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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 $rr_{_{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,\,
\,\,-\infty0}}
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 , $Mt}} |\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{
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\end{document}