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\begin{document}
\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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\end{document}