^{\Lambda}}_+ \left\{{\bf v} \right\}= 0,\,P^{^{\Lambda}}_+
\left[{\bf u} \right]= 0$
    follows from  arbitrariness
   of values of the normal derivatives of the function
$P^{^{\Lambda}}_+ {\bf u} $ on both sides
   of $\gamma^m_{t}$. Similarly  another two boundary  conditions
(\ref{boundcond})
   for  elements ${\bf v} \in D(L^+)$ may be  verified based on the
arbitrariness of values
   of $P^{^{\Lambda}}_- \left\{ {\bf u}\right\}$ and
    $P^{^{\Lambda}}_- \left\{ \frac{\partial {\bf u}}{\partial n} \right\}$.
   Hence the  adjoint operator is symmetric and thus  self-adjoint.

The restriction  $l_{_{\Lambda}}$ of the  operator  ${\cal L}_{\Lambda}$
onto the lower channels is a
self-adjoint operator. Hence the restriction $L_{\Lambda}$ of it  onto the
orthogonal complement of  lower channels is   self-adjoint  too,
with  the  absolutely continuous
spectrum  coinciding with the  spectrum of  the restriction of the
original operator  ${\cal L}$ on  the upper channels  $\sigma
(L_{_{\Lambda}}) =
\left[ \lambda_{_{min}}^{^{\Lambda}},\,\infty \right)$,
\[
   \lambda^{^{\Lambda}}_{_{min}} =  \mbox{min}_{_{_{
   \frac{\pi^2 \, l^2}{2 \mu^{\bot}_{_m}\delta_m} + V_{_m} > \Lambda}}}\,\,\,\,
\frac{\pi^2 \, l^2}{2 \mu^{\bot}_{_m}\delta_m} + V_{_m}.
\]

   $\Box$

   The eigenfunction and  eigenvalues of  the  operator  ${\cal
L}_{\Lambda}$ may be  calculated
    based  on a variational  principle. Consider the quadratic functional
    \[
\int_{\Omega} \left(\frac{1}{2\mu}|\bigtriangledown {\bf u}|^2  + V
|{\bf u}|^2\right) dm +
   H \int_{\Gamma} |P^{^{\Lambda}}_{+}{\bf u}|^2 d \Gamma
    \]
with a  large positive parameter $H$. The  operator ${\cal
L}^{^{H}}_{\Lambda}$ on the orthogonal
complement of  the lower  channels, supplied with the  boundary  condition
\[
P^{^{\Lambda}}_+ \left( {\bf u}_{\Omega} - {\bf u}^{^{\omega}}\right) = 0,
\]
\[
H P^{^{\Lambda}}_{+}{\bf u}_{\Omega} +
P^{^{\Lambda}}_{+}\,\left[\frac{1}{2\mu_t}\frac{\partial{\bf
u}_{\Omega}}{\partial n}
- \frac{1}{2\mu^{\parallel}} \frac{\partial{\bf u}^{\omega}}{\partial
n}\right]= 0,\,\,
\]
\begin{equation}
\label{bcondH}
P^{^{\Lambda}}_{-}\left[{\bf u}_{\Omega}-{\bf u}_{\omega}\right] = 0,\,
   P^{^{\Lambda}}_{-}\,\left[\frac{1}{2\mu_t}\frac{\partial{\bf
u}_{\Omega}}{\partial n}
- \frac{1}{2\mu^{\parallel}} \frac{\partial{\bf u}^{\omega}}{\partial
n}\right]= 0
\end{equation}
is  self-adjoint and  it's eigenfunctions may be  constructed by the
use of mini-max
principle. The eigenfunctions  and the  eigenvalues of  the  operator
$L_{\Lambda}$ may be  obtained from the eigenfunctions and  eigenvalues  of the
operator  $L^{^{H}}_{\Lambda}$ as  limits  when $H\to \infty$. We  assume that
this  is already done\footnote{This theoretical construction and
proper algorithms
will  be described in following publications. Here we  concentrate on 
the derivation
of an approximative formula for the  Scattering matrix}.
   We  assume also that the  eigenfunctions of the   absolutely-continuous
   spectrum of the  operator  $L_{\Lambda}$ and  it's Green function
$G_{\Lambda} (x,s)$
    are  constructed:
\[
-\bigtriangleup_{\mu} G + V G = \lambda G + \delta(x-s).
\]
Then we can construct the Dirichlet-to-Neumann map (DN-map) of  the
operator $L_{\Lambda}$.
The  standard  DN-map  is  described in \cite{SU2,DN01}. The  modified  DN-map
of  the  operator $L_{\Lambda}$ can be  obtained  by
a  {\it projection} onto the  lower channels of  the  boundary  current
$\frac{1}{2\mu_{_{\Omega}}}
P^{^{\Lambda}}_{+}\frac{\partial u}{\partial n}\bigg|_{_{\Gamma}}$ of
the solution of the
homogeneous equation $l u - \lambda u = 0$ with the boundary condition
$u\bigg|_{\Gamma} = u_{\Gamma}\in E_{+}^{^{\Lambda}}$, for regular
values of the
spectral variable $\lambda$. The solution $u$ tends to  zero
at  infinity if $\Im \lambda\neq 0$. It may be  constructed using the
resolvent kernel $G^{^{\Lambda}} (x,x',\lambda)$ of the  operator
$L^{^{\Lambda}}$. The
corresponding  Poisson map  is  given by
\[
{\cal P}^{^{\Lambda}} u_{\gamma}(x) = -\int_{\Gamma} \frac{1}{2\mu_{_{\Omega}}}
\frac{\partial G_{\Lambda}(x,s,\lambda)}{\partial n_s}u_{_{\Gamma}}(s)d s,
\]
\[
P^{^{\Lambda}}_{+}{\cal P}_{\Lambda}: E_{+} \to W_{2}^{3/2 - 0}(\Gamma),
\]
since the finite-dimensional subspace $E_{+}$ belongs
\footnote{ For  Sobolev classes see \cite{Embedding}. Here and  further we use
the Sobolev's notations $W_2^{3/2-\epsilon}$ and assume that
$W_{2}^{3/2 - 0}(\Gamma):=
   \cup_{\epsilon >0} W_{2}^{3/2 - 0}(\Gamma)$} to $W_{2}^{3/2 - 0}(\Gamma)$.
Then  the corresponding DN-map  of the  operator $L_{\Lambda}$ is
\[
P^{^{\Lambda}}_{+}\,\,\frac{1}{2\mu_{_{\Omega}}}
\frac{\partial {\cal P}_{\Lambda} u_{\gamma}}{\partial n} = -
P^{^{\Lambda}}_{+}
\int_{\Gamma} \frac{1}{(2\mu_{_{\Omega}})^2}
\frac{\partial^2 G_{\Lambda}(x,s,\lambda)}{\partial n_x\,\,\partial
n}u_{_{\Gamma}}(s)d s :=
{\cal D}{\cal N}_{\Lambda} u_{\Gamma}.
\]
The  normal  current  $\frac{\partial {\cal P}_{\Lambda}
u_{\gamma}}{\partial n}$ belongs
to $W_{2}^{1/2 - 0}$ hence the formal  integral operator $\int_{\Gamma}
\frac{\partial^2 G_{\Lambda}(x,s,\lambda)}{\partial n_x\,\,\partial 
n}u_{\Gamma}(s)d
s $
is  unbounded in $W_{2}^{3/2-0}(\Gamma)$. However, since
$E^{^{\Lambda}}_{+}\subset W_{2}^{3/2-0}(\Gamma)$
   the projection is   bounded
\[
- P^{^{\Lambda}}_{+}
\int_{\Gamma}
\frac{\partial^2 G_{\Lambda}(x,s,\lambda)}{\partial n_x\,\,\partial
n}u_{\Gamma}(s)d s \in
   W_{2}^{3/2-0}(\Gamma),
\]
\[
{\cal D}{\cal N}^{^{\Lambda}} : W_{2}^{3/2-0}(\Gamma) \longrightarrow
W_{2}^{3/2-0}(\Gamma).
\]
One  may prove by  standard  methods of qualitative spectral
analysis, see for instance
\cite{simon},that there are only a finite number of eigenvalues
$\lambda_r^{\Lambda}$
of the operator $L_{\Lambda}$
in any compact domain of the complex plane $\left\{\lambda \right\}$.
Then  the  following
spectral representation is  valid  for  the  corresponding  DN-map:
\[
{\cal D}{\cal N}^{^{\Lambda}}(\lambda) =  \sum_r
\frac{1}{2\mu^{^2}_{_{\Omega}}}
\frac{P^{^{\Lambda}}_{+}\frac{ \partial \varphi_r (x)}{\partial n_x}\rangle
\,\,\langle P^{^{\Lambda}}_{+}\frac{ \partial \varphi_r (s)}{\partial n_r}}
{\lambda - \lambda^{\Lambda}_r} +
\]
\begin{equation}
\label{DNL}
\sum_{V_{m} + \frac{\pi^2 \,l^2}{2 \mu^{\bot}_m \delta^2_m}> \Lambda}
\int_{\frac{\pi^2 \,l^2}{\delta^2_m}}^{\infty} \frac{1}{2\mu^{^2}_{_{\Omega}}}
\frac{P^{^{\Lambda}}_{+} \frac{\partial \varphi_{\mu,l} (x)}{\partial
n_x}\rangle
\,\,\langle P^{^{\Lambda}}_{+} \frac{\partial \varphi_{\mu,l}
(s)}{\partial n_s}}
{\lambda - \mu} d\mu,
\end{equation}
where the  normal  derivatives  are calculated  on $\Gamma$,
the integration is extended over all branches of the
absolutely-continuous spectrum
of the operator $L_{\Lambda}$  emerging  from the  upper  thresholds
   $\frac{\pi^2 \,l^2}{2 \mu_{_m}\delta^2_m} + V_{_m} > \Lambda$ and the
tensor $2\mu$ of  the  effective mass is diagonal, as defined in the
previous section.
One can check that
$P^{^{\Lambda}}_{_{+}}\frac{\partial \varphi}{\partial
n}\bigg|_{_{\Gamma}} \in W_{_2}^{^{3/2-0}}$
and then prove that  the  series and  integrals  are  convergent due
to the presence
of  the  finite-dimensional projection  $P^{^{\Lambda}}_{_{+}}$ onto
the  entrance subspace
$E^{^{\Lambda}}_+\subset W_2^{3/2 - 0}$. Hence  the  whole  finite-dimensional
matrix-function is  bounded in $W_2^{3/2 - 0}$ with  poles  of first
degree  at the  eigenvalues
$\lambda^r_{\Lambda}$  of  the  operator  $l_{\Lambda}$ and  a  cut
along the upper branches
$\left[ \frac{\pi^2 \,l^2}{2\, \mu^{\bot}_{_m} \delta^2_m}  +
V_{_m}\right.,\left.\infty \right) $.
    \par
Practical  calculation of  the  DN-map  ${\cal D}{\cal
N}^{^{\Lambda}}(\lambda)$ by  the  above  formula
(\ref{DNL}) requires  knowledge of the eigenfunctions and  eigenvalues of
the  discrete spectrum  and
the  eigenfunctions of  the  absolutely-continuous  spectrum of  the
operator $L_{\Lambda}$.
Another  expression for  the  DN-map  may be useful. This is
an  expression of  ${\cal D}{\cal N}_{_{\Lambda}}$ in terms  of
matrix elements of  the
DN-map   ${\cal D}{\cal N}$  of  the  compact part $\Omega_0$ of  the
network with all semi-infinite  wires having zero boundary
condition on their bottom sections.
\par
The  spectrum of the  Schr\"{o}dinger operator $L_0$ on  $\Omega_0$
with  zero  boundary conditions
is  discrete. The  DN-map of  $L_0$  is  an  operator  with  the
generalized  kernel
\[
(2\mu)^2\,\,{\cal D}{\cal N} (x,s,\lambda) = - \frac{1}{(2\mu)_{_{\Omega}}^2}
   \frac{\partial^2 G (x,s,\lambda)}{\partial n_x\,\,\partial n_s}
\]
\[
{\cal D}{\cal N}(\lambda) :
W_2^{3/2 - 0} (\partial \Omega_0)\longrightarrow W_2^{1/2 -
0}(\partial \Omega_0).
\]
The spectral representation of  the  DN-map of the  operator  $L_0$
is  given  by  the  formal  series
\[
{\cal D}{\cal N}(\lambda) =  \sum_r \frac{1}{(2\mu)_{_{\Omega}}^2}
\frac{\frac{ \partial \varphi_r (x)}{\partial n_x}\rangle
\,\,\langle \frac{ \partial \varphi_r (s)}{\partial n_r}}
{\lambda - \lambda^{0}_r}  : =
   \sum_r \frac{1}{(2\mu)_{_{\Omega}}^2}
\frac{ \phi_r (x)\rangle
\,\,\langle \phi_r (s)}
{\lambda - \lambda^{0}_r},
\]
where  $\phi_r (x) = \frac{ \partial \varphi_r (x)}{\partial n_x}$
and $\lambda^{0}_r$ are  the
eigenvalues of  the  operator  $L_0$.
We  present the  DN-map of  $L_0$ as  a  matrix  with respect  to  the
orthogonal decomposition $L_2 (\Gamma) =  E^{^{\Lambda}}_+ \oplus
E^{^{\Lambda}}_{-}$
of  entrance  subspaces  of  the  open
and  closed  channels. Based  on  the  observation  $E_{+}\in
W_2^{3/2-0}$, we  see  that  the matrix
elements  ${\cal D}{\cal N}_{_{+-}}(\lambda)$ are  operators mapping
$E_{\pm}\dots $ into $E_{\pm}\dots$
respectively  (for regular $\lambda$)
   \[
{\cal D}{\cal N} = \left(
\begin{array}{cc}
P^{^{\Lambda}}_+ \left({\cal D}{\cal N}\right) P^{^{\Lambda}}_{+}&
P^{^{\Lambda}}_+
\left({\cal D}{\cal N}\right) P^{^{\Lambda}}_{-}\\
P^{^{\Lambda}}_- \left({\cal D}{\cal N}\right) P^{^{\Lambda}}_{+}&
P^{^{\Lambda}}_- \left({\cal D}{\cal N}\right) P^{^{\Lambda}}_{-}
\end{array}
\right): =
\left(
\begin{array}{cc}
{\cal D}{\cal N}_{_{++}}& {\cal D}{\cal N}_{_{+-}}\\
{\cal D}{\cal N}_{_{-+}}&
{\cal D}{\cal N}_{_{--}}
\end{array}
\right).
   \]
Consider the basis $\left\{ e^m_l \right\}$  of all entrance vectors of closed
channels $\frac{\pi^2 l^2}{2 \mu^{\bot}_{_m}\delta_m^2} + V_{_m}>
\Lambda $ and  introduce the  diagonal
matrix  $K_{_{-}}^{^{\Lambda}}$ with elements
$k^m_l = - \frac{1}{\sqrt{2 \mu^{^{\parallel}}}} \sqrt{\frac{\pi^2 l^2}
{2 \mu^{\bot}_{_m}\delta_m^2} + V_{_m} - \lambda},\,\, $ on the 
entrance  subspaces
$E_{_l,-}^{^{m}}$ of closed  channels
$\frac{\pi^2 l^2}{2 \mu^{\bot}_{_m}\delta_m^2} + V_{_m}> \Lambda $.
The corresponding operator
$K_{_{-}}^{^{\Lambda}} (\lambda): = K_{_{-}} (\lambda)$ has a  bounded inverse
$\left( K_{_{-}}\right)^{^{-1}}$
from  $W_2^{1/2-\varepsilon} \to W_2^{3/2-\varepsilon},\,\,\varepsilon \geq 0$
for  $\lambda $ below the  minimal upper  threshold
   ${\lambda}^{^{\Lambda}}_{_{min}}$.

Let ${\bf u}= \left\{ u_t,\,u^{m}\right\}$ be a  solution
of  the  Schr\"{o}dinger  equation  on the  network with the boundary data
$u_{_{\Gamma}} \in E_{_{+}}$ on the sum $\Gamma$ of bottom sections of the
semi-infinite open  channels and matching boundary  conditions in the
upper  channels
   \[
P^{^{\Lambda}}_{+}{\bf u}_{\Omega}  = u_{_{\Gamma}},\,\,
P^{^{\Lambda}}_{-}\left[{\bf u}_{\Omega}-{\bf u}_{\omega}\right] = 0,\,\,
P^{^{\Lambda}}_{-}\left[\frac{1}{2
\mu_{_{\Omega}}}\,\frac{\partial{\bf u}_{\Omega}}{\partial n} -
\frac{1}{2 \mu^{^{\parallel}}_{_{\omega}}}\frac{\partial{\bf
u}^{\omega}}{\partial n}\right] = 0
\]
and standard  matching boundary  conditions on  all  bottom sections
of the finite  wires.
Denote by $u_{-}$ the  projection  of  the  solution  onto  the
entrance  subspace  $E_-$
in  the  upper  channels in the semi-infinite  wires.
\par
We calculate the  DN-map of the  operator $L_{\Lambda}$ as a  projection
   onto $E_+$  of the boundary  current
\[
{\cal D}{\cal N}^{^{\Lambda}} u_{\Gamma} =
\frac{1}{2\mu_{_{\Omega}}} P_{+} \frac{\partial {\bf u}}{\partial n}
\]
of  the solution  $u$ of  the  homogeneous  equation  ${\cal
L}_{_{\Lambda}} u = \lambda u$ with
the  boundary  condition  $u \big|_{\Gamma} = u_{_{\Gamma}} \in E_{_{+}}$
and the  matching conditions in upper  channels:
   \[
P_{-}\left[{\bf u}_{\Omega}-{\bf u}_{\omega}\right] = 0,\,\,
P_{-}\left[\frac{1}{2\mu_{_{\Omega}}} \, \frac{\partial{\bf
u}_{\Omega}}{\partial n} -
\frac{1}{2 \mu^{^{\parallel}}_{_{\omega}}}\frac{\partial{\bf
u}^{\omega}}{\partial n}\right] = 0.
\]
This  gives the  following  system of  equations:
\[
P_+ \left({\cal D}{\cal N}\right) P_{+} u_{\Gamma} + P_+ \left({\cal
D}{\cal N}\right) P_{+}  u_{-}=
   \left({\cal D}{\cal N}\right)_{_{\Lambda}} u_{\Gamma},
\]
\[
P_- \left({\cal D}{\cal N}\right) P_{+} u_{\Gamma} +
P_- \left({\cal D}{\cal N}\right) P_{-} u_{-} = K_{-} u_{-}.
\]
Eliminating  $u_{-}$ we  obtain the  following  statement:
\begin{theorem}{\it The DN-map  ${\cal D}{\cal N}^{^{\Lambda}}$  of
the intermediate operator $L_{\Lambda}$ is  connected  with  the  DN-map
$\left({\cal D}{\cal N}\right)$ of the operator  $L_0$ on the
compact part $\Omega_0$
of  the  network $\Omega$ by  the  formula
\begin{equation}
\label{DNr}
{\cal D}{\cal N}^{^{\Lambda}} = P^{^{\Lambda}}_+ \left({\cal D}{\cal N}\right)
P^{^{\Lambda}}_+  -
   P^{^{\Lambda}}_+ \left({\cal D}{\cal N}\right) P^{^{\Lambda}}_{_{-}}
\frac{I}{P^{^{\Lambda}}_- \left({\cal D}{\cal N}\right)
P^{^{\Lambda}}_- - K_{-}}
   P^{^{\Lambda}}_- \left( {\cal D}{\cal N} \right) P^{^{\Lambda}}_+ .
\end{equation}
}
\end{theorem}
The irrational part of the  expression for the DN-map  of  the
intermediate  operator
is  contained  in  $K_{-}$ in explicit  form, ie. as  a  family of 
square  roots
$-\frac{1}{\sqrt{2\mu^{\parallel}_{_m}}}\,\,\sqrt{\frac{\pi^2 l^2}
{2 \mu^{\bot}_{_m}\delta_m^2} + V_{_m} - \lambda}$.
Matrix  elements  of  the  DN-map of
   the  Schr\"{o}dinger operator  $L_0$ in the  domain $\Omega_0$ are
rational  functions
   of  the  spectral parameter with  singularities  at  the
eigenvalues of  the  operator $L_0$.
   \par
   Separating  the  resonance term  $\frac{\phi_0\rangle \langle \phi_0}
{\lambda - \lambda_0}$ in  the  DN-map of $L_{_0}$ we  may write it  as
   \[
\left({\cal D}{\cal N}\right) =\frac{1}{(2\mu_{_{\Omega}})^2}
\frac{\phi_0\rangle \langle \phi_0}
{{\lambda} - {\lambda^0_0}} + \sum_{s\neq 0}\frac{1}{(2\mu_{_{\Omega}})^2}
\frac{\phi_s\rangle \langle \phi_s}
{{\lambda} - {\lambda^0_s}} :=\frac{1}{(2\mu_{_{\Omega}})^2}
   \frac{\phi_0\rangle \langle \phi_0}
{\lambda - \lambda^0_0} + {\cal K},
\]
where  ${\cal K}$ is  the  contribution  from the  non-resonance
eigenvalues  $\lambda^0_s \neq \lambda^0_0$.
The  norm of the contribution $\cal K$,  as
an  operator from  $W_2^{3/2- \epsilon}$  to  $W_2^{1/2- \epsilon}$
(for  each $\epsilon >0$),
may be  estimated  by  the  spacing $\rho (\lambda_0)$
at the resonance level $\lambda_0 \approx E_f = $const.
\par
\begin{lemma}{\it If  the  wires are shrinking as  $y \to
\frac{y'}{\varepsilon^{^{\omega}}}$
and the  quantum wells  are  shrinking as $x \to
\frac{x'}{\varepsilon_{_{\Omega}}}$, then the
operator  $K^{^{-1}}_{_{-}}\,\,{\cal K}$  with respect to  the
variables $x',\, y'$ is  estimated
in  $L_2 (\Gamma)$-norm  as
\begin{equation}
\label{shrink}
\parallel K^{^{-1}}_{_{-}}\,\,{\cal K} \parallel \leq
   C_0 \frac{\varepsilon^{^{\omega}}}{\varepsilon_{_{\Omega}}}.
\end{equation}
}
\end{lemma}
{\it Proof} \,\,\, Notice first that the  operator
$K^{^{-1}}_{_{-}}\,\,{\cal K}$ is  a  bounded
operator at the regular values of the spectral parameter. Since the
DN-map is  homogeneous of degree
$-1$ and  $K^{^{\lambda}}_{_-} = K_{_{-}}$ admits an estimate  by
$\varepsilon_{_{\omega}}$ from above,
   see  (\ref{2}) below, the  whole  estimate is  obvious if  the
distance of the spectral point
    $\lambda$  from the  resonance level  $E_{\lambda}$ is strictly
positive. This gives the
required estimate of the contribution from all non-resonance terms
with eigenvalues which
do not approach the Fermi level in the course of the shrinking. The 
number of other
non-resonance terms is finite hence it is sufficient to
   estimate the contribution from a non-resonance term which 
corresponds to the eigenvalue
   $\lambda'_1 = \frac{\lambda_1}{\varepsilon^{^2}_{_{\Omega}}}$
   closest to  the  resonance eigenvalue
   $\lambda'_0 = \frac{\lambda_0}{\varepsilon^{^2}_{_{\Omega}}} \approx E_f$.
   Notice first that the shrinking of the normalized eigenfunction is
described by the
   formula
   \[
\varphi'_{_{1}} (x') = \frac{\varphi_{_{1}}
\left(\frac{x'}{\varepsilon_{_{\Omega}}}\right)}
{\int_{_{\Omega'}}  |\varphi_{_{1}}|^{^2}
\left(\frac{x'}{\varepsilon_{_{\Omega}}}\right) dx'}
\approx \varphi_{_{1}}
\left(\frac{x'}{\varepsilon_{_{\Omega}}}\right) \,\,
\varepsilon^{^{-2}}_{_{\Omega}}.
   \]
Then the  normal derivative of the  shrinking eigenfunction is  transformed as
\[
\frac{\partial \varphi'_{_{1}}}{\partial n} = \frac{\partial
\varphi'_{_{1}}}{\partial n}\,\,
\frac{1}{\varepsilon^{^{3}}_{_{\Omega}}}.
\]
If the  Fermi-level $E_f$ is  constant in the course of the experiment then
the resonance  is  shifted, due
   to the shrinking  if the  wires, to  the  spectral point
   $\lambda'_0\approx E_f \varepsilon^{^{2}}_{_{\omega}}$. Then
   the  spacing $\rho'(E_f)$ on the  resonance level is  calculated as
    $\frac{\rho\left(E_f
\varepsilon^{^{2}}_{_{\omega}}\right)}{\varepsilon^{^{2}}_{_{\Omega}}}$.
Combining
    all  these  facts we  obtain the  estimate for the one-dimensional
  non-resonance term
    \begin{equation}
    \label{1}
\parallel \frac{\frac{\partial \varphi'_{_{1}}}{\partial n}\rangle \,
\langle \frac{\partial \varphi'_{_{1}}}{\partial n}}{\lambda'_{_1} -
\lambda'_{_0}}
\parallel \leq \frac{\sup_{\Gamma}|\bigtriangledown \varphi |^2}
{\rho\left(E_f
\varepsilon^{^{2}}_{_{\omega}}\right)}\,\,\frac{1}{\varepsilon_{_{\Omega}}},
    \end{equation}
in full agreement with the fact that the  DN-map is
homogeneous of order $-1$. Then
we  have  $\parallel {\cal K} \parallel \leq
\frac{C}{\varepsilon_{_{\Omega}}}$.
\par
One can easily obtain the  estimate for $K^{^{\Lambda}}_{-}$ at
the  Fermi-level:
\begin{equation}
    \label{2}
\parallel \left(K^{^{\Lambda}}_{-}\right)^{^{-1}} \parallel \approx
\max_m \sqrt{\mu_m^{^{\bot}} \mu_m^{^{\parallel}}}
    \frac{\varepsilon_{_{\omega}}}{\pi }
\end{equation}
if  $2 \mu_m^{^{\bot}} \varepsilon^{^2} _{_{\omega}} \max_{_m} |E_f -
V_m|  << \pi $. Summarizing
the  estimates  (\ref{1},\ref{2})  we obtain the required statement.

$\Box$

The shrinking of the device, with the constant Fermi level $E_f$, can 
be applied
to each quantum well and each quantum wire
separately. Then the  DN map  on the joining of the  wells  is  an
orthogonal sum  of
the DN-maps  of  quantum wells $\Omega_t,\,\, t=1,2,\dots$,
which  implies:
\[
   \parallel {\cal K} \parallel <
\max_{_{s}}\frac{\sup_{\Gamma}|\bigtriangledown \varphi_s |^2}
{\rho\left(E_f
\varepsilon^{^{2}}_{_{\omega}}\right)}\,\,\frac{1}{\varepsilon_{_{\Omega}}}
\]
and
\begin{equation}
\label{thin}
\parallel K^{^{-1}}_{_{-}}{\cal K}  \parallel \leq \max_{_{s,m}}
\sqrt{\mu_m^{^{\bot}} \mu_m^{^{\parallel}}}
    \frac{\sup_{\Gamma}|\bigtriangledown \varphi_s |^2}
{\pi \rho\left(E_f \varepsilon^{^{2}}_{_{\omega}}\right)}\,\,
\frac{\varepsilon_{_{\omega}}} {\varepsilon_{_{\Omega}}} :=
C_0 \,\, \frac{\varepsilon_{_{\omega}}} {\varepsilon_{_{\Omega}}}
\end{equation}
with an  absolute constant  $C_0$ depending  on the (non-dimensional)
shape of  the  network. We say
that the  network is  ``thin'' if  $\parallel K^{^{-1}}_{_{-}}{\cal
K}  \parallel < 1 $. This condition
is obviously fulfilled for shrinking  networks, if
$\frac{\varepsilon_{\omega}}{\varepsilon_{\Omega}} \to 0$.
The  following  statement  reveals the  structure of  singularities
in the above representation (\ref{DNr})
for thin (but  non necessarily  shrinking) networks:
\begin{theorem}{\it If the  network  is  thin then
the  pole $\lambda^0_0$ of  the  DN-map  ${\cal D}{\cal N}$ of  the operator
$L_0$ on  the compact part $\Omega_0$ of the  network (the
singularity of  the  first  addendum
of (\ref{DNr})) is  compensated by the pole of the
second addendum and disappears as a  singularity of  the  whole  function
${\cal D}{\cal N}^{^{\Lambda}}$ so  that  the whole expression (\ref{DNr})
is  regular at the point $\lambda^0_0$. A new pole appears as a
   zero in the denominator
$P^{^{\Lambda}}_- \left({\cal D}{\cal N}\right) P^{^{\Lambda}}_- -
K_{_{-}} = 0 $.
}
\end{theorem}
{\it Proof}
Separate the  resonance term in  the  DN-map of  the  operator $L_0$
framed  by  projections onto $E_{\pm}$:
\[
P^{^{\Lambda}}_+\left({\cal D}{\cal N}\right) P^{^{\Lambda}}_+ =
\frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
\frac{{\phi}^{^{+}}_0\rangle \langle {\phi}^{^{+}}_0}
{\lambda - \lambda^0_0} + \sum_{s\neq 0}\frac{1}{\left(2
\mu_{_{\Omega}}\right)^{^{2}}}
\frac{\hat{\phi}^{^{+}}_s\rangle \langle \hat{\phi}^{^{+}}_s}
{\lambda - \lambda^0_s} := \frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
   \frac{{\phi}^{^{+}}_0\rangle \langle {\phi}^{^{+}}_0}
{\lambda - \lambda^0_0} + {{\cal K}}_{++}
\]
\[
P^{^{\Lambda}}_+\left({\cal D}{\cal N}\right) P^{^{\Lambda}}_- =
\frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
\frac{{\phi}^{^{+}}_0\rangle \langle {\phi}^{^{-}}_0}
{{\lambda} - {\lambda_0}} + \sum_{s\neq 0} \frac{1}{\left(2
\mu_{_{\Omega}}\right)^{^{2}}}
\frac{{\phi}^{^{+}}_s\rangle \langle {\phi}^{^{-}}_s}
{{\lambda} - {\lambda^0_s}} := \frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
   \frac{{\phi}^{^{+}}_0\rangle \langle {\phi}^{^{-}}_0}
{{\lambda} - {\lambda^0_0}} + {{\cal K}}_{+-}
\]
\[
P^{^{\Lambda}}_- \left({\cal D}{\cal N}\right) P^{^{\Lambda}}_+ =
\frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
\frac{{\phi}^{^{-}}_0\rangle \langle {\phi}^{^{+}}_0}
{{\lambda} - {\lambda^0_0}} + \sum_{s\neq 0}\frac{1}{\left(2
\mu_{_{\Omega}}\right)^{^{2}}}
\frac{{\phi}^{^{-}}_s\rangle \langle {\phi}^{^{+}}_s}
{{\lambda} - {\lambda^0_s}} :=\frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
   \frac{{\phi}^{^{-}}_0\rangle \langle {\phi}^{^{+}}_0}
{{\lambda} - {\lambda^0_0}} + {{\cal K}}_{-+}
\]
with  ${{\cal K}}_{- +} = \left( {{\cal K}}_{+ -} \right)^{^+} $, and
\[
P^{^{\Lambda}}_-\left({\cal D}{\cal N}\right) P^{^{\Lambda}}_- =
\frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
\frac{{\phi}^{^{-}}_0\rangle \langle {\phi}^{^{-}}_0}
{{\lambda} - {\lambda^0_0}} + \sum_{s\neq 0}
\frac{{\phi}^{^{-}}_s\rangle \langle {\phi}^{^{-}}_s}
{{\lambda} - {\lambda^0_s}} :=\frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
   \frac{{\phi}^{^{-}}_0\rangle \langle {\phi}^{^{-}}_0}
{{\lambda} - {\lambda^0_0}} + {{\cal K}}_{--}.
\]
Here  ${\cal K}_{_{++}},\,{\cal K}_{_{\pm}},\,{\cal K}_{_{--}}\, $
are  the matrix  elements
of the contribution $\cal K$ to the  DN-map from the non-resonance
eigenvalues. The  operators
${\cal K}_{_{++}},\,{\cal K}_{_{+-}}$  are  bounded  in $W_2^{3/2-0}
(\Gamma)$,\, ${\cal K}_{_{-+}}=
{\cal K}_{_{+-}}^+$ for  real $\lambda$ and  ${\cal K}_{_{--}}$ acts
from  $W_2^{3/2-0} (\Gamma)$
into $W_2^{1/2-0} (\Gamma)$ as does $K_-$.
Then  the expression  (\ref{DNr}) may
be written  as
\[
{\cal D}{\cal N}^{^{\Lambda}} =\frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
\frac{{\phi}^{^{+}}_0\rangle \langle {\phi}^{^{+}}_0}
{{\lambda} - {\lambda^0_0}} + {{\cal K}}_{+-} \,\,+
\]
\begin{equation}
\label{DNapprox}
\left[\frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
\frac{{\phi}^{^{+}}_0\rangle \langle {\phi}^{^{-}}_0}
{{\lambda} - {\lambda^0_0}} + {{\cal K}}_{+-}\right]
\frac{I}{ \frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
\frac{{\phi}^{^{-}}_0\rangle \langle {\phi}^{^{-}}_0}
{{\lambda} - {\lambda^0_0}} + {{\cal K}}_{--} - K_-}
   \left[\frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
   \frac{{\phi}^{^{-}}_0\rangle \langle {\phi}^{^{+}}_0}
{{\lambda} - {\lambda^0_0}} + {{\cal K}}_{-+}\right].
\end{equation}
The  operator $-K_{_{-}} $
   for selected values of the energy is  positive and  may  be  estimated
    from below by the  distance from $\lambda$ to the  lowest upper
    threshold
   \[
\rho_{_{-}} (\lambda):=\mbox{min}_{_{_{\frac{\pi^2 l^2}
{2 \mu^{\bot}_{_m}\delta_m^2} + V_{_m} > \Lambda}}}
   \,\,\,\,\sqrt{2\mu^{^{\parallel}}_m}\,\,\,
   \sqrt { \frac{\pi^2 l^2}{2 \mu^{\bot}_{_m}\delta_m^2} + V_{_m} - \lambda},
\]
\[
\displaystyle
\langle -K_{_{-}} u,\,u \rangle \geq \rho_{_{-}} \parallel u
\parallel^{^2}_{_{L_2 (\Gamma)}}.
\]
If the  network is  thin, $\parallel K^{^{-1}}_{_{-}}{\cal K}
(\lambda_0) \parallel < 1$, then the  operator
\[
{\cal K}_{_{--}} - K_{_{-}} := k
\]
is  invertible  $k^{^{-1}} =
   \left(1 - K^{^{-1}}_{_{-}} {\cal K}_{_{--}}\right) \left( -
K^{^{-1}}_{_{-}}\right)$.
The  middle  term  of  the  above product (\ref{DNapprox}) can  be
  given  as a  solution of  the  equation
\[
\left[ \frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
\frac{{\phi}^{^{-}}_0\rangle \langle {\phi}^{^{-}}_0}
{{\lambda} - {\lambda^0_0}} + {{\cal K}}_{--} - K_{_{-}}  \right] u = f,
\]
\[
u = k^{^{-1}}\,\,f - \frac{1}{\cal D}\,\,\, k^{^{-1}}\, \,
{\phi}^{^{-}}_{0}\rangle\,\,\langle {\phi}^{^{-}}_{0},\,  k^{^{-1}}\,\, f
\rangle,
\]
where ${\cal D} =\left(2
\mu_{_{\Omega}}\right)^{^{2}}\,\,\left({\lambda} - {\lambda}^0_0
\right) +
\langle {\phi}^{^{-}}_{0},\,
   k^{^{-1}}\,\, \, {\phi}^{^{-}}_{0}\rangle$. Substituting that expression into
(\ref{DNapprox}) we  notice that  all  factors  $\left({\lambda}-
{\lambda^0_0}\right)$
are  cancelled:
\[
{\cal D}{\cal N}^{^{\Lambda}} = \frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
\frac{{\phi}^{^{+}}\rangle\,\langle {\phi}^{^{+}}}{{\lambda}- {\lambda}_0}
\left[1 - \frac{\langle {\phi}^{^{-}},\,k^{^{-1}}\,\, {\phi}^{^{-}} \rangle}
{{\lambda}- {\lambda}_0} + \frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
\frac{\langle {\phi}^{^{-}},\,k^{^{-1}}\,\,{\phi}^{^{-}} \rangle^{^{2}}}
{{\cal} D\left(\hat{\lambda}- {\lambda}_0\right)} \right] -
\]
\[ \frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
\frac{{\cal K}_{+ -} k^{^{-1}}\,{\phi}^{^{-}}\rangle\, \langle {\phi}^{^{+}} }
{{\lambda}- {\lambda}_0}\left[ I - \frac{\langle {\phi}^{^{-}},\,k^{^{-1}}\,\,
{\phi}^{^{-}} \rangle}
{{\cal} D}\right]-
\]
\[
\frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
\frac{{\phi}^{^{+}} \rangle\, \langle {\cal K}_{+ -} k^{^{-1}}\,\,
{\phi}^{^{-}}}
{{\lambda}- {\lambda}_0}
\left[ I - \frac{\langle {\phi}^{^{-}},\,k^{^{-1}}\,\,
{\phi}^{^{-}} \rangle}
{{\cal} D}\right] +
\]
\begin{equation}
\label{second} \frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
\frac{{\cal K}_{+ -}k^{^{-1}}\,\, {\phi}^{^{-}}\rangle\,
   \langle {\cal K}_{+ -} k^{^{-1}}\,\,
{\phi}^{^{-}} }{{\cal D}}= \frac{1}{\left(2 \mu_{_{\Omega}}\right)^{^{2}}}
\frac{{\phi}^{^+} -{\cal K}_{+ -} k^{^{-1}}\,\,{\phi}^{^{-}}\rangle\,\,
   \langle {\phi}^{^+} - {\cal K}_{+ -} k^{^{-1}}\,\,
{\phi}^{^{-}}}{\cal D}.
\end{equation}
One  can see that on the  first  step of the approximation procedure we  obtain
the residue  of  the  DN- map  $\hat{\Lambda}^r$ at the simple zero of
the  denominator ${\cal D}$ proportional to  ${\phi}^{^+}\rangle \,\,
   \langle {\phi}^{^+}$ as  announced. The  residue, with a small correction
   obtained at the  second  step of  the  approximation procedure, is
given by
   \[
{\phi}^{^+} -{\cal K}_{+ -}k\,\,{\phi}^{^{-}}\rangle\,\,\langle
{\phi}^{^+} - {\cal K}_{+ -} k\,\,
{\phi}^{^{-}}.
   \]
In particular this means that the portions of the  eigenfunctions of  the
operator ${l}$ in the  entrance subspace may be  found via
the successive  approximation
   procedure (on the  second  step) as:
\[
   P_+ \frac{\partial \varphi^r_0}{\partial n} \approx
   {\phi}^{^+} - {\cal K}_{+ -}\,\, k\,\,{\phi}^{^{-}}.
\]

$\Box$

{\bf Corollary}
   If the minimal momentum in the wires at the resonance energy
$\lambda_0$:
\[
p_{_{_{\mbox{min}}}} (\lambda_{_0}) =
\mbox{min}_{_{_{\frac{\pi^2 l^2}{2 \mu^{\bot}_{_m}\delta_m^2} +
V_{_m} < \Lambda}}}\,\,\,
   \sqrt {\lambda_0 - \frac{\pi^2 l^2}{2 \mu^{\bot}_{_m}\delta_m^2} - V_{_m}}
\]
   exceeds the contribution to  the matrix element ${\cal D}{\cal N}_{_{--}}$
   of the  DN-map of  $L_0$ from the  neighboring  non-resonance eigenvalues
   then the
   root vector which corresponds to this new pole coincides, at
first order of the perturbation procedure, with the entrance
vector $P_+ \frac{\partial \varphi_0}{\partial n} = \phi_0$
of the DN-map  ${\cal D}{\cal N}$  of the  operator $L_0$ on the  compact part
   $\Omega_0$ of  the network.

The eigenvalues and the eigenfunctions of the  discrete spectrum
of  the  operator  $L^{^{\Lambda}}$ can be  found  by minimizing
the  corresponding Rayleigh ratio, or from the  corresponding
dispersion equation, involving the
DN-map of  the  Schr\"{o}dinger operator $L_{0}$ with  zero boundary
condition on the  border of  the
compact part  $\Omega_0$ of the  network.
    \begin{theorem}{\it The eigenvalues of the operator $L^{^{\Lambda}}$
may be found as vector zeroes $(\lambda,\, \nu_{_{\lambda}})$ of the
dispersion equation
\[
{\cal D}{\cal N}_{--} (\lambda) \nu_{_{\lambda}} -
K^{^{\Lambda}}_{_{-}}(\lambda) \nu_{_{\lambda}} = 0.
\]
In particular for  values  of  the  spectral parameter  between the
maximal  lower threshold
$\lambda^{^{\Lambda}}_{_{max}}$ and  minimal upper threshold
$\lambda^{^{\Lambda}}_{_{min}}$
the  dispersion equation  takes the form
\begin{equation}
\label{dispers}
\nu = K^{^{\Lambda}}_{_{-}}\,\,(\lambda)^{-1}\,\,\,{\cal D}{\cal
N}_{--}\,\,(\lambda) \nu,
\end{equation}
with   bounded  operator-functions
$\left[K^{^{\Lambda}}_{_{-}}\right]^{-1}\,(\lambda)\,\,
{\cal D}{\cal N}_{--} (\lambda)$ in $L_2 (\Gamma)$,
and it may be transformed  to an equation with a  trace class operator
${\cal D} (H,\,\lambda)$:
\[
\nu = \left[K^{^{\Lambda}}_{_{-}} - {\cal D}{\cal N}_{--}\,\,(-H) +
(H + \lambda){\cal P}^{^{+}}_{_{H}}
{\cal P}^{^{+}}_{_{H}}\right]^{^{-1}}(H + \lambda)^{^{2}}\,
{\cal P}^{^{+}}_{_{H}}{\cal R}_{_{\lambda}}{\cal P}_{_{H}}\nu :=
{\cal D} (H,\,\lambda) \nu ,
\]
where  ${\cal R}_{_{\lambda}}$  and ${\cal P}_{_{\lambda}}$ are
  the
resolvent and the Poisson map of the operator $L$ on the  compact part
of the network with zero boundary condition respectively. The  corresponding
scalar  equation may be  presented in the
form
\begin{equation}
\label{Dispersq}
\det \left[ I - {\cal D} (H,\,\lambda)\right] = 0.
\end{equation}
}.
\end{theorem}
{\it Proof}\,\,\,\,The projections $u_{_{_{\Gamma}}} \in
E^{^{\Lambda}}_{_{-}}$ of
the eigen-function $u,\,\, L^{^{\Lambda}} u = \lambda u $ of the
operator $L^{^{\Lambda}}$
onto the cross-sections $\Gamma$ of the open  channels  should
fulfill  the  condition
   $\frac{\partial u}{\partial n} \bigg|_{_{\Gamma}} =
K_{_{-}}^{^{\Lambda}} (\lambda) u_{_{\Gamma}}$.
   On the  other hand  the  restrictions of $u$ onto  the  compact
part of the  network
fulfills  the  corresponding  homogeneous  equation, hence
   $\frac{\partial u}{\partial n} \bigg|_{_{\Gamma}} = {\cal D}{\cal
N}_{--}\,\,(\lambda) $. Matching
   both  conditions with  the boundary  conditions (\ref{bcond}) we
obtain the dispersion relation:
   \[
\frac{I}{{\bf \mu}^{^{\omega}}} \,\,\, K_{_{-}}^{^{\Lambda}}
(\lambda) u_{_{\Gamma}} u_{_{\Gamma}} =
\frac{I}{{\bf \mu}^{^{\Omega}}}\,\,\, P_{_{-}}{\cal D}{\cal
N}\,\,P_{_{-}}\,\,(\lambda) u_{_{\Gamma}}.
\]
Due to the invertibility  of  $ K_{_{-}}^{^{\Lambda}} (\lambda)$ below
$\lambda^{^{\Lambda}}$ we  obtain
the  first  statement (\ref{dispers}) of  the  theorem. The  second
statement requires the following
formula  for  the  DN-map, see  \cite{DN01},
\[
{\cal D}{\cal N}\,\,(\lambda) = {\cal D}{\cal N}\,\,(-H) -
(H + \lambda){\cal P}^{^{+}}_{_{H}}
{\cal P}_{_{H}} - (H + \lambda)^{^{2}}{\cal P}^{^{+}}_{_{H}} R_{_{\lambda}}
{\cal P}_{_{H}},
\]
easily derived from Hilbert identity. Both second and third terms of
the sum in the right hand side
are compact operators in $L_2 (\Gamma)$ if the  compact part
$\Omega_0$ of the  network has
a  piece-wise smooth boundary with  Meixner boundary  conditions at  the
corners. Moreover, the
third  term is an operator with a  finite  trace. Then the  operator
${\cal D} (H,\,\lambda)$
has a finite trace  too and  the  dispersion equation may be
presented  in the form
\[
\det \left[ I - {\cal D} (H,\,\lambda)\right] = 0.
\]

$\Box$

\section{Scattering matrix}
\noindent
We can write the scattered  waves in the lower channels on
semi-infinite wires $\omega^m $ by the
linear combination of modes: the  oscillating exponentials combined
with eigenfunctions of
cross-sections in open channels and the decreasing exponentials combined  with
eigenfunctions of cross-sections in closed channels:
\[
\psi_{+} = \sum_{\frac{\pi^2 l^2}{2 \mu_m^{\bot}\delta_m} + V_m<
\Lambda} e^m_l \,\, \nu^m_l
e^{-i\sqrt{2 \mu_m^{\parallel}}\sqrt{\lambda - V_m \frac{\pi^2 l^2}{2
\mu_m^{\bot}\delta_m}}\,\,x}+
\]
\[
\sum_{\frac{\pi^2 k^2}{2 \mu_m^{\bot}\delta_m} + V_m <\Lambda}
e^m_l\,\,S^{m,l}_{n,k}\nu_k^n\,\,
   e^{-i\sqrt{2 \mu_m^{\parallel}}\sqrt{\lambda - V_m -\frac{ \pi^2
l^2}{2 \mu_m^{\bot}\delta_m}}\,\,x}+
\]
\begin{equation}
\label{SAnsatz}
\sum_{\frac{\pi^2 k^2}{2 \mu_m^{\bot}\delta_m} + V_m > \Lambda}
e^m_l\,\,s^{m,l}_{n,k}\nu_k^n\,\,
   e^{-\sqrt{2 \mu_m^{\parallel}}\sqrt{ V_m + \frac{ \pi^2 l^2}{2
\mu_m^{\bot}\delta_m} - \lambda}\,\,x}.
\end{equation}
   The finite  matrix  $S =
\left\{ S^{m,l}_{n,k}\right\}$ is  a  Scattering  matrix---the main
object  of  our  search.
We present the  above  Scattering  Ansatz in the following short 
form, introducing the notations:
   $\sum_{ml} e^m_l \,\, \nu^m_l = \nu_+ \in E_+,\,
\, \left\{ S^{m,l}_{n,k}\right\} = {\bf S},\,\,
\left\{ s^{m,l}_{n,k}\right\} = {\bf s}$ and  the  diagonal matrices
in $E^{\Lambda}_+,\,E^{\Lambda}_- $:
\[
K_{_{+}}= \left\{\sqrt{2 \mu_m^{\parallel}}\sqrt{\lambda -
V_m \frac{\pi^2 l^2}{2 \mu_m^{\bot}\delta_m}}\right\}
\]
for $\frac{\pi^2 k^2}{2 \mu_m^{\bot}\delta_m} + V_m <\Lambda$ and
\[
K_{_{-}}= - \left\{\sqrt{2 \mu_m^{\parallel}}\sqrt{ V_m +
\frac{ \pi^2 l^2}{2 \mu_m^{\bot}\delta_m} - \lambda}\right\}
\]
for  $\frac{\pi^2 k^2}{2 \mu_m^{\bot}\delta_m} + V_m > \Lambda$.
We  may present  the  above  Ansatz in the form:
\begin{equation}
\label{Sanzatz}
\psi_{+} :=  e^{^{-iK_{_{+}}x}}\nu_{_{+}} + e^{^{ iK_{_{+}}x}} {\bf
S}\nu_{_{+}} +
e^{^{K_{_{-}}x}}{\bf s}\nu_{_{+}}.
\end{equation}
The  scattering matrix  in the open  channels is ${\bf S}$. The order 
of the matrix,
$M\times M$, is  defined  by  the
   number of  open  channels $M$.
To derive the formula for the Scattering matrix of the operator $L$
on the  whole
network $\Omega$ we  should  substitute the  above Ansatz into  the  boundary
conditions for  matching  in  each  channel:
\begin{equation}
\label{matching}
P_{\pm}\left[ \psi \right]|_{\Gamma}= 0,\,
\,P_{\pm}\left[\frac{\partial \psi }{\partial n}\right]|_{\Gamma}= 0,
\end{equation}
where  $\left[ \psi \right]|_{\Gamma} = \left[\psi^{\omega}-
\psi_{_{\Omega}}\right]|_{\Gamma}$ and
$\left[\frac{\partial \psi_{_{\Omega}} }{\partial n}\right]|_{\Gamma} =
\left[\frac{1}{{\bf \mu}^{^{\omega}}}\frac{\partial
\psi_{_{\omega}}}{\partial n} -
\frac{1}{{\bf \mu}_{_{\Omega}}}
\frac{\partial \psi^{\Omega}}{\partial n}\right]|_{\Gamma}$. The
connection between the values of the components of the above Ansatz
in the  wires
is obtained by  straightforward  differentiation. However, the  corresponding
connection between the values of the  components
$\psi_{_{\Omega_t}}$ of
the  eigenfunction in the vertex  domains $\Omega_t$ is  given  by 
the  DN-map:
\[
{\cal D}{\cal N} \psi_{_{\Omega_t}}\bigg|_{\Gamma} =
   \frac{\partial \psi_{_{\Omega}}}{\partial n}.
\]
Then, using the boundary  conditions (\ref{matching}), we  may
obtain an equation
for the  undefined  coefficients of  the  Ansatz  (\ref{SAnsatz}).
\par
In this section we follow the above program assuming that  the
network consists of a  compact
part $\Omega_0$ with a  several  wires  $\omega^m,\, m=1,2,\dots$
width $\delta_m$
attached orthogonally to $\Omega_0$  at  the  flat  pieces $\gamma_m$ 
of  the  boundary
$\partial \Omega_0$,\,\, $\Gamma = \cup_{m} \gamma_m$. The  potential
$V (x)$ of the  Schr\"{o}dinger
   operator
\[
L u = -\bigtriangleup u + V(x) u
\]
takes constant values $V_m$ on the  wires $\omega^m$
   and  is a real bounded measurable function on the  vertex  domains.
Assume  that  the  DN-map of  the operator $l_0$ on the compact part $\Omega_0$
of  the  network $\Omega$ is already constructed and  presented (formally)
   by a spectral  series in terms of the eigenfunctions  $\varphi_r$ and
eigenvalues $\lambda_r$
   corresponding  to  zero  boundary conditions on  $\partial \Omega_0$:
\begin{equation}
\label{DN}
{\cal D}{\cal N}(x,x') =  \sum_s \frac{\frac{ \partial \varphi_r
(x)}{\partial n_x}\rangle\,
\langle  \frac{\partial \varphi_r (s)}{\partial n_{x'}}}
{\lambda - \lambda^0_r} .
\end{equation}
Choosing  $\Lambda$ we  define  the  intermediate operator
$L_{\Lambda}$ and  construct the  corresponding  DN-map  ${\cal
D}{\cal N}_{\Lambda}$ as
described in the previous  section. We  may    calculate
the  Scattering  matrix
of the  operator  $L$ at  the  level $\Lambda$ matching  the
Scattering Ansatz in the  wires
either  using  the  DN-map ${\cal D}{\cal N}$ of the operator
$L_0$ on the compact
sub-domain  $\Omega_0$, or  using the
DN-map ${\cal D}{\cal N}_{\Lambda}$  of  the  intermediate  operator
$l_{\Lambda}$.
    \par
In second  case  we  should  just  match the  exponential Ansatz  in
the open channels
\[
\psi^{\omega} = e^{-i K_+ x} \nu  + e^{ iK_{+}x} S\nu,\,
\]
with an  outgoing solution  $\psi_0$ of the  homogeneous  equation
$l_{\Lambda} u_0 = \lambda u_0$
for  which  the  boundary  data are  connected  via  the
corresponding  DN-map:
\[
{\cal D}{\cal N}^{^{\Lambda}}\psi_0 \bigg|_{\Gamma} = P_{_{+}}
\frac{\partial \psi_0}{\partial n}\bigg|_{\Gamma}.
\]
Writing  the  exponential ansatz as
\[
e^{\pm i K_+ x} = \mbox{diag}\left\{ e^{\pm i \,\sqrt{2\mu^{\parallel}_m}
\sqrt{\lambda - V_m - \frac{\pi^2\, l^2}{2\mu^{\bot}_m\delta_m}}x}
e^m_l\rangle \,\langle e^m_l \right\}
\]
and  assuming that $\nu = \sum_{m,l}\nu_{ml}e^m_l $ we  obtain from
the  matching conditions
\[
\frac{1}{{\bf \mu}^{^{\Omega}}}
{\cal D}{\cal N}^{^{\Lambda}} \left[ I + S \right] =
\frac{1}{{\bf \mu}^{^{\omega}}}\left[ - i K_{+} +  i K_{+} S \right]
\]
so that
\begin{equation}
\label{Smatrix}
S (\lambda) = - \frac{\frac{{\bf \mu}^{^{\omega}}}
{ {\bf \mu}^{^{\Omega}}}{{\cal D}{\cal N}}^{^{\Lambda}} + i K_+}
{ \frac{{\bf \mu}^{^{\omega}}}
{{\bf \mu}^{^{\Omega}}} {{\cal D}{\cal N}}^{^{\Lambda}} - i K_+}.
\end{equation}
Here the  denominator is the  second  factor of  the  product. The
formula  (\ref{Smatrix})
is  convenient  for  calculating  resonances, see \cite{Lax}, and
for the  description of  transition processes. It  may also be  used
for  construction of  Scattered  waves.
One may obtain an  approximate expression for  the
Scattering matrix in an  interval
$\left( \Lambda - T,\,\Lambda + T \right)$ by substituting  into
(\ref{Smatrix}) the reduced
expression  for  the  DN-map  ${\cal D}{\cal N}^{^{\Lambda}}_{_T}$
\[
{\cal D}{\cal N}^{^{\Lambda}}_{_T}(\lambda) =
   \sum_{_{\left(\Lambda - T\,<\lambda^{\Lambda}_r\,<\Lambda + T \right)}}
\frac{P_{+}\frac{ \partial \varphi_r (x)}{\partial n_x}\rangle
\,\,\langle P_{+}\frac{ \partial \varphi_r (s)}{\partial n_r}}
{\lambda - \lambda^{\Lambda}_r} +
\]
\begin{equation}
\label{reduced}
\sum_{V_{m} + \frac{\pi^2 \,l^2}{2 \mu^{\bot}_m \delta^2_m}> \Lambda}
\int_{_{\left(\Lambda - T\,<\mu<\Lambda + T \right)}}
\frac{P_{+} \frac{\partial \varphi_{\mu,l} (x)}{\partial n_x}\rangle
\,\,\langle P_{+} \frac{\partial \varphi_{\mu,l} (s)}{\partial n_s}}
{\lambda - \mu} d\mu,
\end{equation}
\begin{equation}
\label{Sapprox}
S (\lambda) \approx -\frac{{\cal D}{\cal N}^{^{\Lambda}}_{_T} + i K_+}
{{\cal D}{\cal N}^{^{\Lambda}}_{_T} - i K_+}.
\end{equation}
This expression may  contain one  or  several polar  terms. The
resulting  expression for
the  Scattering  matrix  is  actually  the  Scattering  matrix  of  a
solvable  model
similar  to  the  model presented  in  \cite{BF61,Opening,Zero_range},
see  appendix 1.  The  deviation of the  exact
expression (\ref{Smatrix}) from the approximate expression
(\ref{Sapprox}) may be  estimated
based on  the  spectral formula for the  DN-map. It will  be  done in
following publications.
A simple  example  of  one-pole and  few-poles  approximations  for
the  Scattering matrix
is  presented  in Appendix 2.
\par
    One  may  obtain another expression for  the  Scattering  matrix
based  on the DN-map of
the  compact  domain. Denote by  $E_+$ the $M$-dimensional subspace
in  $L_2 (\gamma)=
E$ spanned by the vectors $e^1_s,\, s=1,2,3,\dots M$, $E_+$ plays  the
role of  the  ``entrance  subspace" of  open channels.
The orthogonal complement of  $E_+$, $L_{2}(\Gamma)\ominus E_+ = E_- $ is the
entrance subspace of  closed channels. The above ansatz
(\ref{Sanzatz}) is already decomposed in terms of exponential solutions in
open $f^-_1,f^+_1 $ and  closed $f^l_s$
channels:
\[
    \Psi_s = \left[\delta_{s1}
f^-_1 + S^1_{s1} f^+_1\right] + \sum_{closed}^{\infty} S^l_{s1}
f^l_s = \Psi_s^+ + \Psi_s^-.
\]
   Our aim is to find the  coefficients  $S^1_{st}$ of  the  ansatz which form
the Scattering  matrix. We  may  find  them from  the condition
(\ref{boundcond}) of
the continuation of  the  Scattering  ansatz  inside  the  domain.
Denoting  by  $\Psi_{\gamma},\frac{\partial \Psi_\gamma}{\partial
n_{\gamma}}$ the  boundary  data of  the  above Scattering ansatz on
$\Gamma$ and assuming that the  boundary values of the component
of the  scattered  wave  inside  the  vertex  domain coincide on the
bottom sections of  wires  with the  boundary  values of the
Scattering ansatz we may write these conditions using the DN-map
    of  the  Schr\"{o}dinger operator in  $\Omega_0$:
\begin{equation}
\label{matching}
   \frac{1}{\mu^{\parallel}} \,\,\frac{\partial \Psi_{\gamma}}{\partial
n_{\gamma}} =
   \frac{1}{\mu_{\Omega}}\frac{\partial \Psi}{\partial n_{\gamma}}=
\Lambda^{^{^0}} \Psi_{\gamma}.
\end{equation}
Denote by  ${\cal D}{\cal N}$ the  DN-map of  the  operator  $L_0$ on
the  compact
part of the  network. Then  based  on  the representation
(\ref{DNr}) of  the  DN-map of
the  intermediate  operator and  the  formula  (\ref{Smatrix}) we  obtain

\begin{theorem}{\it  The Scattering Matrix on the  network
$\Omega$
   may  be  presented  in terms of  the  Dirichlet-to-Neumann map
   $\Lambda^{^{^0}}$  of  the operator $L_0$ on the compact part $\Omega_0$ as
\begin{equation}
\label{Scatt} S(\lambda)= - \frac{ \left({\cal D}{\cal N}\right)_{++} -
   \left({\cal D}{\cal N}\right)_{+-}
\frac{I}{\left({\cal D}{\cal N}\right)_{--} - K_{-}}\left( {\cal
D}{\cal N} \right)_{-+} + iK_{_{+}}}
{\left({\cal D}{\cal N}\right)_{++} -
   \left({\cal D}{\cal N}\right)_{+-}
\frac{I}{\left({\cal D}{\cal N}\right)_{--} - K_{-}}\left( {\cal
D}{\cal N} \right)_{-+} - iK_{_{+}}}.
\end{equation}
}
\end{theorem}
{\bf Corollary}  Summarizing  the  statements  proved in the last
section we  see  that
   the  step-wise  structure of  the  continuous  spectrum  of  the
quantum network permits us
   to  reduce  the  calculation of  the  Scattering  matrix in a
certain interval of  the
   spectral parameter (and hence the asymptotic description  of  the
corresponding  non-stationary
   processes)   to  the  construction of  the
eigenfunctions  of  the
   discrete  spectrum and the corresponding  eigenvalues  in  the
interval. In particular, one  may
   use  the  variational method  for the construction  both of the 
eigenfunctions
and eigenvalues and
   then obtain  the  Scattering  matrix  based  on one  of the formulae
(\ref{Scatt},\ref{Smatrix}).
   \vskip1cm
   \section{Appendix 1. A solvable  model for the Quantum  Switch}

In the previous section we constructed a representation
(\ref{Sapprox}) for the Scattering
matrix of the Quantum Network near the resonance. It  was  mentioned
there that this formula
coincides  with  the  Scattering Matrix  of a  solvable  model. Our
  aim is  the  construction
of the  corresponding  solvable model  for  the  simplest 
network---the resonance  quantum
switch which consists of a single  vertex  domain and  several  quantum
wires  attached  to  it.
We  assume  that the  quantum  wires are equivalent and
quasi-dimensional, that is  the  exponential modes
on them satisfy the  homogeneous  matrix  Schr\"{o}dinger equation
\[
L \psi = -\psi'' + V \psi = \lambda \psi,
\]
where  $\psi$ is a vector  function which takes values in the space
$E= E_{_1}\oplus E_{_2}\oplus
E_{_3}\oplus\dots $. The potential  $V =$ diag$\left\{
V_{_1},\,V_{_2},\,V_{_3},\dots\right\}$
is a  constant block-diagonal matrix  with blocks
proportional to the projections $P_{_{s}}$ onto the corresponding
subspaces $E_{_{s}}$
of  the  cross-section subspaces  of
the  wires: $V = \sum V_{_{s}}\,\, P_{_{s}}$. In the  simplest  case
of  a  triadic
Resonance Quantum Switch  dim$P_{_{1}} = 4 $. This  corresponds to
one input  wire
and  three  terminals, see for instance \cite{Helsinki2}.
   We  assume  that  the  Fermi-level  $\Lambda$ lies  on the
first spectral band  of  the  Scr\"{o}dinger operator $L$
in the  wires  $\x \in [\left.0,\,\infty \right.)$
with some  self-adjoint  boundary  condition at the  origin,
$V_{_{1}} < \Lambda < V_{_{2}} $. Hence
the  bounded  solutions of  the  homogeneous  equation  may be
presented  as  a  combination of
   oscillating
   exponentials  in the  first  channel  and  decreasing  exponentials
in the  upper  channels:
   \[
\psi (x) = e^{^{-i K_{_{+}}x}}\nu  + e^{^{+i K_{_{+}}x}} S(\lambda)
\nu  + s(\lambda) e^{^{ K_{_{-}}x}}\nu
   \]
   where  $\nu \in E_{_1}$ is  a   cross-section vector in the  open
channels, $S(\lambda)$ is  a
   matrix coefficient  $S : E_{_{+}} \to E_{_{+}} := E_{_{1}}$
  obtained from proper  boundary  conditions (the Scattering
matrix) and  $s(\lambda)$
   is a  matrix  from the  cross-section space  $E_{_{-}} = E_{_{2}}
\oplus E_{_{3}} \oplus\dots$
   of the closed  channels  into  $ E_{_{+}} := E_{_{1}}$. The
exponents  $K_{_{\pm}}$ are equal to
$\sqrt{\lambda - V_{_1}}$ and  diag$_{_{s}}\left\{ - \sqrt{V_{_s} -
\lambda}\right\}$ respectively.
We  assume that the  inner  Hamiltonian  $H$ of the  electron in the  vertex
domain  is  given
   by  a  finite-dimensional  hermitian operator acting in the space $E_{_{H}}$.
   We  restrict both $L$ and  $H$ to  symmetric operators on  the  corresponding
   domains $L\to L_{_0} = L \bigg|_{_{D^{^{L}}_{_{0}}}},\,\, H\to
H_{_{0}} = H \bigg|_{_{D^{^{H}}_{_{0}}}}$.
   The subset $D^{^{H}}_{_{0}} = \frac{I}{H - iI}\,\,[E_{_{H}} \ominus 
N_{_{i}}]$
   of $H_{_{0}}$, defined  by
   the  deficiency  subspace  $N_{_{i}}$,
   is  not  dense \footnote{Concerning the definition of  the  formal adjoint
   and  extension of  non-densely defined  Hermitian operators see
\cite{KP00} and  references therein.}.
    Nevertheless, the  extension procedure  for the  orthogonal sum
   $L_{_0}\oplus H_{_{0}}$ may be achieved, using the  symplectic formalism,
by the selection  of a Lagrangian
    plane in the sum of the corresponding  boundary  forms, see for
instance \cite{Extensions,KP00}.
The  boundary  form  is  obtained for  the  operator $L_{_{0}}^{^{+}}$ via
conventional integration by parts:
    \[
{\cal J}_{_0} (\psi,\, \varphi) =  \langle L_{_{0}}^{^{+}}
\psi,\,\varphi\rangle - \langle \psi ,\,
L_{_{0}}^{^{+}} \varphi\rangle =
\]
\[
   \langle \psi',\, \varphi\rangle  -
\langle \varphi',\, \psi\rangle
    \]
where  $\psi = \psi (0)$ and  the  derivative  $\psi'$
   at  the  node $x = 0$ is  taken in the outgoing direction.
The  corresponding boundary  form for the  inner, formally  adjoint,
   operator  $ H_{_{0}}^{^{+}}$ may be  obtained
via  abstract  integration by parts, see  \cite{KP00}:
\[
{\cal J}_{_1} (\psi,\, \varphi) = \langle
\xi_{_{+}}^{^{\psi}},\,\xi_{_{+}}^{^{\varphi}}\rangle -
\langle \xi_{_{-}}^{^{\psi}},\,\xi_{_{+}}^{^{\varphi}}\rangle.
\]
The sum of boundary forms ${\cal J}_{_0} (\psi,\, \varphi) + {\cal
J}_{_1} (\psi,\, \varphi) $
vanishes on the Lagrangian plane which  may be  defined by the
boundary condition
\begin{equation}
\label{bcondsw}
\left(
\begin{array}{c}
\psi'\\
\xi_{_{+}}^{^{\psi}}
\end{array}
\right)
=
\left(
\begin{array}{cc}
0 & \beta_{_{01}}\\
\beta_{_{10}}& 0
\end{array}
\right)
\,\,
\left(
\begin{array}{c}
\psi\\
\xi_{_{-}}^{^{\psi}}
\end{array}
\right).
\end{equation}
Using the  connection between the  boundary  data  $\xi_{_{\pm}}$ of
the  inner  component of the
wave  function of the  switch $ \xi_{_{-}} = - {\cal M}\, \xi_{_{+}}$,
calculated using  Krein's  Q-function  ${\cal M} = P_{_{N_{i}}}\frac{I +
\lambda H}{H - \lambda I}P_{_{N_{i}}}$,
we  may eliminate the  inner
component of the  wave  function
\[
\xi_{_{+}}^{^{\psi}} = \beta_{_{10}} \left[ (I + S) \nu + s \nu\right]
\]
and  obtain an equation  for $S,\, s$ with any $\nu \in E_{_{+}}$:
\[
- i K_{_{+}} (I - S) \nu  +  K_{_{-}} s \nu =  - \beta_{_{01}} {\cal
M} \beta{_{10}}.
\]
Projecting this equation onto  $E_{_{\pm}}$ respectively  and
denoting  by  ${\cal D}{\cal N}^{^{H}}$
the  analog  $\beta_{_{01}} {\cal M} \beta_{_{10}}$  of the
Dirichlet-to-Neumann map
of the  inner  operator  and  by  ${\cal D}{\cal N}^{^{H}}_{_{\pm}}$
the same operator-function
bordered  by  the  projections  $P_{_{\pm}}$ onto the open and
closed channels, we  obtain a  pair
of  equations from which the  Scattering  matrix $S$ may be  found  as:
\begin{equation}
\label{smatrixsw}
S(\lambda) = - \frac{ {\cal D}{\cal N}^{^{H}}_{_{++}} - {\cal D}{\cal
N}^{^{H}}_{_{+-}}
\frac{I}{{\cal D}{\cal N}^{^{H}}_{_{--}} - K_{_{-}}} {\cal D}{\cal
N}^{^{H}}_{_{-+}} + i K_{_{+}}}
{{\cal D}{\cal N}^{^{H}}_{_{++}} - {\cal D}{\cal N}^{^{H}}_{_{+-}}
\frac{I}{{\cal D}{\cal N}^{^{H}}_{_{--}} - K_{_{-}}} {\cal D}{\cal
N}^{^{H}}_{_{-+}} - i K_{_{+}}}.
\end{equation}
Here the  denominator precedes the numerator. The  structure of  the
above expression  (\ref{smatrixsw})
is  equivalent to  the  structure of  the  expressions
(\ref{Smatrix},\ref{Sapprox},\ref{Scatt}) where
an important  role  is  played by  the  functional parameter  ${\cal
D}{\cal N}_{_{\Lambda}}$---the
   Dirichlet-to-Neumann map of the  intermediate operator. One may also
introduce the intermediate operator
   in the solvable model considered above and show that the
observed coincidence is not arbitrary.
   \par
Consider  the  operator  $L_{_{0}}^{^{+}} \oplus H_{_{0}}^{^{+}} $
subjected to  the  extended
   boundary condition:
   \begin{equation}
\label{bcondex}
\left(
\begin{array}{c}
\psi'\\
\xi_{_{+}}^{^{\psi}}
\end{array}
\right)
=
\left(
\begin{array}{cc}
0 & \beta_{_{01}}\\
\beta_{_{10}}& 0
\end{array}
\right)
\,\,
\left(
\begin{array}{c}
\psi\\
\xi_{_{-}}^{^{\psi}}
\end{array}
\right),\,\, P_{_{+}}\psi(0) = 0 .
\end{equation}
Here the additional  assumption $P_{_{+}}\psi(0) = 0$ is of the vanishing  of
the  wave-function at  the
origin  in the  first (open)  channel. It  is  obvious  that
the  component  of  this  operator  ${\cal L}_{_{\beta}}$ in the 
first  channel
coincides  with
   the  self-adjoint  operator in $L_2(R_+,E_{_{1}})$,
$ -\psi_{_{1}}'' + V_{_{1}}\psi : = l_{_{1}}$,  with zero boundary
condition at  the  origin
and may be orthogonally split from ${\cal L}_{_{\beta}}$:
${\cal L}_{_{\beta}} \ominus l_{_{1}} = {L}_{_{\beta}} $.
\begin{theorem}
{\it The  operator ${L}_{_{\beta}}$ is  self-adjoint  in  $L_2
(E_{_{2}}\oplus E_{_{3}}\dots,\, R_{_{+}} )
\oplus E_{_{H}}$ and  the continuous  spectrum fills  the interval
   $\left[ \left. V_{_{2}},\, \infty \right.\right) $ with
variable multiplicity  defined
   for  given $\lambda \in  (V_{_{t}},\, V_{_{t+1}})$ by
    the dimension  of  the  space  $\sum_{_{s< t+1,\,s\neq 1}} $ dim
${E_s}$. The   discrete  spectrum
    of  the  operator  ${L}_{_{\beta}}$ is  defined  by  the
dispersion  equation
    \[
\psi' (0,\lambda) + P_{_{-}}\,\beta_{_{01}} {\cal
M}\beta_{_{10}}\,P_{_{-}}\psi (0,\lambda) = 0
    \]
with  $\psi' (0,\lambda) = K_{_{\lambda}} \psi (0,\lambda)$ to
match with  exponentially
decreasing solutions in the  wires  on the  first spectral band. The
DN-map  of  the  operator
${L}_{_{\beta}}$  for  the  boundary  data  from  the  subspace
$E_{_{1}}$ at  the  origin
coincides  with  the  expression  in the  numerator and  the
denominator  of  the  Scattering  Matrix
   (\ref{smatrixsw})
   \[
{\cal D}{\cal N}^{^{\Lambda}}_{_{L_{_{\beta}}}} =
    {\cal D}{\cal N}^{^{H}}_{_{++}} - {\cal D}{\cal N}^{^{H}}_{_{+-}}
\frac{I}{{\cal D}{\cal N}^{^{H}}_{_{--}} - K_{_{-}}} {\cal D}{\cal
N}^{^{H}}_{_{-+}}.
\]
Consequently, the  Scattering  matrix  (\ref{smatrixsw}) can be  represented as
\[
S^{^{\beta}} (\lambda) = -\frac{{\cal D}{\cal
N}^{^{\Lambda}}_{_{L_{_{\beta}}}}  + i K_{_{+}}}
{{\cal D}{\cal N}^{^{\Lambda}}_{_{L_{_{\beta}}}}  - i K_{_{+}}}
\]
where the  denominator precedes the  numerator.
}
\end{theorem}
{\it Proof} can be  obtained via  straightforward  calculation of
the solution of  the  Dirichlet
problem for  the  corresponding  homogeneous  equation.
\par
It  follows from the above theorem that choosing the  operator  $H$,
its  deficiency  subspace $N_{_{i}}$
and  the  boundary  parameters---the  matrices $\beta$---such that
the  DN-map of  the  model operator
$L_{_{\beta}}$ coincides with the  DN-map  of  the  intermediate
operator of  the  network we  obtain
   the  solvable  model  corrresponding to the  network. In 
particular, the  poles  of  the
DN-map of the  network are  sitting
at the  eigenvalues $\lambda_{_{s}}$ of  the  intermediate  operator
so  should  be the  eigenvalues
   of  the  model. The corresponding  residues are  related by
   \[
\left[1 + \lambda_{_{s}}^{^{2}}\right] \,\, P_{_{1}}\beta_{_{01}}e_s\rangle\,
\langle P_{_{1}}\beta_{_{01}}e_s  =
   P_{_{1}} \frac{\partial \varphi_{_{s}}}{\partial n}\rangle\,\langle
   P_{_{1}} \frac{\partial \varphi_{_{s}}}{\partial n}.
   \]
  This  formula gives  necessary data to  select  proper boundary
parameters $\beta_{_{01}}$
   once the eigenfunctions of the  intermediate operator on the
network are known.
  This  will be  discussed in a following  publication.
\vskip1cm
\section{Appendix 2. Intermediate operator techniques for  the
Friedrichs model}

\noindent
The spectral structure of  the classical model of Friedrichs with  simple
absolutely continuous spectrum
   is  different from  the  step-wise spectral
structure of the  Schr\"{o}dinger operator on  a  quantum network.
   Nevertheless,  it also  allows us to construct an intermediate
operator and  ``one-pole approximation" of  the  corresponding
Scattering  matrix. The one-pole
approximation  corresponds
   to the separation of  the resonance  factor near the resonance  energy.
We  suggest here an
analytically  convenient version of the classical Friedrichs model
which is   obtained as
a one-dimensional perturbation of the orthogonal
sum of  the  momentum  $P = \frac{1}{i}\,\, \frac{d}{dx}$ in $L_2
(R)$ and a finite-dimensional
self-adjoint operator $A: E \to E$,\,dim$E = N < \infty$. Restriction
$P \to P_{_{0}} $ of the
momentum onto the domain $D_0 = W_2^{1,0} (R)$ of all  functions
vanishing at the origin is
a symmetric operator with deficiency indices $(1,1)$.The
corresponding adjoint  operator $P_0^+$
is defined on $W_2^{1} (R_-) \oplus W_2^{1} (R_+)$ without any
boundary condition at  the  origin.
The boundary  form, see for  instance \cite{Extensions,Gorbachuk},
   of  the adjoint  operator $J(u,v) = \langle P_0^+ u,\,v\rangle -
\langle u,\, P_0^+ v\rangle = i\left[ u \bar{v} (0^+) - u \bar{v}
(0^-)\right]$ my be written
in terms of  symplectic variables $\xi_{\pm}$:
\[
\xi^u_+ = \frac{u(0^+) + u(0^-)}{2},\,\, \xi^u_- = \frac{u(0^+) - u(0^-)}{2i}
\]
as
\begin{equation}
\label{bf_p}
J_p (u,v) = 2 \left[ \xi^u_+ \bar{\xi^v_-} - \xi^u_- \bar{\xi^v_+} \right].
\end{equation}
For the restricted operator  $A \to A_0 = A\bigg|_{_{D_0}}$
   with  deficiency vectors ${\bf e_i} = {\bf e},\, {\bf e_{-i}} =
\frac{A+iI}{A-iI} {\bf e}$ the boundary form of the adjoint
operator on  elements
${\bf u},\, {\bf v} \in D(A^+_0)$
\[
{\bf u} = {\bf u}_0 + \eta^u_+ \,\frac{{\bf e_i} + {\bf e_{-i}}}{2} +
\eta^u_- \,\frac{{\bf e_{i}} - {\bf e_{-i}}}{2i},\, {\bf u}_0 \in D_0
\]
   may be  presented, see  for  instance \cite{Extensions}, in terms of
   the corresponding  symplectic  coordinates  $\eta_{\pm}$ as
\begin{equation}
\label{bf_A}
J_A (u,v) = 2 \left[ \eta^u_+ \bar{\eta}^v_- - \eta^u_- \bar{\eta}^v_+ \right].
\end{equation}
One may  consider the  orthogonal sum $P_0 \oplus A_0$ of  restricted
operators and
construct a Lagrange plane parametrized by a $2\times 2$ hermitian
matrix $\Gamma$
\[
B= \left(
\begin{array}{cc}
\beta_{00} & \beta_{01}\\
\beta_{10} & \beta_{11}
\end{array}
\right)
\]
\begin{equation}
\label{Lagplane}
\left(
\begin{array}{c}
\xi_-\\
\eta_-
\end{array}
\right) =
   B \left(
\begin{array}{c}
\xi_+\\
\eta_+
\end{array}
\right),
\end{equation}
where the joint boundary form  $J_p(u,v)+ J_A ({\bf u},\, {\bf v})$
vanishes. This Lagrange plane
defines a   self-adjoint extension ${\bf P}_{_{\beta}}$ of  $P_0
\oplus A_0$. The  components
of the eigenfunctions of the combined extended  operator  ${\bf P}_{_{\beta}}$
   in the sum of the inner space  $E$ and the outer space
$L_2 (R)$ are solutions of the adjoint homogeneous equations. The
symplectic coordinates
$\eta_{\pm}$ of  the  corresponding  solution are  connected  via the 
analog of the
Weyl function ${\cal M} (\lambda)= \langle
\frac{I + \lambda A}{A - \lambda I}{\bf e},\,{\bf e} \rangle$, see for
instance \cite{DN01}:
\[
   \eta_{_{-}} = - {\cal M} (\lambda)\eta_+.
\]
Then, presenting the outer component of the eigenfunction by the ansatz
\[
\Psi(x)= e^{i\lambda x}\,\, \mbox{for}\,\, x<0,\,\,
\Psi(x)=S(\lambda) e^{i\lambda x}\,\, \mbox{for}\,\, x>0 ,
\]
we can  rewrite the above boundary condition (\ref{Lagplane}) as
\[
\left(
\begin{array}{c}
\frac{I-S}{2i}\\
- {\cal M}\eta_{_+}
\end{array}
\right) =
   \left(
\begin{array}{cc}
\beta_{00} & \beta_{01}\\
\beta_{10} & \beta_{11}
\end{array}
\right)
   \left(
\begin{array}{c}
\frac{I+S}{2}\\
\eta_+
\end{array}
\right).
\]
Hence  $\xi_{_{+}} = - \left( \beta_{_{11}} + {\cal
M}\right)^{^{-1}}\,\,\,\beta_{_{10}}\frac{I+S}{2}$
and $\frac{I-S}{2i} =
\beta_{_{00}}\, \frac{I+S}{2} -
\frac{|\beta_{_{00}}|^2}{\beta_{_{11}} + {\cal M}}
\,\,\frac{I-S}{2i}$.
This  implies the  following expression for  the  Scattering matrix $S$:
\begin{equation}
\label{S}
S(\lambda) = - \frac{\left[\beta_{00}-\beta_{01}\frac{1}{{\cal M}
+\beta_{11}}\beta_{10}\right] +i}
{\left[\beta_{00}-\beta_{01}\frac{1}{{\cal M}
+\beta_{11}}\beta_{10}\right] - i}.
\end{equation}
If $B = 0$ then $S=1$ which corresponds to the unperturbed
operator. But {\it it is
impossible} to construct an analytic branch of the eigenfunctions of the
perturbed operator with
$B\neq 0$ that coincides with  the  eigenfunction $e^{i\lambda x}$ of
the unperturbed operator
at $B = 0$. Nevertheless, one may observe the singularities of  the
solutions---the resonances.
\par
Consider the  extended operator ${\bf P}_{_{\epsilon}}$  defined  by
the above boundary
condition with small off diagonal elements
\[
B_{\varepsilon}= \left(
\begin{array}{cc}
0 & {\varepsilon}\\
{\varepsilon} & 0
\end{array}
\right)
\]
\begin{equation}
\label{Lagp}
\left(
\begin{array}{c}
\xi_-\\
\eta_-
\end{array}
\right) =
   B_{\varepsilon} \left(
\begin{array}{c}
\xi_+\\
\eta_+
\end{array}
\right).
\end{equation}
Then we obtain, according to  {\ref{S}}
\begin{equation}
\label{SFried}
S_{\varepsilon}(\lambda) = - \frac{i - \frac{\varepsilon^2}{M(\lambda)}}
{i + \frac{\varepsilon^2}{M(\lambda)}}.
\end{equation}
For  small values of  $\varepsilon$ the zeroes and  poles  of the
Scattering matrix $S_{\varepsilon}$
and the complex  poles  of the Scattered waves
\begin{equation}
\label{SwaveF}
\psi_{_{\varepsilon}} =
\left\{
\begin{array}{cc}
e^{^{i\lambda x}} & \mbox{if} \,\,\, x\in \, (-\infty,\,0)\\
S_{_{\varepsilon}}(\lambda)\,\,\, e^{^{i\lambda x}} \,\,\, &
\mbox{if}\,\,\, x \in \, (0\,,\,\,\infty),
\end{array}
\right.
\end{equation}
sit near the zeroes $\alpha_s$ of  the  Weyl-function $M(\lambda)$
where  ${\cal M}(\lambda) + i\varepsilon^{^2} = 0$.
Since the operator  $A$ is  self-adjoint the  Weyl-function  is from
the Nevanlinna class,
$\Im M (\lambda)\,\,\, \Im \lambda > 0$ and  $M'(\lambda) > 0$ on
the  real  axis $\lambda$. Then for
small  $\varepsilon $ the  scattered  matrix  is  almost  trivial
everywhere on real  axis  of
the  spectral parameter  $\lambda$. However,  near zeros $\alpha_{_s}$ of
the  Weyl-function  it may be
approximated  by  functions
\[
S (\lambda) \approx S_{_s}(\lambda) = \frac{ (\lambda - \alpha_{_s}) +
   i \frac{\varepsilon^2}{M'(\alpha_{_s})}}
   {(\lambda - \alpha_{_s}) - i
\frac{\varepsilon^2}{M'(\alpha_{_s})}}\,\,\,\,\, \theta(\alpha_{_s})
\]
where $\,\, |\theta (\alpha_{_s})|=1$. The last  formula for the
Scattering matrix
   may be  interpreted as a ``one-pole  approximation''  near  $\alpha_s$.
   The  resonance may be  found  as the
   only zero of the Scattering matrix  which is situated  according to the
Rouche theorem  near
   $\alpha_s $. It can be calculated approximately as
   \[
\lambda^{^{\varepsilon}}_{_s} \approx \alpha_{_0}  -  i
\frac{\varepsilon^2}{M'(\alpha_{_s})}.
   \]
    The  complex-conjugate  point  $\bar{\lambda}^{^{\varepsilon}}_s$
in the upper half-plane is
a simple pole of the Scattering matrix\footnote{In the physical
literature the  poles of the
Scattering matrix  are  called resonances. On the other hand we  follow
here the terminology of \cite{Lax}
where the  resonances  are  associated  with  zeroes of  the
Scattering matrix.}. In  our  case
dim$E = N < \infty$    the Scattering
matrix and the  functions $\cal M$ are  rational  functions,
and  the  Scattering  matrix  may be  decomposed  into  a  product
\[
S_{_{\varepsilon}}(\lambda) = C
   \prod_{_{s=1}}^{^{N}}
\frac{\lambda - \lambda^{^{\varepsilon}}_{_{s}}}{\lambda
-\bar{\lambda}^{^{\varepsilon}}_{_{s}}}.
\]
The  coefficient  $C$ may  be  found  from the  asymptotic of  the
Weyl-function ${\cal M}$   at
    infinity:
    \[
    {\cal M}(\lambda) \longrightarrow - \langle A e,e \rangle
\,\,\,\,\mbox{when} \,\,\lambda \to \infty,
    \]
    \[
C =
   -\frac{ \varepsilon^2 + i \,\,\langle A{\bf e},\, {\bf e}\rangle}
   {\varepsilon^2 - i \,\,\langle A{\bf e},\, {\bf e}\rangle}\,\,\, .
    \]
In particular the Scattering  matrix has physically reasonable
behavior at infinity  $S(\lambda)\to 1$
for  $\varepsilon \to 0$, if  $\langle A{\bf e},\, {\bf e}\rangle \neq 0$. If
$\langle A{\bf e},\, {\bf e}\rangle = 0$ then  for  each
$\varepsilon$ the asymptotic of
the Scattering matrix is  $-1$ which is not typical for real physical systems.
\par
One may guess that the one-pole  approximation  is the Scattering matrix
of  a  model  quantum  system with  Hamiltonian  ${\bf
P}^{^{1}}_{_{\varepsilon}}$  which
reveals typical features of the resonance behavior
of the  original  system at  the given resonance energy $\alpha_{_1}$.
At the  end  of  this  section we will construct an intermediate
model quantum system  such
that the Scattering matrix between the non-perturbed
operator ${\bf P}$ and  the constructed  model  ${\bf
P}^{^{1}}_{_{\varepsilon}}$ coincides with
the  resonance  factor up to  a  constant modulo $1$:
\[
   S^{^1}_{_{\varepsilon}}(\lambda) = \frac{\lambda -
\lambda^{^{\varepsilon}}_{_1} }
   {\lambda - \bar{\lambda}^{^{\varepsilon}}_{_1} }  .
\]
Based on this  construction  we  may consider
two  successive  Scattering processes  $P \longrightarrow {\bf
P}^{^{1}}_{_{\varepsilon}}
\longrightarrow {\bf P}_{_{\varepsilon}}$. The  transformation of
the  Scattered  waves in these
two processes  is  confined to  the multiplication  of  the  second
component  by  the
factor $S^{^1}_{_{\varepsilon}}(\lambda)$  at the  first  step, and to
multiplication  by  the  complementary  factor
\[
S^{^{\varepsilon}}_{_{{1}}} := -
\frac{ \varepsilon^2 + i \,\,\langle A{\bf e},\, {\bf e}\rangle}
   {\varepsilon^2 - i \,\,\langle A{\bf e},\, {\bf e}\rangle}\,\,\,
\prod_{_{s\neq 1}}
\frac{\lambda - \lambda^{^{\varepsilon}}_{_{s}}}{\lambda
-\bar{\lambda}^{^{\varepsilon}}_{_{s}}}
\]
at the  second step:
\[
\left(
\begin{array}{cc}
e^{^{i\lambda x}} & \mbox{if} \,\,\, x\in \, (-\infty,\,0),\\
e^{^{i\lambda x}} \,\, & \mbox{if}\,\,\, x \in \, (0\,,\,\,\infty)
\end{array}
\right) \longrightarrow
\left(
\begin{array}{cc}
e^{^{i\lambda x}} & \mbox{if} \,\,\, x\in \, (-\infty,\,0),\\
S^{^{1}}_{_{\varepsilon}}(\lambda) e^{^{i\lambda x}} \,\, &
\mbox{if}\,\,\, x \in \, (0\,,\,\,\infty)
\end{array}
\right)
\longrightarrow
\]
\begin{equation}
\label{Swaves}
\left(
\begin{array}{cc}
e^{^{i\lambda x}} & \mbox{if} \,\,\, x\in \, (-\infty,\,0),\\
S^{^{\varepsilon}}(\lambda) e^{^{i\lambda x}} \,\, & \mbox{if}\,\,\,
x \in \, (0\,,\,\,\infty)
\end{array}
\right).
\end{equation}
The  complementary  factor  is the
   scattering matrix  between the  intermediate  operator ${\bf
P}^{^{1}}_{_{\epsilon}}$
    and  the perturbed  operator
   ${\bf P}_{_{\epsilon}}$. This  factor  is an analytic function
   with respect to  $\varepsilon, \lambda$ for small $\varepsilon$
in a  neighborhood of  the  resonance   value  $\alpha_{_0}$ of  the
spectral parameter $\lambda$.
This  construction may be  continued to  an extended  domain
containing  two
   zeroes  of the  Weyl  function  by separating  two  resonance
factors and  so  on.
\begin{theorem}{\it Consider  the  set $\left\{
\lambda^{^{\varepsilon}}_{_{s}},
   s= 1,2,\dots N \right\}$ of  all  zeroes of
the  denominator of  the  Scattering  matrix (\ref{SFried}) with a
finite-dimensional
self-adjoint operator  $A$:
\[
1 - i \frac{\varepsilon^2}{{\cal M}(\lambda_s)} = 0.
\]
If  all eigenvalues $\alpha_s$ of the operator $A$ are simple then, for  small
values of the  perturbation parameter $\varepsilon$, near each
eigenvalue $\alpha_s$ there
exists exactly  one  zero $\lambda^{^{\varepsilon}}_{_{s}}$ of
   the  denominator---the corresponding simple pole of the  Scattering 
matrix---and  the
scattering matrix may be written as a product of  one-pole terms:
\[
S(\lambda)=
   \frac{{\cal M}(\lambda)+ i\varepsilon^2}{{\cal M}(\lambda)- i\varepsilon^2}=
   \frac{\langle A{\bf e},\, {\bf e}\rangle - i\varepsilon^2}
   {\langle A{\bf e},\, {\bf e}\rangle + i\varepsilon^2}\,\,\,
   \prod_{_{s=1}}^{^{dim A}}
\frac{\lambda - \bar{\lambda^{^{\varepsilon}}_{_{s}}}}{\lambda -
\lambda^{^{\varepsilon}}_{_{s}}} .
\]
Each resonance term is the Scattering  matrix of a model system which
may be constructed
via the operator-extension procedure.
}
\end{theorem}

{\it Proof} Only the last  statement of  the  theorem needs an
additional proof. Recall first
that the space $E$ is  finite-dimensional. Consequently, the 
Weyl-function $M (\lambda)$
is  a
rational function of  Nevanlinna  class, $\Im {\cal M} (\lambda)
\,\,\Im \lambda >0$.
The  derivative of  it  at  every real zero $\alpha= \alpha_0$
is  positive, $M' (\alpha) > 0$. Hence there  exists an  analytic branch
$\lambda_{_{\alpha}} (\varepsilon)$ of the  function $\lambda =
\lambda_{_{\alpha}} (\varepsilon)$
defined  by  the  equation ${\cal M}(\lambda) + i \varepsilon^2 = 0$
and  the  initial
condition  $\lambda (0) = \alpha $. This  function
takes values in the lower  half-plane $\Im \lambda < 0$ and  fulfills
the initial condition $\lambda(0) = \alpha$. Consider  now  a 
one-dimensional operator
$A'$ with an eigenvalue  $\alpha'$.
We  will specify   the analytic function  $\alpha' (\varepsilon)$ and choose
the parameters  in the boundary conditions (\ref{Lagplane})
connecting  the the inner  and  outer
spaces  the  following way:
\[
\beta_{11} = \alpha',\,\, \beta_{00} = 0,\,,\, \alpha' = \Re \lambda
(\varepsilon),\,\,
   |\beta_{01}|^2 = \frac{1 + (\alpha')^{^ 2}}{\Im \bar{\lambda} (\alpha)},
\]
with  $\alpha' (\varepsilon)$ approaching  $\alpha$ when $\varepsilon \to 0$,
and $\beta_{01} (\varepsilon)$ approaching  infinity. The  model Hamiltonian
   ${\bf P}^{^{\alpha}}_{_{\varepsilon}}$ defined by
the operator $A'(\varepsilon)$  has  the  following  scattering
matrix  with  respect  to  the
unperturbed  Hamiltonian (the momentum):
\[
S{{1}}_{_{\varepsilon}} =
-\frac{-\frac{|\beta_{01}|^2}{\frac{1 + \alpha'
\lambda}{\alpha'-\lambda} + \alpha'} + i}
{-\frac{|\beta_{01}|^2}{\frac{1 + \alpha' \lambda}{\alpha'-\lambda} +
\alpha'} - i} =
-\frac{\lambda-\lambda_{_{\alpha}}(\varepsilon)}{\lambda-\bar{\lambda_{_{\alpha}}(\varepsilon)}},
\]
which coincides with the  resonance factor.
It is not an analytic function of $\varepsilon$ near the  origin
$\varepsilon = 0$.
Then the  corresponding  scattered  waves  are given by
\begin{equation}
\label{Swalpha}
\psi_{_{\alpha}}(\varepsilon) =
\left\{
\begin{array}{cc}
e^{^{i\lambda x}} & \mbox{if} \,\,\, x\in \, (-\infty,\,0)\\
S_{_{\alpha}}(\varepsilon)(\lambda)\,\,\, e^{^{i\lambda x}}
\,\,\, & \mbox{if}\,\,\, x \in \, (0\,,\,\,\infty),
\end{array}
\right. .
\end{equation}
The scattered  waves are not analytic in $\varepsilon$ either. 
However, the complementary factor
$- C \prod_{_{s\neq s_0}} \frac{\lambda - \lambda_s}{\lambda -
\bar{\lambda_s}}$ of the complete
Scattering matrix $S_{\varepsilon}$ is an analytic function in any
neighborhood of $\alpha=\alpha_0$
which does not contain other zeroes $\alpha_s$ of the Weyl-function.

$\Box$

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