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\begin{document}
   \title
   { Symplectic operator-extension techniques and \\
    zero-range quantum models }

\author
{Boris S. Pavlov} \email {pavlov@math.auckland.ac.nz} \affiliation
{Department of Mathematics, University of Auckland, Private Bag
92019, Auckland, New Zealand}


\author
  {Vladimir I.Kruglov}
 \email
{vik@phy.auckland.ac.nz}
 \affiliation
{Department of Physics, The University of Auckland, Private Bag
92019, Auckland, New Zealand}


       \vskip 1truecm
            % \date{}

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  \begin{abstract}
\begin{small}
F. Berezin and L. Faddeev  interpreted Fermi zero-range model as a
self-adjoint extension of the Laplacian. Various modifications of
this model in conventional Hilbert space possess rich spectral
properties, but unavoidably have the negative  effective radius
and contain numerous parameters which do not have a direct
physical meaning. We suggest, for spherically-symmetric
scattering, a generalization of the Fermi zero-range model
supplied  with an indefinite metric in the inner space and a
Hamiltonian of the inner degrees of freedom. Effective radius of
this model may be both positive or negative. We propose also a
general {\it principle of analyticity} formulated  in terms  of
$k{\rm cot}\delta (k)$ as a function  of the scattering phase
shift $\delta(k)$ depending on the wave-number $k$. This principle
allows us to evaluate all parameters of the  model, including the
indefinite metric tensor of the inner space, once the  basic
parameters of the  model: the spectrum $\sigma_{_{p}}$ of the
inner Hamiltonian, the scattering length and the effective radius,
are fixed, such that the sign of the effective radius is connected
with the spectrum $\sigma_{_{p}}$  by an appropriate consistency
condition. The absolutely-continuous  part  of  the extension
plays a role of the quantum Hamiltonian of the special model.
\end{small}
\end{abstract}

\pacs {~03.65.Nk,~ 82.20.Fd,~ 28.20.Cz} \keywords {zero-range
potential}

\maketitle

       \vskip10.1cm
             \section{Introduction}
In \cite{BF61} a realistic description of quantum scattering by  a
point-wise object was interpreted, based on \cite{Fermi}, in terms
of J. von Neumann operator extensions. From the point of view
developed in \cite{BF61,Albeverio}, the solvable zero-range model
suggested by Fermi in \cite{Fermi}, is actually a self-adjoint
extension, see \cite{Neumann}, of the Laplace operator defined on
smooth functions which have a singularity at the origin
\begin{equation}
\Psi (r) = \frac{{\cal A}^{\psi}}{4\pi r} + {\cal B}^{\psi} +o(1),
\label{i6}
\end{equation}
with the asymptotic boundary values ${\cal A}$ and ${\cal B}$
submitted to the real {\it boundary condition}, $\gamma =
\bar{\gamma}$
\begin{equation}
\gamma {\cal A} + {\cal B}=0. \label{i8}
\end{equation}
This  operator has only trivial spectral structure (one negative
eigenvalue $-\kappa^{^2}$, for $\kappa = 4\pi \gamma > 0$ or one
resonance $\kappa' = i 4\pi \gamma $ for $\gamma < 0$). In attempt
to extend the construction to solvable models with rich spectrum,
Wigner suggested in \cite{Wigner51} to calculate scattered waves
for the Schr\"{o}dinger operator with compactly supported
potentials $V({\bf r})$, ($V({\bf r})=0$ for $r= |{\bf
r}|>r_{_0}$) imposing an appropriate energy-dependent boundary
condition at $r=r_{_0}$. From the modern point of  view, the
suggestion of Wigner is equivalent to matching the scattering
Ansatz in the outer space $r>r_{_0}$ with solutions of the
Schr\"{o}dinger equation on the inner space $r<r_{_0}$ based on
Dirichlet-to-Neumann map (DN-map) $\Lambda$ of the Schr\"{o}dinger
operator on the inner space, see \cite{SU2}. In particular, for
the spherically-symmetric potential $V$, when using the
spherically symmetric Ansatz for the scattered waves (in
``s-channel''):
\begin{equation}
\psi(r,k)=\frac{C}{r}{\rm sin}(kr+\delta (k)) := \frac{C}{r} \chi
(r, k), \,r>r_{_{0}} \label{i12}
\end{equation}
in the  outer space, one can present the  matching  condition in
form
\begin{equation}
\left(\frac{\partial}{\partial
r}\chi(r,k)+\alpha(k)\chi(r,k)\right)\bigg|_{_{r=r_{0}}} = 0.
\label{i11}
\end{equation}
Here $\alpha$ is the corresponding  Weyl-Titchmarsh function, see
\cite{Titchmarsh} - the  component of  DN-map in s-channel.
Unfortunately, this condition contains the full spectral data (
encoded in $\alpha(k)$ ) of the inner problem, thus reducing the
scattering problem to the inner spectral problem. Nevertheless the
suggestion of Wigner inspired numerous attempts to construct
energy-dependent potentials characterized by the energy-dependent
boundary conditions. In \cite{Zero_range} the ``zero-range
potential with inner structure" was suggested based on
replacement, for spherically-symmetric scattering, the
Schr\"{o}dinger operator in the inner space by  a finite matrix
which plays a  role of the Hamiltonian of inner degrees of
freedom. Then the role of the component $\alpha $ of the DN-map is
played by the analog  $F (k)$ of the corresponding Krein-function,
see \cite{K}. The operator extension procedure suggested  in
\cite{Zero_range} is equivalent to the ``asymptotic boundary
condition'' at the origin :
\begin{equation}
\left(\frac {1}{\chi({\bf r},k)}\frac{\partial}{\partial
r}\chi({\bf r},k)\right)\bigg|_{_{r=0}} = k{\rm cot}\delta(k) : =
F(k) \label{i15}.
\end{equation}
Here the  function $F (k)$ is defined by the spectral structure of
the Hamiltonian of the inner degrees of freedom and by the
boundary parameters of the model, see \cite{Zero_range}. The
operator extensions approach allows  to construct simple solvable
models for few - channels and few - body scattering systems with
interesting resonance properties, without solving sophisticate
boundary problems, see \cite{Extensions,Novikov,Exner,Ad_Pav} and
more references in \cite{AK00}. The behavior of the function
$F(k)$ which corresponds to  the zero-range model
\cite{Zero_range} looks similar to  the  low-energy behavior of
the corresponding function in s-channel for conventional rapidly
decreasing potentials, see \cite{Landau}:
\begin{equation}
F(k) = k{\rm cot}\delta(k) = -\frac{1}{a}+\frac{r_{0}}{2}k^{2}
\dots. \label{i10}
\end{equation}
The  parameters $r_{0},\, a$ in (\ref{i10}) are called,
respectively, the {\it effective radius} of the scatterer  and the
scattering length, see \cite{Landau}. They  are measured in
experiment. Unfortunately the zero-range model  \cite{Zero_range}
with inner structure and {\it positive metric in the inner  space}
\vskip0.3cm a. unavoidably has a negative effective radius,
\cite{MMM95}, because in that case $- F $ is a Nevanlinna class
function of $\lambda = k^{^2}$, and \vskip0.3cm b. It contains
numerous ``extra parameters'' which {\it have no straightforward
physical interpretation} and hence are not {\it fitting
parameters}. These are: the deficiency vector in the inner space
and the boundary parameters.  \vskip0.3cm  We propose a``special
zero-range model'' which does not have  above deficiencies a,b.
Our model is supplied with an {\it indefinite metric on the inner
space}, and  the  corresponding  function $F$ is an entire
function of  $\lambda = k^{^{2}}$:
\begin{equation}
F (k) = k \cot \delta (k)= \sum_{l=0}^{\infty}g_{l}k^{2l}.
\label{i17}
\end{equation}
In case of finite-dimensional ``inner space''  dim$E = N $ the
function $F (k)$ submitted to this {\it principle of analyticity}
is just a polynomial degree $2N$. The condition (\ref{i17})
enables us to evaluate all ``extra parameters'' of the special
zero-range model with inner structure, if the {\it fitting data} :
the scattering length, the effective radius and the spectrum of
the Hamiltonian of the inner degrees of freedom are connected by
the appropriate consistency condition. The standard self-adjoint
quantum Hamiltonian of the model is obtained via restriction of
the model onto the absolutely continuous subspace which has the
positive metric. More physical discussion and fitting of the model
to experimental data can be found in \cite{JPA}.
  \vskip0.1cm
\section{Preliminaries: symplectic extension procedure}

\noindent Zero-range  model with inner  structure can be
constructed via operator  extension procedure applied to an
orthogonal sum of a restricted bounded inner Hamiltonian and the
restricted Laplacian in $L_{_{2}}(R_{_{3}})$. But the classical
version of the operator-extension procedure, \cite{Neumann}, is
inconvenient for differential operators. The extension procedure
for them is usually reduced  to choosing of a Lagrangian plane of
the corresponding symplectic {\it boundary form} e.g
$\frac{\partial u }{\partial r }(r_{_{0}})\bar{v}(r_{_{0}})-
u(r_{_{0}})\frac{\partial \bar{v} }{\partial r }(r_{_{0}})$,for
Laplacian in the outer space $r>r_{_{0}}$. I.M. Gelfand in
sixties, \cite{Novikov1}, attracted attention of specialists to
necessity of developing a simplectic version of the operator
extension procedure for abstract operators. It was done in
\cite{Zero_range} for solvable quantum models with bounded inner
Hamiltonian and positive metric in the inner spaces. In this
preliminary section we describe the symplectic version of the
extension procedure for operators in Pontryagin space, based on an
abstract analog of the boundary form.
\par
Consider the scattering problem for two quantum particles with
masses $m_{_{1}},\, m_{_{2}}$ and  the reduced mass $\mu =
m_{_{1}}\, m_{_{2}}(m_{_{1}} + m_{_{2}})^{^{-1}}$. Choosing some
characteristic wave number $k_{_{0}} > 0$ and the corresponding
characteristic energy
$E_{0}=(2\mu)^{^{-1}}{\hbar^{^2}k_{_0}^{^2}}$, we introduce the
dimensionless coordinates and the spectral parameter, connected
with the standard Euclidean coordinates ${\bf r}$ in $R_{3}$ and
the conventional energy $E =(2\mu)^{^{-1}}{\hbar^{^2}k^{^2}}$ as
\begin{equation}
\label{s3} {x}=k_{0}{\bf r},\, \lambda = k^{^2}\,k_{_0}^{_{-2}} =
E (E_{_0})^{^{-1}}.
\end{equation}
Then we  re-write the conventional Schr\"{o}dinger equation as
\begin{equation}
\left(\bigtriangleup +\lambda -\tilde{V}({x})\right)u({ x}) = 0.
\label{s1}
\end{equation}
Here $ \tilde{V}({ x}) = E_{0}^{-1}V({\bf r})$ is the
dimensionless interaction potential. The wave functions $u({ x})$
are dimensionless. We  will replace the equation (\ref{s1}) by the
zero-range model, assuming that the component of the  model in the
outer space is defined by the Schr\"{o}dinger equation with zero
potential. The ``free'' Green's function which corresponds to  the
equation with zero potential on the  whole space  $R^{^{3}}$,
satisfies the equation
\begin{equation}
-(\triangle +\lambda)G_{\lambda}(x,x') = \delta^{(3)}(x-x')
\label{s4}
\end{equation}
and has a  form of an  outgoing wave generated at the  pole  $x'$:
\begin{equation}
G_{\lambda}(x,x') = -(\triangle + \lambda)^{-1}\delta^{(3)}(x-x')
= \frac{e^{i \sqrt{\lambda} |x-x'|}}{4\pi|x-x'|}. \label{s5}
\end{equation}
We assume that the outer space is  supplied with the standard
$L_{_{2}} (R^{^{3}})$ - dot-product $\langle u, \alpha v\rangle =
\alpha \langle u, v\rangle = \langle \bar{\alpha}u, v\rangle$  and
the non-perturbed two-body Hamiltonian is defined  by the
three-dimensional self-adjoint Laplacian with respect to
dimensionless coordinates
\begin{equation}
\label{nonpert} lu = -\bigtriangleup u,
\end{equation}
on  the  Sobolev class $W_2^2(R^{^{3}})$. Variables $\tilde {x}$
are reserved for the  inner space $E$ of  the model.
\par
  Following \cite{BF61} we
restrict the Laplacian $l$ onto the domain consisting of all
smooth functions vanishing near the point $x_{_0}$. The closure
$l_0$ of the restricted operator is defined on the domain $D_0$
which consists of all $W^2_2$-functions vanishing just at
$x_{_0}$. This operator is symmetric and  has deficiency indices
$(1,1)$:
\[
\displaystyle \overline{(l_0 - i I){D}_0} = L_2(R_3)\ominus
N_i,\,\, \overline{(l_0 + i I){D}_0} = L_2(R_3)\ominus N_{-i},
\]
dim $N_i = $ dim $N_{-i}= 1$. The  adjoint operator $l_0^{+} =
(-\bigtriangleup)^{^{+}} $ is defined, due  to the von Neumann
theorem, see \cite{Neumann,Akhiezer}, by the same differential
expression (\ref{nonpert}) on the domain
\begin{equation}
\label{domain} D_0^+ = D_0 + N_i + N_{-i}.
\end{equation}
The one-dimensional deficiency subspaces $ N_i ,  N_{-i}$ are
spanned by the corresponding non-perturbed free Green functions
\[
N_i = \left\{G_{-i}(*,x_{_0})\right\},\,\,N_{-i} =
\left\{G_{i}(*,x_{_0})\right\},
\]
\begin{equation}
\label{defect} G_{-i} (x,x_{_0})=  \frac{e^{i \sqrt{-i}
|x-x_{_0}|}}{4\pi|x-x_{_0}|},\,G_{i} (x,x_{_0}) = \frac{e^{i
\sqrt{i} |x-x_{_0}|}}{4\pi|x-x_{_0}|} = \frac{l + iI}{l-iI} G_{-i}
(x,x_{_0}),
\end{equation}
with branches of square roots defined by the condition ${\rm Im}
\sqrt{\lambda} > 0$.  We assume further that the  restriction
point is $x_{_0} = 0$, if another point is  not selected .
\par
The  above representation  (\ref{domain}) of  the  domain of the
adjoint operator  was used  in \cite{Akhiezer,BF61}. Another
representation of  the  domain  of the adjoint Laplacian as a set
of singular  elements  described above (\ref{i6}) was suggested by
Fermi \cite{Fermi} and is now commonly used in physical
literature, see for instance \cite{Demkov,Albeverio}. It
characterizes each element $u$ from the domain of the adjoint
operator  by  {\it dimensionless asymptotic boundary values }
$A,\,\,B$ for $|x|\to 0$:
\begin{equation}
\label{fermivar} u (x) = \frac{A^u}{4\pi |x|} + B^u + o(1).
\end{equation}
The boundary form of  $l_{_{0}}^{^+}$ on elements from the  domain
of the adjoint Laplacian is
\begin{equation}
\label{i7} {\cal J}(u,v) = \langle (l_{_{0}})^{^{+}}u,\, v \rangle
- \langle u,\,(l_{_{0}})^{^{+}} v \rangle = \bar{B}^{^{u}}
{A}^{^{v}}- \bar{A}^{^{u}}{B}^{^{v}}.
\end{equation}
Due to the connection (\ref{s3}) between dimensional and
dimensionless coordinates the corresponding asymptotic values are
connected by the dimensional factor $k_{_{0}}:\, A = {\cal A}
k_{_{0}},\, B = {\cal B}$.
\par
  It is well known that two descriptions
(\ref{domain},\ref{fermivar}) of the domain of the adjoint
Laplacian are equivalent, \cite{BF61}. They  form a basement,
respectively, of the classical von-Neumann and symplectic version
of the extension procedure for the restricted Laplacian. Planning
to develop the symplectic version for abstract operators we will
derive the equivalence  of both above  representations based on
decomposition of elements of the defect $ N = N_i + N_{-i}$ with
respect to  the {\it symplectic basis}
\[
W_+ (x)= \frac{1}{2}\left[G_{-i}(x,0) + G_{i}(x,0) \right] =
\frac{l}{l - iI}G_{-i}(x,0),
\]
\begin{equation}
\label{W}
  W_{-} (x)= \frac{1}{2i}\left[ G_{-i}(x,0)  -  G_i (x,0)
\right ]= - \,\,\, \frac{I}{l - iI}G_{-i}(x,0).
\end{equation}
We  will present the von-Neumann  formula for  elements of the
domain of the adjoint operator in terms of the decomposition  with
respect to the  basis $W_{\pm}$~:
\begin{equation}
\label{sympl3} u = u_0 + \eta^u_+ W_+ + \eta^u_- W_-,
\end{equation}
where $u_0$ is  an  element from the  domain of  the closure $l_0$
of the restricted operator and $\eta^u_{\pm}$ are complex
coefficients. Deficiency elements $G_{\pm i}(*,0)$ are
eigenvectors of the adjoint  operator with eigenvalues $\mp i $
respectively. Then  the elements $W_{\pm}$ are transformed by  the
adjoint  operator as:
\begin{equation}
\label{Wpm}
       l_0^+ W_+ = W_{-},\,\,
        l_0^+ W_{-}= - W_{+}.
\end{equation}
\begin{lemma}{ The  boundary form (\ref{i7})
of the adjoint operator $l_0^+$ depends only on components  $u_d =
\eta^u_+ W_+ + \eta^u_- W_- , \, v_d = \eta^v_+ W_+ + \eta^v_- W_-
$ of the elements $u,\, v$ in the  defect. It is an Hermitian
symplectic form of the variables $\eta_{\pm}$ and can be presented
alternatively as
\begin{equation}
\label{intbp}
       {\cal J}(u,v)  =  {\cal J}_{_l}(u_d,v_d)=  \langle l_0^+u_d
,\,\, v_d \rangle - \langle u_d,\,\, l_0^+ v_d \rangle =
\,\,\frac{1}{4\pi\sqrt{2}} \left(\bar{\eta}^u_+ {\eta}^v_- -
\bar{\eta}^u_- {\eta}^v_+\right).
\end{equation}
The above  variables $\eta_{\pm}$ are connected with the
asymptotic boundary values $A,\, B$ via the transformation:
\begin{equation}
\label{J-unit}
\left(
\begin{array}{c}
\eta_+\\
\eta_-
\end{array}
\right) = \left(
\begin{array}{cc}
1&0\\
-1& -4\pi\sqrt{2}
\end{array}
\right) \left(
\begin{array}{c}
A\\
B
\end{array}
\right).
\end{equation}
}
\end{lemma}
{\bf Remark}\,\, Both pairs of  variables  $\eta_{_{\pm}}$ and
$A,B$ are {\it symplectic coordinates} of  the  element $u$ in
representations (\ref{fermivar},\ref{sympl3}). The representation
(\ref{i7}) of the  boundary form in terms of the symplectic
variables $A,B$ is usually obtained, see for instance
\cite{Fermi,BF61, Demkov, Albeverio} via straightforward
integration by parts. An alternative  calculation  based  on
(\ref{Wpm}) goes  in line with an abstract analog (Lemma II.2
below) of the above statement .

       {\it Proof}\,\,\, of  the  formula  (\ref{intbp}) is obtained
via the direct application of the adjoint operator $l_0^+$ to the
above version (\ref{sympl3}) of the von-Neumann  decomposition of
elements from $D_0^+$ in the  boundary  form  of  the adjoint
operator: $ \langle l_0^+ u,\,\, v \rangle - \langle u,\,\, l_0^+
v \rangle := {\cal J}_{l}(u,v).$ One can easily  see that  the
boundary form depends only on the parts of the elements $u,\,\,v$
in the sum of the deficiency subspaces (in the ``defect''), since
$l_{_0}^{^+} u_{_{0}} = l$ and $[l_{_0}^{^+} u_{_{d}},v_{_{0}} ] =
[ u_{_{d}},l_{_0}\,v_{_{0}} ] $ and can be calculated using
(\ref{Wpm}).Then due to $\int_{R_3} |G_{i}(\xi,0)|^2 d^3\xi =
1/(4\pi \sqrt{2})$  we obtain:
\begin{equation}
\label{bforml} {\cal J} (u,v) = \langle l_0^+u_d ,\,\, v_d \rangle
- \langle u_d,\,\, l_0^+ v_d \rangle = \left(\bar{\eta}^u_+
{\eta}^v_-  - \bar{\eta}^u_- {\eta}^v_+\right)\,1/(4\pi \sqrt{2}).
\end{equation}
The second formula  (\ref{J-unit}) is based on the asymptotic
behavior of the non-perturbed Green function for $x \to 0$ :$ W_+
(x)= \frac{1}{4\pi |x|} - \frac{1}{4\pi\sqrt{2}}+ o(1),\,\, W_-
(x)= - \frac{1}{4\pi\sqrt{2}}+ o(1)$.
\par
Self-adjoint extensions of $l_0$ were obtained in \cite{BF61} via
submitting $A,\,B$ to the Fermi boundary conditions $ B = \gamma A
$. For general construction of extensions of densely defined
operators in terms of boundary forms see also \cite{Gorbachuk}.
Though the boundary forms (\ref{i7}) and (\ref{intbp}) are
equivalent, our version (\ref{intbp}) can be  used for the whole
class of all differential operators with square-integrable
Green-functions, in particular for the Schr\"{o}dinger operators
in a bounded  domain with Robin boundary conditions, even
restricted at the boundary point $x{_{0}}$.
\par
We choose the non-perturbed  Hamiltonian of inner degrees of
freedom ( we call it  further just inner Hamiltonian) as a
J-self-adjoint operator in the inner Pontryagin space supplied
with an indefinite metric. After introducing  the dimensionless
spectral parameter $\lambda$  the  zero-range model  of two
particles Hamiltonian  will be obtained as an extension of the
orthogonal sum $l_0 \oplus {\cal H}_0 $ of the restricted
Laplacian and the restricted inner Hamiltonian ${\cal H}$ in the
Pontryagin space $L_{_2} (R^{^{3}}) \oplus E_{_{J}}$. The inner
space $E_{_{J}}$ is obtained via equipping  a finite-dimensional
Hilbert space $E$,\, dim$E = N $, with an indefinite metric.
Consider two complementary orthogonal projections $P_{\pm}$ with
respect to the conventional dot-product $\langle*,*\rangle$ in
$E$. Then the indefinite dot product in $E$ ($J$-dot product) is
defined as :
\[
\langle J\tilde {x},\,\tilde {y} \rangle = [\tilde {x},\tilde
{y}]= \langle P_+ \tilde {x},P_+ \tilde {y} \rangle - \langle P_-
\tilde {x},P_- \tilde {y} \rangle.
\]
Obviously $ [\bar{\alpha} \tilde {x},\tilde {y}] = {\alpha}[\tilde
{x},\tilde {y}]= [\tilde {x},\alpha \tilde {y}]$.The space $E$
supplied with the above $J$-dot product is denoted by $E_{_J}$.
For general properties of Pontryagin spaces see
\cite{Krein-Langer}. We sketch below a symplectic version of the
operator-extension technique in the  inner component $ E_{_{J}}$.
\par
A bounded operator ${\cal H}$ is called $J$-symmetric (symmetric
with respect to the above $J$-dot product) if $[{\cal H} \tilde
{x},\tilde {y}]=[\tilde {x},{\cal H}\tilde {y}]$. In
finite-dimensional space each $J$-symmetric operator is
$J$-self-adjoint. We will apply the symplectic scheme to the {\it
bounded} operator which is also {\it symmetric with respect to the
conventional dot-product in $E$ and commutes with $J$}. We assume
that the spectrum of the $J$- symmetric operator ${\cal H}$
consists of a finite number of simple {\it positive} eigenvalues
$\lambda_s = k^{^2}_{_{s}} \,k^{^{-2}}_{_{0}},\, \left({\cal H} -
\lambda_s I\right) e_s= 0$ and ${\cal H} - iI $ is invertible.
Choose a normalized {\it generating} vector $e\in
E_{_{J}},~~[e,e]= 1$  ( which is non-orthogonal to each
eigenvector $e_s$ of the operator ${\cal H}$, $[e,e_s]:=e^{^s}\neq
0 ,\, s = 1,2,\dots N)$. For selected
 vector $e$ define the domain $D_{{\cal H}_0}$ of the
restricted operator as $D_{{\cal H} _0}=\left[{\cal H} -
iI\right]^{-1} \,\,\, \left(E_{_{J}} \ominus_{_J} \left\{e
\right\}\right)$ and set ${\cal H}_0 = {\cal H} \big|_{D_{{\cal
H}_0}}$, where  the $J$-orthogonal difference is denoted by $
\ominus_{_J} $. The vectors $e,e'$ play roles of deficiency
vectors of the restricted operator at  the spectral points $\pm
i$: $ [({\cal H} - iI)D_{{\cal H}_0}, e]= 0,\,\, [({\cal H} +
iI)D_{{\cal H}_0}, e']=0 $. Then the vectors $e$ and $e' = \frac{
{\cal H}+iI}{{\cal H}-iI} e$ form a linearly-independent pair,
with a positive  angle between them. The  subspaces $M_{_{i}}: =
\bigvee_{_{\alpha}}\left\{ \alpha e\right\} ,\,M_{_{-i}} : =
\bigvee_{_{\alpha}}\left\{ \alpha e' \right\}$ will play roles of
the deficiency subspaces.
\par
Generally the $J$-adjoint operator ${\cal H}_0^+$ is defined by
the formula $[{\cal H}_0 \tilde {x},\tilde {y}]= [\tilde {x},{\cal
H}_0^+ \tilde {y}]$ on elements  $\tilde {y}$ for  which the
$J$-dot product in the left hand side may be  continued  onto the
whole space $E$ as an ``anti-linear" functional  of $\tilde {x}$.
For densely-defined operators this condition implies $
 ({\cal H}^+_0 + iI)e = 0,\,\, ({\cal H}^+_0 - iI)e'
= 0.$ For a bounded operator ${\cal H}$ {\it we just  define} the
formal adjoint operator ${\cal H}_{_{0}}^{^+}$ on the defect  $M =
M_{_{i}} + M_{_{-i}}$ by the above formulae. Then, for complex
values of $\lambda$, the deficiency vectors
\[
e_{\lambda}= \frac{{\cal H} + iI}{{\cal H} - \bar{\lambda} I}
e,\,\, e_{\bar{\lambda}}= \frac{{\cal H} + iI}{{\cal H} -
{\lambda} I} e.
\]
are eigenvectors of the corresponding formal adjoint operator:
\[
\left({\cal H}_0^+ - \bar{\lambda} I\right) e_{\lambda} = 0,\,\,
\left({\cal H}_0^+ - \lambda I\right) e_{\bar{\lambda}}=0 .
\]
M. Krasnosel'skii noticed, see \cite{Krasn49}, that the extension
construction for non-densely defined closed operators is actually
developed in the  defect and can be accomplished similarly  to
von-Neumann construction, if the deficiency subspaces do not
overlap. We obtain a  symmetric extension of the restricted
operator ${\cal H}_{_0}$ on the defect via restriction of the
formal adjoint ${\cal H}_{_{0}}^{^+}|_{_{\cal L}}$ onto some
Lagrangian plane $ {\cal L}\subset M$ where the corresponding
boundary form (\ref{boundf}) vanishes. After that the
corresponding extension in the whole space is obtained as a direct
sum ${\cal H}_{_0} + {\cal H}_{_{0}}^{^+}|_{_{\cal L}}$. An analog
of the von-Neumann representation of the domain of the adjoint
operator remains true in Ponryagin space, see \cite{Krein-Langer}.
The described above approach \cite{Krasn49} to  extension of
non-densely  defined operators is applicable to non-densely
defined operators in Pontryagin space.

We develop here the operator-extension construction with
one-dimensional deficiency subspaces assuming that dim $E_{_{J}}
\geq 2$. In case dim $E_{_{J}} = 1 $ the Krein formulae for the
resolvent and eigenfunctions of the extended operator also remain
true and may be verified via direct calculation, see \cite{MPP}
and references therein.
\par
 Consider a new  basis in the defect $M  = M_i  +
M_{-i}$ which is similar to the  above  basis (\ref{W}) combined
of the Greens-functions:
\begin{equation}
\label{w}
  w_+ = \frac{e + e'}{2} = \frac{{\cal H}}{{\cal H} - iI} e
,\,\,w_- = \frac{e - e'}{2i} = \frac{- I}{{\cal H} - iI} e.
\end{equation}
Due to above definition of the formal adjoint operator we  have:
\begin{equation}
\label{Wabs} {\cal H}^+_0 w_+ =  w_-,\,\, {\cal H}^+_0 w_- = -
w_+.
\end{equation}
We use the new  basis $\left\{ w_+ ,\, w_-\right\}$ to represent
elements from the domain of the formal adjoint operator via  new
dimensionless symplectic variables $\xi_{\pm}$ which play a role
similar to  the above pair $\eta_{\pm}$:
\[
\tilde {x} = \xi^x_+ w_+ + \xi^x_- w_-.
\]
\begin{lemma}
{The  boundary  form of  the  (formal) adjoint  operator in terms
of the variables $\xi_{\pm}$ is given by an Hermitian symplectic
form :
\begin{equation}
{\cal K}(\tilde {x},\tilde {y})=\left[{\cal H}_{_0}^+ \tilde
{x},\,\tilde {y}\right]- \left[\tilde {x},\, {\cal H}_{_0}^+
\tilde {y}\right]= \bar{\xi}^x_+ {\xi}^y_- - \bar{\xi}^x_-
{\xi}^{y}_{+}. \label{boundf}
\end{equation}
}\end{lemma}
 {\it Proof}  exactly   follows the pattern of  the
previous lemma (II.1). The  vector $e$ is  normalized in
$E_{_{J}}$, hence the normalization constant  similar  to $(4\pi
\sqrt{2})^{^{-1}}$  in (\ref{intbp}) is  equal to 1.

Note that the  formula (\ref{boundf}) can  be  interpreted as  an
``abstract   formula of integration by parts '' for  the operator
${\cal H}^{^{+}}_{_{0}}$. The symplectic version of the extension
procedure allows to  obtain the $J$-self-adjoint extension of the
formal adjoint operator via restriction of it onto the Lagrangian
plane ${\cal L }_{_{\gamma}}$ in the defect defined by the
boundary conditions as\,\, $\xi_{_{-}} = \gamma
\xi_{_{+}},\,\,\gamma = \bar{\gamma}$. Then the construction of
the $J$-self-adjoint  extension in $E_{_{J}}$ is accomplished via
forming of the direct sum ${\cal H}_{_{0}} + {\cal
H}_{_{0}}^{^+}|_{_{\cal L_{\gamma}}}$.
\par
Our  aim is  the  construction of the  joint extension ${\cal
A}_{_{\Gamma}}$ of $l_{_{0}}\oplus {\cal H}_{_{0}}$. Consider the
orthogonal sum of operators $l\oplus {\cal H}$ in the Pontryagin
space  $L_2 (R^{^3})\oplus E_{_J} $ with elements $U = (u,\,\tilde
{x}),\, u\in L_2(R^{^3}), \,\tilde {x}\in E_{_J} $. Restricting
both operators as above we obtain the symmetric operator $l_0
\oplus {\cal H}_0 $ with the defect $N\oplus M$ and the deficiency
index $(2,2)$. We define the adjoint operator as an orthogonal sum
of ``adjoint'' operators $l_0^+ \oplus{\cal H}_0^+$ and calculate
the boundary form of  it as a sum of boundary forms of the  outer
and inner components:
\[
{\bf J}(U,V)= {\cal J}(u,v)+ {\cal K}(\tilde {x},\tilde {y}) =
(\bar{B}^u {A}^v - \bar{A}^u {B}^v )+(\bar{\xi}^x_+{\xi}^y_- -
\bar{\xi}^x_- {\xi}^y_+).
\]
The boundary form ${\bf J}(U,V)$ of the orthogonal sum  of
operators $l_0^+ \oplus {\cal H}_0^+$ in the Pontryagin  space
$L_2 (R^{^3})\oplus_{_J} E_{_J}$ is a symplectic Hermitian form.
Hermitian extension ${\cal A}_{_{\Gamma}}$  of  the operator
${\cal A}_{_{0}} = l_0 \oplus {\cal H}_0$ (or Hermitian
restrictions of the adjoint operator ${\cal A}_{_{0}} = l_0^+
\oplus {\cal H}_{_{0}}^{^+} $ is constructed via imposing proper
boundary conditions on symplectic variables with an hermitian
matrix ${\bf \Gamma}$:
\begin{equation}
\label{bcond}
\left(
\begin{array}{c}
B \\
-\xi_-
\end{array}
\right) =
\left(
\begin{array}{cc}
\gamma_{_{00}} & \gamma_{_{01}}\\
\gamma_{_{10}} & \gamma_{_{11}}
\end{array}
\right)
\left(
\begin{array}{c}
A \\
\xi_+
\end{array}
\right).
\end{equation}
The  joint  boundary form ${\bf J}(U,V)$ vanishes on the
corresponding Lagrangian plane ${\cal L}_{_{\Gamma}} \subset M +
N$ defined by the above boundary condition. All Lagrangian planes
of the joint boundary form may be constructed either with a help
of various Hermitian matrices $\Gamma$, or obtained from already
constructed planes by proper $ J$-unitary transformation. As
previously, the  extended operator ${\cal A}_{_{\Gamma}}$ is
obtained  as the direct sum of the  constructed restriction of the
adjoint ${\cal A}^{^+}_{_{\gamma}} = l_{_{0}}^{^+}\oplus {\cal H
}^{^+}_{_{0}}\, |_{{\cal L}_{\Gamma}}$ onto the Lagrangian plane
${\cal L}_{\Gamma}$ in the  defect and the closure of the
restricted operator , ${\cal A}_{_{\Gamma}} = \left(l_{_{0}}
\oplus {\cal H}_{_{0}}\right) + {\cal A}^{^+}_{_{\gamma}}$.
\vskip0.3cm
\section{The spectral structure of ${\cal A}_{_{\Gamma}}$ and the  positive  subspace}

\noindent One can show, combining \cite{Krasn49,Ad_Pav}, that the
constructed operator ${\cal A}_{_{\Gamma}}$ is a J-self-adjoint
operator in the Pontryagin space $L_{_{2}}(R^{^3}) \oplus E_{_J}$.
According to general description of  the spectral structure of
self-adjoint operators in Pontryagin space, see
\cite{Krein-Langer}, the operator ${\cal A}_{_{\Gamma}}$ has a
{\it positive } invariant subspace $H \subset L_{_{2}}
(R^{^3})\oplus E_{_J}$ of a finite co-dimension in
$L_{_{2}}(R^{^3}) \oplus E_{_J}$, where the  part of the operator
is a conventional self-adjoint operator with respect to the
positive metric $P_{_{H}}[I \oplus J ]\, P_{_{H}}$. This
self-adjoint operator may play a role of a conventional quantum
mechanical Hamiltonian of the constructed model. Normally,
description of the  positive  subspace  is  complicated task since
the indefinite metric can be  degenerated. In our case we can
provide an alternative description of the subspace by showing that
it coincides  with the subspace of the absolutely-continuous
spectrum of ${\cal A}_{_{\Gamma}}$.
\par
Assume that the  extension procedure is developed as described in
the  previous section  based on a  generating vector $e =
e_{_{i}}$ with use of the  boundary conditions (\ref{bcond}).  The
spectrum of the constructed operator ${\cal A}_{\Gamma}$ is
defined by  the  classical Krein formula, see \cite{K}. In our
case it coincides, see for instance \cite{Ad_Pav}, with
singularities (the cut and the poles) of the scattering matrix
(\ref{SS}) and consists of a finite number of eigenvalues and
Lebesgue absolutely continuous spectrum filling the positive
semi-axis $0<\lambda <\infty$ with infinite multiplicity. The
generalized eigenfunctions of the absolutely continuous spectrum
of ${\cal A}_{_{\Gamma}}$ are obtained  via fitting to the above
boundary conditions (\ref{bcond}) the conventional Ansatz for
scattered waves :
\begin{equation}
\label{ET} {\Psi}_{_{\lambda}} =
 \left(
 \begin{array}{c}
 {\psi}_\lambda (x,\nu)\\
\tilde{\psi}_{_{\lambda}}
\end{array}
\right) = \left(
\begin{array}{c}
e^{i \sqrt{\lambda} (\nu,x)} + k_{_{0}}\,f(\sqrt
{\lambda})\frac{e^{i \sqrt{\lambda} |x|}}{|x|}\\
\xi_{+}\frac{{\cal H} + i I}{{\cal H} - \lambda I} e
\end{array},
\right)
\end{equation}
with components $ {\psi}_\lambda, \tilde{\psi}_{_{\lambda}}$ in
the outer and inner spaces. The  symplectic variables
$\xi_{_{\pm}}$ of the inner component of the solution of the
adjoint homogeneous equation are connected $\xi_{_{-}} = -
Q(\lambda) \xi_{_{+}}$ via the Krein $Q$-function  of  the  inner
Hamiltonian - an abstract analog  of the  Weyl-Titchmarsh
$m$-function, see \cite{Titchmarsh}:
\begin{lemma} \label{QF}
{The  $Q$-function for the operator ${\cal H}_{_0}^{^{+}}$ has the
form:
\[
Q(\lambda) =  \left[e,\frac{I +  \lambda {\cal H}}{{\cal H} -
\lambda I} e \right]. \] }
\end{lemma}
{\it Proof} of the  statement, similarly to the  proof of the
corresponding statement in case of  positive metric in
\cite{Extensions}, is reduced  to  the  calculation of  symplectic
coordinates  of the deficiency element $e_{\bar{\lambda}}$.
Consider the $J$-orthogonal projections $P_{_e} = e ] [ e$ and $
{I}- P_{_e} $ onto the one-dimensional deficiency subspace $N_i$
and onto the complementary subspace $E \ominus_{_J} N_i$,
respectively. Then  the solution of the adjoint homogeneous
equation $e_{\bar{\lambda}}$ can be  be presented as:
\begin{equation}
\label{qfunc} e_{\bar{\lambda}} =\frac{{\cal H} + i I}{{\cal H} -
\lambda I} e = \frac{{\cal H}}{{\cal H} - i I}e + \frac{I}{{\cal
H} - i I} e\,\, \left[e,\frac{I + \lambda {\cal H}}{{\cal H} -
\lambda I}\,\, e\,\,\right] +
       \frac{I}{{\cal H} - i I} ({ I}- P_{_e} )\frac{I +
\lambda {\cal H}}{{\cal H} - \lambda I}e.
\end{equation}
The  last term in the  right-hand   side is an element $u_0$ from
the domain of  the restricted operator ${\cal H}_0$, but the first
two terms belong  to the defect  $M$ so  the  whole linear
combination  with use of the  notation  $Q (\lambda) \equiv
[e,\frac{I + \lambda {\cal H}}{{\cal H} - \lambda I} e] $ can  be
presented as
\[
\frac{{\cal H}+iI}{{\cal H} - \lambda I}e = w_+  -  Q (\lambda)
w_- + u_0.
\]
Hence  the  symplectic coordinates $\xi_{_{\pm}}$ of
$e_{\bar{\lambda}}$ are $\xi_{_{+}} = 1,\, \xi_{_{-}} = -
Q(\lambda)$.
\par
We consider an  indefinite  metric tensor $J$  defined by a
diagonal matrix, for  instance  $\left\{J_{ss}\right\}= \pm 1,\,
s= 1,2,\dots N$, commuting with a  positive diagonal matrix ${\cal
H} = \left\{\lambda_{s}\right\},\,\, s = 1,2,\dots, N$. Recall
that $J$-normalized  deficiency  vector  $e$ $([e,e]=1)$  has
non-zero components $e^s$ with respect  to the standard basis in
$E$. Then the $Q$-function has the form
\begin{equation}
\displaystyle
       \label{q_indef} Q(\lambda) =
\sum_{s=1}^{N} \frac{1 + \lambda \lambda_s}{\lambda_s -\lambda}
J_{ss}|e^s|^2 = \sum_{s=1}^{N} \frac{1 + \lambda
\lambda_s}{\lambda_s -\lambda}P_s,
\end{equation}
with {\it real} weights $P_s = J_{ss}|e^s|^2$ and poles of first
order at the eigenvalues $\lambda_s$ of  ${\cal H}$. In case of
positive metric $J_{ss}>0$ the $Q$- function belongs to Nevanlinna
class.
\par
The previous lemma allows us to solve the adjoint non-homogeneous
equation and obtain the Krein formula \cite{K} for the resolvent
of the $J$-self-adjoint  extension of the symmetric operator $l_0
\oplus {\cal H}_0$  with the boundary condition (\ref{bcond}). The
scattered waves in s-channel may be derived from it in a rather
standard way, see \cite{Ad_Pav}. One may also show, see  for
instance \cite{Albeverio,AK00}, that the boundary conditions
(\ref{bcond}) formulated  for elements of the domain of  the
extension ${\cal A}_{_{\Gamma}}$ are fulfilled  for the
corresponding scattered waves. We focus now on the straightforward
derivation of the expressions for the scattered waves $\left\{
\Psi \right\}$ based on the  above  Ansatz (\ref{ET}). The
symplectic variables in the  ansatz  for components of the
scattered wave in the outer and the inner spaces are:
\begin{equation}
B = \left( 1 + i\sqrt{\lambda} k_{_{0}} f \right),\,\, A = 4\pi
k_{_{0}} f,\, \xi_- = - Q \,\xi_+ . \label{coeff}
\end{equation}
This gives, due to (\ref{bcond}), the following equation for the
amplitudes $f,\,\xi_{_+}$ :
\begin{equation}
\label{ampeq}
\left(
\begin{array}{c}
     1 + i{\sqrt{\lambda}} k_{_{0}} f\\
Q \xi_{_{+}}
\end{array}
\right) = \left(
\begin{array}{cc}
\gamma_{00} & \gamma_{01} \\
\gamma_{10} & \gamma_{11}
\end{array}
\right)
       \left(
\begin{array}{c}
   4\pi k_{_{0}} f \\
      \xi_{_{+}}
\end{array}
\right).
\end{equation}
Solving this equation we obtain the expression for the amplitude
$f$:
\begin{theorem}
\label{samplitude}{ The  amplitude  $f$, for  real positive
$\lambda = k^{^2}\,\,k^{^{-2}}_{_{0}}$, is equal to
\begin{equation}
f (k) = \left(4\pi k_{0}\gamma_{00}-\frac{4\pi
k_{0}|\gamma_{01}|^{2}}{\gamma_{11}-Q(\lambda)} - ik\right)^{-1}.
\label{Scat1}
\end{equation}
}
\end{theorem}
{\it Proof}\,\, follows from the  previous discussion if the
dimension  of the  inner  space is $\geq 2$,and  $\xi_{_{+}} =
\gamma_{_{01}}\frac{4\pi k_{_{0}} f}{Q-\gamma_{_{11}}}$. In case
of the one-dimensional inner space the formulae (\ref{Scat1}) are
verified by the direct calculation. \vskip0.3cm {\bf Corollary}
The scattering  matrix in s-channel connected to the amplitude
$f(\sqrt{\lambda}) $  via the  formula, see \cite{AK00}:
\begin{equation}
\label{SS}
 S(\sqrt{\lambda}) = 1 + 2ik\,f(\sqrt{\lambda})  = \frac{\gamma_{00}-
\frac{|\gamma_{01}|^2}{\gamma_{11} - Q} + i
\frac{\sqrt{\lambda}}{4 \pi }}{\gamma_{00}-
\frac{|\gamma_{01}|^2}{\gamma_{11} - Q} - i
\frac{\sqrt{\lambda}}{4 \pi }},
\end{equation}
with  the  branch of $\sqrt{\lambda}$ defined  by  the condition
$\sqrt{\lambda} > 0$ for $\lambda > 0$. The dimensionless
eigenvalues $\lambda_{_s}$ of ${\cal A}_{_{\gamma}}$ are  found
as poles of the scattering matrix on the  spectral sheet  $\Im
\sqrt{\lambda}
> 0$:
\[
\gamma_{00}- \frac{|\gamma_{01}|^2}{\gamma_{11} - Q(\lambda)} - i
\frac{\sqrt{\lambda}}{4 \pi } = 0.
\]
When considering the non-stationary scattering problem, see for
instance \cite{Lax}, it is  convenient to use the complex
conjugate scattering matrix, instead of (\ref{SS}).
Non-pure-imaginary zeroes of our scattering matrix
${S}(\sqrt{\lambda})$ sit in the upper half-plane $\sqrt{\lambda}
> 0$, and  the corresponding complex-conjugate poles - in the
lower  half-plane. Vice versa, the non-pure-imaginary zeroes of
the complex conjugate  matrix sit in the lower half-plane
$\sqrt{\lambda}< 0$, and the corresponding poles sit in
complex-conjugate points in the upper half-plane.
\par
 In the next section we consider zero-range models with scattering
 matrix approaching  $1$ at  infinity. For these models
the scattering matrix is presented as  a  ratio of two  finite
Blaschke products $S(k) = S_{_{+}}(k) \,\, S_{_{-}} (k)$ with
zeroes in upper  and  lower half-planes correspondingly:
$S_{_{+}}(k) =
\prod_{_{l}}\frac{k-k_{_{l}}}{k-\bar{k}_{_{l}}}.\,\,\Im
k_{_{l}}>0,\,\,\, S_{_{-}}(k) =
\prod_{_{n}}\frac{k-\kappa_{_{n}}}{k-\bar{\kappa}_{_{n}}}.\,\,\Im
\kappa_{_{n}}<0$. We proceed  assuming that the  Scattering matrix
has this  form.
\par
In the remaining part of this section we  describe the positive
subspace  of the operator ${\cal A}_{_{\Gamma}}$. In case of
s-scattering, only eigenfunctions with spherically - symmetric
outer part differ from the non-perturbed exponentials. The whole
absolutely-continuous  subspace is  split orthogonally into the
trivial part constituted by eigenfunctions  with trivial inner
components and the complementary part $H_{_{a}}$ characterized by
spherically-symmetric outer components of the corresponding
(generalized) eigenfunctions. Consider the restriction of ${\cal
A}_{_{\Gamma}}$ onto subspace $H_{_{a}}$. The scattered waves from
$H_{_{a}}$  are obtained via  fitting an appropriate  ansatz to
(\ref{bcond}). This  gives:
\begin{equation}
\label{s-waves} \Psi_{_{k}}(r) = \left(
\begin{array}{c}
\frac{1}{4\pi r} \left[e^{^{-ikr}} - S(k)\, e^{^{ikr}}\right]\\
\xi(k)\, \frac{{\cal H} + iI}{{\cal H}-\lambda I} e
\end{array}\right),
\end{equation}
$S(k)$ coincides with (\ref{SS}) and $\xi(k) = \gamma_{_{01}} \,
\frac{2 i \sqrt{\lambda}}{Q - \gamma_{_{11}}}\, f(k)$. Further
construction of the  positive subspace is developed in $H_{_{a}}$.
\begin{theorem}{The sperically - symmetric subspace $H_{_{a}}$ of the  absolutely-
continuous  spectrum of ${\cal A}_{_{\Gamma}}$ is  positive, and
the restriction of the  operator ${\cal A}_{_{\Gamma}}$ onto that
subspace is a conventional self-adjoint operator which is
equivalent to the multiplication by $k^{^2}$ in $L_{_{2}} (R)$}.
\end{theorem}
{\it Proof} We suggest an indirect proof of this  statement based
on analysis of the energy norm associated with the wave equation
$u_{_{tt}} + {\cal A}_{_{\Gamma}} u = 0$. Introduce the
corresponding energy norm  on a dense linear set of all Cauchy
data ${\bf U} = \left\{ u,\,u_{_{t}}\right\}$ with both components
from $H_{_{a}}$:
\[
{\bf U}(t) = \left( \begin{array}{c} u\\u_t\end{array}\right)=
\int_{_{-\infty}}^{^{\infty}} \left( \begin{array}{c}
\frac{1}{ik}\Psi
\\  \Psi\end{array} \right) e^{^{ikt}} h(k) dk,\,
\,\,-\infty<t<\infty.
\]
We can  choose $h = h_{_{-}}\in S^{^{-1}}_{_{+}} m_{_{-}}
H^{^{2}}_{_{-}}$, with a  rapidly  decreasing at infinity outer
function $m_{_{-}} (k)$ in lower  half-plane. Then  both
components of ${\bf U}(t)$ belong to the domain of ${\cal
A}_{_{\Gamma}}$. The closure of $\bigvee_{_{t>0}}
e^{^{ikt}}S^{^{-1}}_{_{+}} m_{_{-}} H^{^{2}}_{_{-}}$ in
$L_{_{2}}$-norm coincides with $L_{_{2}} (R)$ and gives the
spectral representation of the ``whole'' absolutely-continuous
subspace. As usual, see \cite{Lax}, the evolution of the above
wave equation on the space of Cauchy data is energy-preserving:
$\frac{1}{2}\left\{ \left[ A u,u\right] +
\left[u_{_{t}},u_{_{t}}\right] \right\} = Const$. It can be
presented as exp$(i {\cal L})$ with the  corresponding matrix
generator:
\[
{\cal L} = i \left(
\begin{array}{cc}
0& -1\\
{\cal A}_{_{\Gamma}} & 0
\end{array}
 \right).
\]
The  absolutely continuous  spectrum of the  operator ${\cal L}$
is  simple on the  whole  real axis $k$, and corresponding
generalized  eigenfunctions are two-component vectors $ \left(
\frac{1}{ik}\Psi ,\Psi\right)$. To prove that the energy
$\frac{1}{2}\left\{ \left[ A u,u\right]+
\left[u_{_{t}},u_{_{t}}\right] \right\}$ is positive, it suffice,
due  to the energy conservation, to verify the positivity  of
energy on elements ${\bf U}(t) $ for  $t\to -\infty$. Note that,
due to the special choice of $h_{_{-}}$,the  parts  of each
component $u,u_{_{t}}$ in the inner subspace vanish for large
negative $t$, so that the total energy coincides with one  of the
component of ${\bf U}$ in the outer space and is equal to the
standard energy  for the wave equation. Hence it is positive for
large negative $t$. Then due to the energy conservation, the
energy  is  positive on elements ${\bf U}(t)$ for  each $t$. Hence
the generator ${\cal L}$ of the evolution is a self-adjoint
operator in  the space of Cauchy data ${\bf U}$ with the positive
energy. The square of it is defined on smooth elements by the
diagonal matrix in the space of energy-normed Cauchy data
\[
{\cal L}^{^2} =  \left(
\begin{array}{cc}
{\cal A}_{_{\Gamma}}& 0\\
0 &{\cal A}_{_{\Gamma}}
\end{array}
 \right).
\]
It can be  extended by the Friedrichs procedure onto the maximal
domain in  the energy normed space. The square norm of the first
component of the corresponding decomposition of the  Cauchy data
coincides with the quadratic form of ${\cal A}_{_{\Gamma}}$. The
restriction of the energy onto the subspace of second components
$u_{_{t}}$ of Cauchy data is still positive and equivalent to the
$L_{_{2}}$ - norm on $k h_{_{-}}$. Closure of the subspace of the
second components of Cauchy data gives the  subspace of the
absolutely-continuous spectrum  of ${\cal A}_{_{\Gamma}} $. Hence
the corresponding absolutely-continuous subspace is positive, and
the part of ${\cal A}_{_{\Gamma}}$ in it  is a conventional
self-adjoint operator. \vskip0.3cm
\section{Fitting of parameters for zero-range model}

In this  section we define the metric of the inner  space, and
evaluate the boundary parameters and the moduli of components of
the deficiency vector in the finite-dimensional case dim$\, E =
N$, assuming that the spectrum of the inner Hamiltonian and the
real values of the scattering length and of the effective radius
are given. \par
 The scattering matrix $S(k)=\exp 2i\delta (k)$
can be presented as the Caley-transform of the function ${\rm
cot}\,\,\delta(k)$:
\begin{equation}
S(k) =\frac{{\rm cot}\,\,\delta(k)+i}{{\rm cot}\,\,\delta(k)-i} =
1 + 2ik f (k). \label{Scat2}
\end{equation}
Hence the  scattering amplitude $f(k)$ is presented as  a function
of  the  scattering phase  $\delta(k)$:
\begin{equation}
f(k)= [k{\rm cot}\,\,\delta(k)-ik]^{^{-1}}. \label{Scat3}
\end{equation}

Combining Eqs.(\ref{Scat1}), (\ref{Scat2}) and (\ref{Scat3}) one
can present the S-scattering matrix in the form:
\begin{equation}
S(k)=  1 + \frac{2 i k}{4 \pi k_0
       [\gamma_{00}-\frac{|\gamma_{01}|^{2}}{\gamma_{11}-Q(\lambda)}]-ik} =
\frac{F(k) + ik}{F(k)- ik},
\label{A25}
\end{equation}
with
\begin{equation}
F(k)= 4 \pi k_0
\left[\gamma_{00}-\frac{|\gamma_{01}|^{2}}{\gamma_{11}-Q(\lambda)}\right]
= k \cot \delta(k). \label{A26}
\end{equation}
Here we  used the above notation (\ref{q_indef}) and the
dimensionless energy $\lambda = (k/k_{0})^{2}$
\par
In this section we consider a special class of zero-range  models
for which  the scattering matrix  tends to $1$ at infinity. This
implies
\begin{equation}
\gamma_{11}+\sum_{s=1}^{N}\lambda_{_{s}}P_{s}=0, \label{A29}
\end{equation}
and the  representation of the  denominator  in (\ref{A26}) as a
ratio of two polynomials ${\cal P}_{_M},{\cal P}_{_N}$ with real
coefficients , $M<N$:
\begin{equation}
\label{prod}
 D(\lambda) = \gamma_{11}- Q (\lambda) = -
\sum_{s=1}^{N}\frac{1 + \lambda^{^2}_{_{s}} }{\lambda_{_s} -
\lambda} P_s = \Lambda_{_{NM}}\,
\frac{\prod_{_{r=1}}^{^M}\left(\lambda - h_{_r}\right)}
{\prod_{_{s=1}}^{^N} \left(\lambda - \lambda_{_{s}}\right)} :=
\Lambda_{_{NM}}\,\frac{{\cal P}_{_M}}{{\cal P}_{_M}}.
\end{equation}
Points  $\left\{h_{_r}\right\}$ sit in  complex plane $\lambda$
symmetrically with respect to complex conjugation. The real
coefficient $\Lambda$  will be found later. Consider the Laurent
series
\begin{equation} \label{D}
D(\lambda) = \Lambda_{_{NM}} \,\left[\frac{1}{\lambda^{^{(N-M)}}}
+ \frac{d_{_{N-M+1}}}{\lambda^{^{(N-M +1 )}}} + \dots
\frac{d_{_{N}}}{\lambda^{^N }} + \dots \right],
\end{equation}

with $d_{_{l}} = 0$ for $l= 1,2,\dots, N-M-1,\,\, d_{_{N-M}} = 1$,
and
\begin{equation} \label{1/D}
D^{^{-1}}(\lambda) = \frac{1}{\Lambda_{_{NM}}}
\frac{\prod_{_{r=1}}^{^M}\left(\lambda - h_{_r}\right)}
{\prod_{_{s=1}}^{^N} \left(\lambda - \lambda_{_{s}}\right)} =
\frac{1}{\Lambda_{_{NM}}} \left[\lambda^{^{(N-M)}} + \dots +
q_{_{1}}\lambda + q_{_{0}} + \frac{q_{_{-1}}}{\lambda} \dots
\right].
\end{equation}
The real coefficients $d_{_{l}},\,q_{_{l}}$ are uniquely defined
by the poles $\lambda_{_{s}}$ and zeroes $h_{_{s}}$ of $D$. For
instance, integrating $D^{^{-1}}$ on a large  circle with proper
weights $z^{^{-1}},\, z^{^{-2}}$ we  obtain
\begin{equation}
\label{qqq} q_{_{0}}= \frac{{\cal P}_{_{N}}(0)}{{\cal
P}_{_{M}}(0)} +
\sum_{_{r=1}}^{^M}\frac{\prod_{_{s=1}}^{^N}\left(h_{_{r}}-\lambda_{_s}\right)}
{h_{_{r}}\prod_{_{t\neq r}}\left(h_{_{r}}- h_{_t}\right)},\,\,
\label{q} q_{_{1}}= \frac{d}{d \lambda}\left(\frac{{\cal
P}_{_{N}}}{{\cal P}_{_{M}}} \right) (0) +
\sum_{_{r=1}}^{^M}\frac{\prod_{_{s=1}}^{^N}\left(h_{_{r}}-\lambda_{_s}\right)}
{h^{^2}_{_{r}}\prod_{_{t\neq r}}\left(h_{_{r}} -
h_{_t}\right)},\,\,
\end{equation}
Comparing coefficients in front  of powers of $\lambda$ in
(\ref{prod},\, \ref{D}) we obtain a linear system for the
variables $p_{_{s}} = \left(1 + \lambda^{^{2}}_{_{s}} \right)
P_{_s} $:
\begin{equation}
\label{Wsystem} \sum_{_1}^{^N}\left(1 +
\lambda^{^{2}}_{_{s}}\right)\lambda^{^{l-1}}_{_{s}} P_{_s} =
\Lambda_{_{NM}} d_{_{l}},\,\,l= 0,1,\dots,N-1.
\end{equation}.
 This  system has Vandermond determinant, which is  positive if
the eigenvalues are enumerated in increasing order
$W(\lambda_{_{1}},\,\lambda_{_{2}},\dots \lambda_{_{N}}) =
\prod_{_{s>t}} |\lambda_{_s}- \lambda_{_{t}}| := W $. The solution
$\left\{p_{_{s}}\right\}$ is  unique and is presented  by the
Cramer formula. Denote by $W_{_{NM}}^{^{s}}$ the determinant with
the column $\left(1,\lambda_{_{s}},\, \lambda^{^2}_{_{s}},\,
\lambda^{^3}_{_{s}},\, \dots \lambda^{^{N-1}}_{_{s}}\right)$
replaced  by the column $\left( 1,\, d_{_1},\,\dots d_{_{N-1}}
\right)$. Then the parameter $\Lambda_{_{NM}} $ is  defined by
$\lambda_{_1},\,\lambda_{_2}\dots$ from the normalization
condition $[e,e]= \sum_{_{s=1}}^{^N} P_{_s}=1$ :
\begin{equation}
\Lambda_{_{NM}} = \left(\sum_{_{s=1}}^{^N} (1 +
\lambda_{_{s}}^{^2})^{^{-1}}\, W^{^{-1}}
W_{_{NM}}^{^s}\,\right)^{^{-1}} \label{normconst}
 \end{equation}
We call $\Lambda_{_{NM}}$ the normalization constant. The role of
fitting parameters  will play : \vskip0.2cm 1.The coefficients
$\frac{r_{_{0}}}{2},\,\frac{1}{a}$ in front of the powers $
k^{^2},\, k^{^0}$ in the Laurent expansion
\begin{equation}
\label{qq} F(k) = 4\pi k_{_{0}}\left\{\gamma_{_{0}}-
\frac{|\gamma_{_{01}}|^{^2}}{D(\lambda)}\right\} = 4\pi
k_{_{0}}\left\{\gamma_{_{0}}-
\frac{|\gamma_{_{01}}|^{^2}}{\Lambda_{_{NM}}}
\left[\lambda^{^{(N-M)}} + \dots + q_{_{1}}\lambda + q_{_{0}} +
\frac{q_{_{-1}}}{\lambda} \dots \right]. \right\}
\end{equation}
of the function $F(k)$ at infinity, with $\lambda =
k^{^{2}}\,k^{^{-2}}_{_0}$. \vskip0.2cm 2. The eigenvalues
$\lambda_{_1},\,\lambda_{_2},\dots$ of ${\cal H }$ and the
corresponding normalization constant $\Lambda_{_{NM}}$.
\vskip0.2cm 3. The zeroes $h_{_1},\, h_{_2},\,\dots h_{_M}$ of
$D(\lambda)$ \vskip0.2cm \vskip0.2cm We assume  that the  above
data  are {\it consistent} in the following  sense:
\begin{equation}
\label{consist} \mbox{sign}\,\, r_{_{0}} = - \mbox{sign}\,\,
\Lambda_{_{NM}} \,\, q_{_{1}}
\end{equation}
\begin{lemma}{ If the  consistency condition (\ref{consist}) is
fulfilled, then the boundary parameters $\gamma_{_{st}}$ and
weights $P_{_s} = \,\, J_{_{ss}} |e^{^s}|^{^2}$ of the model
${\cal A}_{_{\Gamma}}$ are defined in several steps:\par 1.
Calculate  $P_{_s} = \,\, J_{_{ss}} |e^{^s}|^{^2} =
\Lambda_{_{NM}}\,W^{^{s}}\, W_{_{NM}}^{^{-1}} (1 +
\lambda^{^2}_{_{s}})^{^{-1}} $ via solving the system
(\ref{Wsystem}). \par 2. Define the boundary parameters
$\gamma_{_{00}},\, |\gamma_{_{01}}|^{^2}$, for given $r_{_{0}},\,
a$ from the  equations:
\begin{equation}
\label{boundpar} \frac{r_{_{0}}}{2} = - 4\pi k^{^{-1}}_{_{0}}
|\gamma_{_{01}}|^{^2}\,\Lambda_{_{NM}}^{^{-1}}\, q_{_{1}}\,\,\,
-\frac{1}{a}= 4\pi k_{_{0}}\left[\gamma_{_{00}} -
|\gamma_{_{01}}|^{^2}\, \Lambda_{_{NM}}^{^{-1}}\, q_{_{0}}\right].
\end{equation}
\par3. Set $\gamma_{_{11}} = - \Lambda_{_{NM}}\,\sum_{_{s=1}}^{^N}\lambda_{_s}\,W_{_{NM}}^{^{s}}\,
W^{^{-1}}\, (1 + \lambda^{^2}_{_{s}})^{^{-1}}$.}
\end{lemma}
Note  that  the  poles  $h_{_{r}}$ of  $F$ are  the points where
the  scattering matrix is  equal to  $1$. In real physical
problems in $3$-d space  this is never observed for finite real
energy. In \cite{JPA} we suggested extending of this observation
to  complex  values of energy in form of {\it the principle of
analyticity }, assuming that $F$ is   analytic on the whole  plane
$\lambda$. In finite-dimensional case this  means that $F $ is
just a polynomial degree $N$ with $M=0$. The corresponding
operator ${\cal A}_{_{\Gamma}}$ is called ``the  special
zero-range model''. Arranging the eigenvalues of the inner
Hamiltonian in increasing order and  denoting by $W_{_s}$ the
Vandermond determinant with  $\lambda_{_{s}}$ just omitted, we
obtain via direct calculation the following consistency condition:
\begin{equation}
\label{consist1} \mbox{sign}\,\, r_{_{0}} = \mbox{sign}\,\,\Lambda
,\,\, \mbox{where}\,\,\, \Lambda = (-1)^{^{n}}\Lambda_{_{N0}} =
\sum_{_{s = 1}}^{^{N}} \frac{W_{_s}}{W} (-1)^{^s}\frac{1}{1 +
\lambda^{^{2}}_{_{s}}},
\end{equation}
and  fit the model based on equations:
\[  P_{_{s}} = \frac{W_{_s}}{W}
(-1)^{^s}\frac{ \Lambda}{1 + \lambda^{^{2}}_{_{s}}},\,\,\,
|\gamma_{_{01}}|^{^2} = \frac{r_{_{0}}\,k_{_{0}} \Lambda}{8\pi
k_{_{0}} \prod_{_{s=1}}^{^{N}}\lambda_{_{s}}^{^N}
\sum_{_{s}}^{^N}\lambda^{^{-1}}_{_{s}} },\,\gamma_{_{11}} +
\sum_{_{s=1}}^{^N}\lambda_{_{s}}\,\, P_{_{s}},\,\,=0,
\]
\begin{equation}
\label{PA} -\frac{1}{a}= 4\pi k_{_{0}}\left[ \gamma_{_{00}} -
|\gamma_{_{01}}|^{^2} \sum_{_{s = 1}}^{^{N}} \frac{W_{_s}}{W}
(-1)^{^s}\frac{1}{1 +
\lambda^{^{2}}_{_{s}}}\,\prod_{_{s=1}}^{^{N}}\lambda_{_{s}}^{^n}
\sum_{_{s}}^{^n}\lambda^{^{-1}}_{_{s}}\right].
\end{equation}
Physical meaning of the  above  formulae becomes more  transparent
if we return to  dimensional wave number $k$ and spectrum
$k^{^2}_{_s}= k^{^2}_{_{0}}\,\lambda_{_{s}}$. In particular, the
 consistency condition (\ref{consist1}) can be  presented as {\rm
sign} $ r_{_{0}} = ${\rm sign} $\,\sum_{_{s = 1}}^{^{N}}
\frac{(-1)^{^s}}{\prod_{_{l\neq s}}|k^{^2}_{_{s}} -
k^{^2}_{_{l}}|} \frac{k^{^{2N}}_{_{0}}}{k^{^{4}}_{_{0}} +
k^{^{4}}_{_{s}}}$. This gives the following statement:
\begin{theorem}
\label{t1} {For scattering  systems which can be modelled by
special zero-range potentials the only fitting parameters are: the
scattering length, the effective radius, and the spectrum of the
inner Hamiltonian. The {\rm sign}$\,r_{_{0}}$ of the effective
radius is necessarily connected to the spectrum of the inner
Hamiltonian by the  consistency condition. The components of the
metric tensor for those models are defined as $
J_{ss}=(-1)^{^{s}}{\rm sign}\,r_{_{0}} $. Other essential
parameters of the model like: boundary parameters
$\gamma_{_{00}},\gamma_{_{11}},|\gamma_{01}|$, and
 components $|e^{^s}|^{^2}$ of  the  deficiency  vector are defined as
functions of the scattering length $a$, effective radius $r_0$,
and $\lambda_{_s}~~(s=1,2,\ldots,n)$
\begin{equation}
\label{equat}
  a =\left(-4 \pi \gamma_{00}
k_0+\frac{4\pi|\gamma_{01}|^{2}}{k_{0}^{2N-1}\Lambda}\prod_{s=1}^{N}k_{s}^{2
}\right)^{-1} ,\,\, r_{0}=\frac{8 \pi |\gamma_{01}|^2}{\Lambda \,
k_0^{^{2N-1}}}
\left(\prod_{t=1}^{N}k_{t}^{2}\right)\sum_{s=1}^{N}k_{s}^{-2}.
\end{equation}
In particular, the effective radius is  positive, $r_0 >0 $, if
and only if the consistent normalization constant is  positive
too, $\Lambda
> 0$, and $J_{ss} = \left( -1 \right)^{s}$. }
\end{theorem}
Introducing the notations $~~\varepsilon =
4\pi\gamma_{00}k_{0},~~\gamma = 4\pi
\gamma_{01}|^{^2}\,\,k^{^{-2N+1}}_{_0}\,\, \Lambda^{^{-1}}$ we
obtain for the function $F(k)$ the following representation:
\begin{equation}
F(k)=k{\rm cot}\delta(k)= \varepsilon  - \gamma
\prod_{s=1}^{N}(k_{s}^{2}-k^{2}). \label{A40}
\end{equation}
Sign of the effective radius is the same as the  sign $\gamma$ :
\begin{equation}
a=\left(-\varepsilon
+\gamma\prod_{s=1}^{N}k_{s}^{2}\right)^{-1},\,\,\, r_{0}=2\gamma
\sum_{s=1}^{N}\prod_{t(t\neq s)}k_{t}^{2}. \label{A42}
\end{equation}
\vskip0.3cm {\bf Corollary 2} Using (\ref{A25},\ref{A40}) and
above parameters $\varepsilon,\,\gamma$, we may obtain a
convenient representation for the scattering matrix $S$ in terms
of the resonance parameters $k_{s}$:
\begin{equation}
S(k)= 1+\frac{2ik}{\varepsilon -ik-\gamma\prod_{s=1}^{N}(k_{s}^{2}-k^{2})}.
\label{A43}
\end{equation}
The total scattering cross-section is $\sigma(k)=4\pi|f(k)|^{2}$
or can be written in explicit form (see next section) using
equations (\ref{s3}) and (\ref{A40}). This expression for the
scattering matrix in s-channel describes the resonance scattering
of particles with resonances  defined  by  the spectral properties
of the Hamiltonian ${\cal H}$ of the inner degrees of freedom.
\par
  {\bf Re-normalization.} We already proved
that our resonance scattering model depends only the scattering
length $a$, effective radius $r_{0}$, the spectrum $k_{s}$
($s=1,2,\ldots,N$) of the inner Hamiltonian and  the  typical wave
number. Now  we introduce instead  of the  scattering length and
effective radius a new parameter $a_{_0}$ with dimension
($[a_{_0}]=cm$) and a dimensionless parameter $\alpha$ by the
formulae:
\begin{equation}
\label{50}
k_0: =- \frac{1}{4 \pi \gamma_{00} a_0},~~~\alpha: = - \frac{4 \pi
|\gamma_{01}|^2}{\Lambda} \left( 4
\pi \gamma_{00} \right)^{2N-1},
\end{equation}
Then  the function $F(k)= k \cot \delta (k)$ for our zero-range
potential is:
\begin{equation}
\label{51} F(k) =  -\frac{1}{a_{0}}- \frac{\alpha}{a_{0}} \prod_{s =1}^{N}
\left(a_0^2 k_s^2 - a_0^2 k^2\right).
\end{equation}
The  effective radius and the  scattering length  can be  found
from (\ref{51}) as first coefficients of the  polynomial $F (k) =
-a^{-1} + \frac{r_0}{2} k^{^{2}}$:
\begin{equation}
\label{53} \frac{r_0}{2} = \alpha  a_0 \sum_{n=1}^N \prod_{s(s\neq
n)}^{N} a_0^2 k_s^2,\,\,\, a = \frac{a_{0}}{1 + \alpha \prod_{s
=1}^{N} a_0^2 k_s^2}.
\end{equation}
Then the parameter $a_0$ can be  interpreted as a  {\it non-
re-normalized  scattering length} for the zero-range  potential
without inner structure, which corresponds to $\alpha = 0 $. The
above equation connects the re-normalized scattering length $a$ to
the {\it non-re-normalized scattering length} $a_{0}$, taking into
account resonance scattering.

Thus the function $F(k)$ given by Eq.~(\ref{51}) depends on the
typical wave-number and  $N+2$ parameters: $a_0, \alpha$ and $
k_s,\, s = 1,2,\dots N$. Consequently, the  non-re-normalized
scattering length $a_0$, the dimensionless parameter  $\alpha$ and
the spectrum $k_{s}$ ($s=1,2,\ldots,N$) of the Hamiltonian  define
the scattering length $a$ and effective  radius $r_{0}$ of the
model. \vskip0.3cm \noindent {\bf Resonance cross-section.} The
``total'' cross-section  for spherically-symmetric scattering is
generally calculated  as
\[
\sigma (k)= 4\pi |f(k)|^2 = \frac{4\pi }{|F(k)- ik|^2}.
\]
This gives due to (\ref{51}) an explicit formula for total
cross-section of the special zero-range model:
\begin{equation}
\sigma (k) = \label{cross} \frac{4 \pi a_0^2}{1 + a_0^2k^2 +
2\alpha\prod_{s=1}^N \left(a_0^2 k_s^2 - a_0^2 k^2\right) +
\alpha^2 \prod_{s=1}^N \left(a_0^2 k_s^2 - a_0^2 k^2\right)^2}.
\end{equation}
One can see from  (\ref{cross},\ref{51}) that the maxima of the
total cross-section $\sigma (k)$ are shifted from the eigenvalues
of the inner Hamiltonian. They can be interpreted again as {\it
re-normalized} eigenvalues  of the inner Hamiltonian. The
re-normalization is caused  by  the interaction introduced  via
the  boundary condition (\ref{bcond}). Note  that the  final
formula obtained via substitution of (\ref{51}) into (\ref{cross})
is not a phenomenological formula, but an exact formula derived
for certain Hamiltonian. The corresponding solvable model has
equal rights with other quantum solvable models, but unlike them
it may have resonances at positive energy, the sign $\pm$ of the
effective radius defined by the  spectrum of the  inner
Hamiltonian via (\ref{consist1}) and allows complete fitting of
all parameters of the  model from the experimental data, thus
prescribing to the parameters a certain physical meaning.
\vskip0.3cm

 \small{
\begin{thebibliography}{99}

\bibitem{BF61} F.A.Berezin, L.D. Faddeev {\it A  remark on
Schr\"{o}dinger equation with a  singular potential} Soviet Math.
Dokl. {\bf 2} pp 372-376 (1961)

\bibitem{Fermi} E.Fermi {\it Sul  motto dei  neutroni nelle
sostance idrogenate} (in Italian) Richerka Scientifica {\bf 7} p
13 (1936)

\bibitem{Neumann} J. von Neumann {\it Mathematical foundations
of quantum mechanics} Twelfth printing. Princeton Landmarks in
Mathematics. Princeton Paperbacks. Princeton University Press,
Princeton, NJ, (1996)

\bibitem{Albeverio} S.Albeverio, F.Gesztesy, R.Hoegh-Krohn, H.Holden,
{\it Solvable models in quantum mechanics}, (Springer-Verlag, New
York, 1988).

\bibitem{Wigner51}  E.P.Wigner, {\it On a class of analytic functions
 from the quantum theory of collisions} Annals  of  mathematics,  {\bf 2},
N53, 36 (1951)

\bibitem{SU2}  J.Sylvester,\thinspace G. Uhlmann \textit{The Dirichlet to
Neumann map and applications.} In: Proceedings of the Conference
`` Inverse problems in partial differential equations
(Arcata,1989)'', SIAM, Philadelphia, 101 (1990)

\bibitem{Zero_range} B.Pavlov, {\it A  model of  zero-range potential
with an internal  structure}(in Russian) Teor. Mat. Fizika,{\bf
59}, pp 345-353 (1984) (English Translation : Theor. and  Math.
Physics {\bf 59} pp 544-550) (1984)

\bibitem{K} M.Krein {\it Concerning the resolvents of an Hermitian
operator with deficiency index (m,m)} (in Russian) Doklady  AN
USSR,{\bf 52}, 651 (1946)

\bibitem{Extensions} B.Pavlov,  {\it The theory of extensions and
explicitly solvable models} (in Russian)  Uspekhi Mat. Nauk, {\bf
42}, 99 (1987). English translation in : Russian Math. Surveys
42:6, 127-168 (1987)

\bibitem{Novikov} S.P.Novikov, {\it Schr\"odinger operators on
graphs and symplectic  geometry}, Proceedings of the Fields
Institute Conference in Honour of the 60th Birthday of Vladimir I.
Arnol'd, eds. E. Bierstone, B. Khesin, A. Khovanskii and J.
Marsden, Communications of Fields Institute, AMS, 1999.

\bibitem{Exner} P.Exner, P.Sheba, {\it A new  type  of quantum
interference transistor} Phys. Lett. A {\bf 129}, N:8,9, pp
477-480 (1988).

\bibitem{Ad_Pav}  V.M.Adamyn,B.S.Pavlov {\it Zero-
radius potentials and M.G.Krein's formula for generalized
resolvents}, Zap. Nauch. Sem. Leningrad. Otdel. Mat. Inst.
Steklov. (LOMI), {\bf 149}, Issled. Linein. Teor. Functsii. XV,
186, p.7-23 (1986)

\bibitem{AK00} S.Albeverio, P.Kurasov, {\it  Singular
Perturbations of Differential  Operators}, (London  Math. Society
Lecture  Note Series  271. Cambridge  University Press, 2000).

\bibitem{Landau} L.Landau, E.Lifschitz, {\it Physique theoretique
("Landau-Lifchitz"), Tome III: Mechanique quantique. Theorie non
relativiste } (French) Deuxieme edition. Mir, Moscow, 1967.

\bibitem{MMM95} K.Makarov, V. Melezhik, A. Motovilov, {\it Point interactions
in the  problem of three quantum particles with internal
structure}  Teor. Mat. Fiz. {\bf 102}, 2, 258-282 (Translation in
:Teor.and Math. Phys. {\bf 102} p. 188-207 (1995))

\bibitem{Novikov1} Communicated by S.P.Novikov.

\bibitem{Akhiezer} N.I.Akhiezer, I.M.Glazman, {\it Theory of  Linear
Operators in Hilbert Space}, Frederick Ungar Publ., New-York, vol.
1, 1966.

\bibitem{Demkov} Yu.N.Demkov, V.N.Ostrovskij, {\it Zero-range
potentials and  their applications in Atomic Physics},  Plenum
Press, NY-London, 1988 .

\bibitem{Krein-Langer} I.S.Iohvidov, M.G. Krein, H.Langer, {\it
Introduction to the  spectral theory of  operators in spaces with
an inderinite metric}, Mathematical Research, 9. Akademie-Verlag,
Berlin,  1982.

\bibitem{MPP}A. Mikhailova, B. Pavlov, L. Prokhorov,
math-ph/0312038, December 2003.

\bibitem{Krasn49} M. Krasnosel'skij {\it On selfadjoint extensions of
Hermitian Operators} (in Russian) Ukr. Matem. Journ {\bf 1} pp
21-38 (1949)

\bibitem{Gorbachuk} V.I.Gorbachuk {\it Boundary problems for differential-
operator equations}(in Russian) Kiev, Naukova Dumka 284 p, 1984.

\bibitem{Titchmarsh} E.C.Titchmarsh, {\it Eigenfunctions  expansions
associated with second  order differential equations}, Vol. 1,
Clarendon Press, Oxford, 1962.

\bibitem{Lax} P. Lax, R. Phillips {\it Scattering theory} Academic
press,New York,1967.

\bibitem{Pavlov89} B.Pavlov {\it Boundary conditions on thin
manifolds and the semiboundedness of the three-particle
Schroedinger operator with pointwise potential} Math. USSR
Sbornik, v64,1,pp 161-175 (1989).

\bibitem{JPA} V. Kruglov, B. Pavlov {\it Operator extension techniques
for resonance scattering of neutrons by nuclei}. arXiv math-phys.



\end{thebibliography}
}
\end{document}
