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\begin{document}

\title[On Rank One $ {\mathcal H}_{-3} $-Perturbations]{On Rank One $ {\mathcal H}_{-3} $-Perturbations\\
of Positive Selfadjoint Operators}

\author{P.\,Kurasov}

\address{ Dept. of Math., Stockholm University, 10691 Stockholm,
Sweden }

%\curraddr{}

\email{pak@matematik.su.se}

\thanks{Supported by The Royal Swedish Academy of Sciences,
Sweden-Japan Sasakawa Foundation and Gakushuin Univ. Foundation}

\author{K.\,Watanabe}

\address{Dept. of Mathematics, Gakushuin Univ.,
1-5-1 Mejiro, 
 Toshima-ku,
  Tokyo, 171-8588 Japan}

%\curraddr{}
\email{watanabe@math.gakushuin.ac.jp}
%\thanks{}





\begin{abstract}
	Rank one $ {\mathcal H}_{-3} $ perturbations of positive
selfadjoint operators are constructed using a certain
extended Hilbert space and regularization procedures.
Applications to Schr\"odinger operators with point interactions
are discussed.
\end{abstract}

\maketitle




\section{Introduction}
Selfadjoint operators with complicated spectral structure
can be obtained using finite rank perturbations of
well studied operators with rather simple structure of the
spectrum.
Nevertheless these perturbed operators are exactly solvable
in the sense that all eigenfunctions, spectral
decomposition and scattering matrix can be written
explicitly, sometimes using analytic functions.
Suppose that $ A $ is a certain selfadjoint operator
acting in the Hilbert space $ H $. Then its finite
rank perturbation is formally defined by the following
formula
\begin{equation} \label{haha}
A_{V} = A + V ,\end{equation}
where $ V $ is a finite dimensional operator.
The domains of the perturbed and unperturbed
operators coincide if the perturbation is a bounded
operator in the Hilbert space $ V \in {\mathcal B} ({\mathcal H}).$
But it is possible to consider more  general perturbations determined
by operators
 $ V $ acting in the scale of Hilbert
spaces associated with the original operator $ A$
\begin{equation} \label{haow}
 V : {\mathcal H}_{2} (A) \rightarrow {\mathcal H}_{-2} (A).
\end{equation}
In the latter case the perturbed operator can be defined
using the form perturbation technique and the extension
theory for symmetric operators. Really the operators
$ A $ and $ A_{V} $ restricted to the domain
$$ {\rm Dom} (A^0) = \{ \psi \in {\rm Dom} (A): \psi \in {\rm Ker} 
(V) \} $$
coincide. If the operator $ V $ is symmetric and finite dimensional
then the restricted operator has finite deficiency indices
and the resolvents of the perturbed and original operators
are related via Krein's formula.
This approach has been used to construct so-called
operators with point interactions 
\cite{albeverio,demkov,pavlov}.
Perturbations with the support on a discrete set of
points and/or on low dimensional manifolds have been studied
in detail, especially in applications to Schr\"odinger
operators.
Most general finite rank perturbations of
selfadjoint operators have been investigated recently.
It has been shown that standard perturbation theory allows one
to define only perturbations determined by operators
acting as follows $ {\mathcal H}_{1} (A) \rightarrow {\mathcal H}_{{-1}} 
(A) $,
so-called $ H_{-1} $ perturbations.
It is possible to generalize this approach to include more general
perturbations (\ref{haow}). The perturbed operator cannot be
determined uniquely in this case, only a finite parameter
family corresponding to the formal expression
(\ref{haha}) can be established. To determine the unique operator
one needs to use extra assumptions, for example scaling
properties of the original operator and perturbation.
This approach does not work for $ {\mathcal H}_{-3} $
perturbations determined
by the operators acting as follows
\begin{equation} \label{howo}
 V : {\mathcal H}_{3} (A) \rightarrow {\mathcal H}_{-3} (A).
\end{equation}
Really consider the one dimensional perturbation
formally determined by
\begin{equation} \label{oper}
A_\alpha = A + \alpha \langle \varphi, \cdot \rangle \varphi,
\end{equation}
where $ \varphi \in {\mathcal H}_{-3} (A)\setminus {\mathcal H}_{-2} (A).$
The restriction of the operator $ A $ to the domain
$ {\rm Dom} (A^0) = \{ \psi \in {\rm Dom} (A): \langle \varphi, \psi 
\rangle = 0 \} $
is essentially selfadjoint. Every selfadjoint
extension of this symmetric
operator coincides with the original operator $ A.$
Therefore the operator corresponding to (\ref{oper}) cannot
be defined in the original Hilbert space using standard
methods of extension theory. The aim of the current paper
is to define operator $ A_{\alpha} $ as a selfadjoint
operator in a certain extended Hilbert space.

Operators formally defined by (\ref{oper}) are important
in different applications. For example consider Laplace
operator with one point interaction acting in $ L_{2} ({\bf R}^3).$
The deficiency indices of Laplace operator restricted
to the set of functions vanishing at one point
are equal to $ (1,1) $.
All selfadjoint extensions of this symmetric operator
are rotationally invariant. Therefore no non spherically
symmetric point interaction for Laplace operator in $ {\bf R}^3 $
can be constructed. Different modifications of this approach
has been suggested. The first approach mathematically rigorous
uses  perturbations in
Pontryagin spaces, where the perturbed operator has been
defined in a certain extension of the original Hilbert space
with non positive definite scalar product.
Another approach has been used by physicists who suggested to consider
certain abstract boundary conditions 
connecting coefficients in the asymptotics of  wave functions
 without
paying any attention to the fact that the wave functions
corresponding to
the $p$-state have nonnormalizable
singularities of order $ r^{-2} $.
Therefore no scalar product space has been taken into
consideration.
Another constructive approach has been suggested by Yu.E.Karpeshina. 
In the first step a set of functions having the form of $p$-scattering
states is constructed. Then the selfadjoint operator
and the Hilbert space are defined using this set of scattering waves.
Recently I.Andronov suggested to use extended Hilbert spaces
to model nonspherically symmetric scatterers in $ {\bf R}^2 $. 
The latter model has the advantage of being exactly
solvable. It is very similar to generalized point interactions
introduced by B.Pavlov. We are going 
to use some of the ideas suggested to construct rank one
$ {\mathcal H}_{-3} $ perturbations.


It was proven that $ {\mathcal H}_{-2} $
perturbations can be defined uniquely using a certain regularization
procedure. It appears that a similar approach works for 
$ {\mathcal H}_{-3} $ perturbations. In our paper we are going to
show how to define arbitrary rank one $ {\mathcal H}_{-3} $-perturbations
of selfadjoint operators restricting our consideration to the case
of semibounded (positive) operators. The approach suggested can easily
be modified to include finite rank and not necessarily semibounded
operators. We are not going to discuss how to construct regularizations
and relations to symmetry properties of the problem. We are planning to
present these result in one of the forthcoming publications.
















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H.Holden, Solvable models in quantum mechanics, Springer, 1988.



\bibitem{demkov} Yu.N.Demkov and V.N.Ostrovsky,
      {Zero-range potentials and their applications in atomic
                physics},
   {Plenum Press},
  {New York and London},
  1988.

  \bibitem{pavlov} B.Pavlov, The theory of extensions and explicitly solvable
models, {\it Uspekhi Mat. Nauk}, {\bf 42}, 99-131, 1987.
\bibitem{pavlov2} B.Pavlov, Boundary conditions on thin manifolds and the
semiboundedness of the three-body Schr\"odinger operator with point potential, 
{\it Mat. Sb. (N.S.)}, {\bf 136}, 163-177, 1988. 

\bibitem{pavlovpopov}
B.S.Pavlov and I.Popov, 
   {An acoustic model of zero-width slits and the hydrodynamic
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 {\bf 86},
     {1991},
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  \bibitem{shondin}
Yu.Shondin,
   {Perturbation of differential operators on high-codimension
             manifold and the extension theory for symmetric linear
             relations in an indefinite metric space},
  {\it Teoret. Mat. Fiz.},
{\bf 92},
    {1992},
  {466--472}.
  
  \bibitem{simon}
B.Simon,
  {Spectral analysis of rank one perturbations and applications},
in: {Mathematical quantum theory. II. Schr\"odinger operators
             (Vancouver, BC, 1993)},
   {109--149},
  {\it CRM Proc. Lecture Notes},
  {\bf 8},
{AMS},
{Providence, RI},
1995.


  
   
   

\end{thebibliography}




\end{document} 

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