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Weyl-Titchmarsh function, boundary value problems, open systems, operator colligations, linear systems with boundary control, Schrodinger operators
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\begin{document}
\title{A Weyl-Titchmarsh Function \\
of an Abstract Boundary Value Problem, \\
Operator Colligations,
and Linear Systems \\
with Boundary Control}
\author{Vladimir Ryzhov}
\date{April 24, 2007}
\maketitle
%\begin{center}
%\textit{ Dedicated to the memory of Moshe~S.~Liv\v{s}i{c}
%(1917-2007),
% \\
%pioneer in nonselfadjoint operator theory}
%\end{center}
%
%\bigskip
\begin{abstract}
The paper defines the
Weyl-Titchmarsh function for an
abstract bo\-un\-da\-ry value
problem and shows that it coincides with
the transfer function of some explicitly
described linear
boundary control system.
%
%
%
On the ground of obtained results
we explore interplay among boundary value
problems, operator colligations, and the linear systems theory
and
suggest an approach to the study of boundary value problems
based on
the open systems theory due to M.~S.~Liv\v{s}i{c}.
%
%
Examples of boundary value problems for partial
differential equations and
calculations of their Weyl-Titchmarsh
functions are
offered as illustration.
%
%
In particular, we give an independent derivation
of the Weyl-Titchmarsh
function for the three dimensional Schr\"odinger operator
introduced by W.~O.~Amrein and D.~B.~Pearson.
%
Relationships to the Schr\"odinger operator with
singular potential supported by the unit sphere
are clarified and
other possible applications of the developed approach
in mathematical physics are noted.
%
\end{abstract}
%%% ----------------------------------------------------------------------
%%% ----------------------------------------------------------------------
\section*{Introduction and notation}
One of the mathematical folklore beliefs is the possibility to
interpret the Weyl-Titchmarsh function known in the theory of
Sturm-Liouville equation~\cite{Titch} as a transfer function of some
linear system.
%
Indeed, as follows from the definition, say for the formally
selfadjoint one-dimensional Schr\"odinger differential expression on
the half line
\begin{equation}\label{intro:1}
l[y] := -y''
+ q(x)y, \qquad x\in[0,\infty)
\end{equation}
in the case of limit circle at the infinity~\cite{Titch}, the
Weyl-Titchmarsh function~$m(\lambda)$, $\lambda \in \Complex$
relates values of solutions~$y_\lambda\in L^2(0,\infty)$ to the
equation~$l[y] = \lambda y$ and values of their
derivatives~$y_\lambda'$ at the origin~$x=0$ by the
formula~$y_\lambda'(0) = m(\lambda)y_\lambda(0)$.
%
The resemblance with the setting of system theory becomes clear if
one interprets the solution~$y_\lambda$ as the internal state of
some system with the state space~$L^2(0,\infty)$, and values
of~$y_\lambda$ at $x =0$ as the system's input.
%
Then~$y'_\lambda(0)$ are regarded as the system output obtained from
the input~$y_\lambda(0)$ by the transfer function~$m(\lambda) :
y_\lambda(0)\mapsto y_\lambda'(0)$.
%
%
These more or less empiric arguments are justified by the theory of
boundary control systems~\cite{CuZw, Lions, Sta} and by its counterpart,
the so-called BC-method of the inverse
problem theory~\cite{Belishev,Is}.
%
%
These theories deal in particular with linear systems governed by
partial differential operators defined on some domain~$\Omega$ in
the Euclidian space with the control and observation taking place at
the domain's boundary~$\partial\Omega$.
%
%
Applied to~(\ref{intro:1}),
the positive half-axis~$(0,\infty)$ is treated as the domain~$\Omega$,
whereas the role of the boundary~$\partial\Omega$ is played by the point~$x =0$.
%
%
Then the inverse problem for the Schr\"odinger
expression~(\ref{intro:1}) consisting in the recovery of
potential~$q$ given the Weyl-Titchmarsh function~$m(\lambda)$ is
interpreted as reconstruction of a linear system from the knowledge
of its transfer function.
%
%
The generalization of this concept to the multidimensional setting is
quite obvious, at least for smooth bounded domains~\cite{SU}.
%
However, a simple argument based on the imbedding theorems shows
that in contract with the one-dimensional situation,
values of the Weyl-Titchmarsh function in this case
ought to be unbounded
operators acting in~$L^2(\partial\Omega)$.
%
Indeed,
the Weyl-Titchmarsh function maps boundary traces
of solutions of the second order partial differential equation
in the domain~$\Omega$ to traces of their normal derivatives.
%
According to imbedding theorems and properties of the trace map,
values of the Weyl-Titchmarsh function decrease smoothness,
therefore
are unbounded operators in~$L^2(\partial\Omega)$.
%%==========================================================
The concept of Weyl-Titchmarsh function is known for the general
Sturm-Liouville problem, difference operators, orthogonal
polynomials, Hamiltonian systems~\cite{At, BE, HS}, various classes
of extensions of symmetric operators studied within the theory of
boundary and quasi-boundary triplets~\cite{BL, DM1}, and some elliptic
partial differential operators~\cite{BM, GMZ, SU}, where it is
conventionally called the Dirichlet-to-Neumann map.
%
The recent remarkable work~\cite{AmP} of W.~O.~Amrein and
D.~B.~Pearson
develops the notion of Weyl function for the three dimensional
Schr\"odinger operator defined in the whole space.
%
Their approach differs from mentioned above in that there is no
boundary
given
\textit{a priori}, and the authors have to introduce it
artificially.
%
By doing so they arrive at the Weyl-Titchmarsh function
defined as bounded operator-function acting in~$L^2(S_1)$, where
$S_1$ is the unit sphere in~$\Real^3$.
%
From the systems theory point of view their constriction
is equivalent to the introduction of some means
of control and observation into the otherwise closed
system described by the ``unperturbed'' Schr\"odinger
operator.
%
With these control and observation in place,
the Weyl-Titchmarsh function
again can be likened to the transfer function of so created linear
system.
%
We elaborate more on this example
in the last section of paper.
%
%
%%===================================================
%
%
All instances mentioned above share one feature that again hints at
the close relationship between Weyl-Titchmarsh functions and linear
systems.
%
%
% It is the Herglotz property of Weyl-Titchmarsh functions.
%
%
%
They all are analytic and possess the positive
imaginary part in the upper half plane.
%
Functions with this property
are called Herglotz or $R$\nobreakdash-functions
and
an extensive theory has been developed not only for scalar
$R$\nobreakdash-functions, but also for their operator-valued analogues.
%
In the
cases cited above all respective Weyl-Titchmarsh functions are Herglotz
functions, either scalar, or matrix-, or operator-valued, depending
on the nature of the problem.
%
As it turns out, $R$\nobreakdash-functions are also well known in
the
system theory.
%
Namely, any operator-valued $R$\nobreakdash-function is the transfer
function of a certain linear
system of special type, the passive conservative resistance systems
studied in the theory of electrical circuits~\cite{Ar}.
%
%%=========================================================
These observations suggest that the relationship
between the theory of Weyl-Titch\-marsh functions and the theory of
linear systems is of general nature.
%
%
%
The systematic treatment
of this
topic is hindered by
the lack of a
convenient representation of Weyl-Titchmarsh functions.
%
%
%
Indeed, neither abstract forms of Herglotz integral, nor more
detailed representations found in works~\cite{AmP, BM} for some
special cases of partial differential operators, are sufficiently
explicit to serve as a foundation of the general theory.
%
%
The present paper links the Weyl-Titchmarsh functions theory with
the theory of linear systems in precise manner.
%
%
Our study is based on the definition
of abstract boundary value problems (BVPs)
that the notion of Weyl-Titchmarsh function
is applicable to.
%
All examples mentioned above belong to this class.
%
We show that
the Weyl-Titchmarsh functions under consideration
can be identified with characteristic functions
of certain operator colligations~\cite{Bro,Liv}.
%
This very fact allows
us to pass the treatment
to the setting of systems theory.
%
%
More precisely, we show that any BVP from the class
studied in the paper
corresponds to some
linear boundary control systems such that
the Weyl-Titchmarsh fucntion of
the former coincides
with the transfer
function of the latter.
%
%
%
%
%%==========================================================
%
%
In order to dispel possible confusion the reader may have at this
point with regard to the paper's content, a clarification is needed.
%
As mentioned above,
the relationship between boundary value problems
and the systems theory has been known for a long time and was
successfully captured in the notion of boundary control systems, see
for instance~\cite{CuZw, Lions, Sta}.
%
%
The main premise of the paper, however, is not based on these
results.
%
Instead, the research conducted below
makes use of the
open systems theory due to
M.~S.~Liv\v{s}i{c}~\cite{Liv}.
%
Nevertheless, in the course of investigation we discover
a natural connection between the
theory of open systems~\cite{Liv}
and the ``standard''
linear boundary control system theory~\cite{CuZw, Lions,Sta}.
%
It is not without interest to notice that
this connection is
the gist of the method of
reciprocals recently suggested in~\cite{Cur,Sta}.
%
This method
allows one to reduce the study of
a system with boundary control, which
is described by three typically unbounded mappings,
the interior,
the control and the observation operators,
to the study of another linear system with
the same properties, but whose operators are bounded.
%
%
To give an illustrative example,
let us turn to the Schr\"odinger
expression~(\ref{intro:1}) again.
%
%
The boundary control system associated with~(\ref{intro:1}) consists
of two Hilbert spaces, $ H = L^2(0,\infty)$ and $E = \Complex$, and
three operators $L : y\mapsto l[y]$, $C : y \mapsto y(0)$, $ O :
y\mapsto y'(0)$ defined on sufficiently smooth functions~$y \in
L^2(0,\infty)$, the interior,
the control and the observation operators, respectively.
%
It turns out that if the restriction of~$L$ to the null
set~$\ker(C)$ is a selfadjoint
boundedly invertible operator~$L_0$, and~$C$ restricted to~$\ker(L)$
possesses a bounded left inverse~$C^{[-1]}$, then
the study of system~$\{L, C, O\}$
can be reduced to the study of four operators~$L_0^{-1}$,
$C^{[-1]}$, $O L_0^{-1}$, and $O C^{[-1]}$
that in turn define another boundary control system,
called the reciprocal.
%
These considerations are not limited to the one-dimensional setting
of~(\ref{intro:1}).
%
We show that any BVP of the class introduced
in the paper simultaneously defines
a linear boundary
control system and its reciprocal.
%
The latter is described
as an open system of M.~S.~Liv\v{s}i{c}.
%
The transfer functions of these systems
essentially coincide with the Weyl-Titchmarsh function
of
the original BVP.
%
%
%%===============================================================
%
%
Interconnections among boundary value problems, open systems, and
linear boundary control theory
clarify in what sense the Weyl-Titchmarsh function can be interpreted
as a transfer function.
%
At the same time the study
provides alternative perspective on the
individual topics involved in the research.
%
%
Apart from the mentioned above method of reciprocals, our
considerations link the boundary control systems theory
with the theory
of almost solvable extensions
of symmetric operators~\cite{DM1, GG},
clarify some principles of the inverse spectral
theory~\cite{BRF, Is},
and demonstrate applicability
of the null extensions approach utilized in the paper
to the study of BVPs
for partial differential operators
traditionally regarded as singular~\cite{AlbevKur}.
%
%
%
In particular, below we derive the Weyl-Titchmarsh function of
the Schr\"odinger
operator introduced by
W.~O.~Amrein and D.~B.~Pearson
in~\cite{AmP}
by independent considerations
based on the obtained results.
%
%%============================================================
Let us give a brief outline of the paper.
%
%
The Section~1 introduces an important notion of the so-called null
extensions of linear operators that appears to be a convenient
abstraction for our purposes.
%
The null extensions based approach
provides a coherent methodological framework that unifies
diversity of topics studied in the paper.
%
Here we describe a class of boundary value problems under
consideration, define strong and weak solutions, discuss the Green's
identity and solvability criteria, give a few equivalent
definitions of the Weyl-Titchmarsh function accompanied by a brief
discussion of its properties, and briefly consider the extension
theory of symmetric operators~\cite{AS,Birman,Kr,Vi} within
in the paper's
context.
%
%
Preliminary variant of some results contained in the Section~1
appears in the paper~\cite{Ryz2}.
%%===============================================================
Sections~2 is devoted to linking the null extension
associated with a BVP
to the so-called Brodski\u{\i}-Liv\v{s}ic operator
colligation~\cite{Bro,Liv}
with the characteristic
function uniquely determined by
the BVP's Weyl-Titchmarsh function.
%
%
Properties of colligations corresponding to
null extensions under consideration are described.
%
Guided by the works of M.~S.~Liv\v{s}ic on the
theory of open systems~\cite{Liv} we show that the
studied BVPs can be put into an
one-to-one correspondence with
a certain class of open systems.
%
%
%
It is shown in Section~3 that this type of open systems
is comprised of reciprocals~\cite{Cur, Sta}
of linear boundary control systems whose transfer functions
coincide with the Weyl-Titchmarsh functions of the original BPVs.
%
%
Thus, we establish connections between null extensions (with
their corresponding BVPs), the open system theory,
and with theory of
linear systems with boundary control.
%
%
With this result in place, the question of interpretation
of the Weyl-Titchmarsh function
as the transfer functions of some boundary control system
becomes settled on the abstract level.
%
As seen from this explanation and will be elucidated more in the
main text, the theory of open systems due to M.~S.~Liv\v{s}ic's
is a crucial component of the study.
%
It brings together the theory of
boundary value problems and the
theory of boundary control systems
in a fruitful
manner that emphasizes
the unifying role the principal object of the study, the
Weyl-Titchmarsh function, plays in both fields.
%
%
%
%%===============================================================
%
The last section is devoted to
the Schr\"odinger operator~$\mathscr L = -\Delta + q(x) $
in the three
dimensional space.
%
The Weyl-Titchmarsh function of~$\mathscr L$
under assumption~$q\in L^\infty(\Real^3)$
was devised by W.~O.~Amrein and D.~B.~Pearson
in~\cite{AmP}
where it is called the $M$-function.
%
We show that the same result can be obtained within the
paper's framework if in addition
the function~$q$ is smooth.
%
%
To that end
we introduce the external control
over the system
described by~$\mathscr L$.
%
The control is realized
as
single layer potentials with densities
supported on a smooth closed surface.
%
Then we explicitly calculate all objects
of the general theory, including the transfer
function of the obtained linear system.
%
%
%
By
direct comparison with the research of
W.~O.~Amrein and D.~B.~Pearson
we show that it coincides with the
$M$-function obtained in their work~\cite{AmP}.
%
This fact allows one to calculate
the Weyl-Titchmarsh function
of multidimensional Scr\"odinger
without resorting to the
limiting procedure analogous to
the one-dimensional case~\cite{Titch}
and constructed in~\cite{AmP}.
%
More precisely, we show that
the $M$-function from~\cite{AmP}
is in fact the operator of single layer potential
associated with the Green's function of~$\mathscr L = -\Delta + q(x)$
acting on the space~$L^2(S_1)$, where $S_1$
is the unit sphere in
$\Real^3$.
%
The author is grateful to Prof. R.~Froese who pointed out
that this result can be derived
directly from the properties
of Dirichlet-to-Neumann maps of boundary value
problems
for the interior and the exterior of the unit ball.
%
%
Irrespective of possible
consequences for the linear system theory
and the theory of multidimensional Schr\"odinger operator,
this may indicate some relevance
of the approach employed in the
paper to the theory of
partial differential equations.
%
%
%
A few remarks regarding the relationship
between singular perturbations as per~\cite{AlbevKur} and the
linear boundary control systems theory conclude the section.
%
%
%%===============================================================
Results of the paper have common points with other disciplines.
%
%
%
%
One of them is the extension theory of symmetric operators,
particularly based on the theory of almost solvable extensions and
boundary and quasi-boundary triplets~\cite{BL, DM1, GG}.
%
The framework of the paper furnishes an abstract
foundation for the study of characteristics of
such extensions.
%
It expands
the existing theory to cover more
generic situations where assumptions
of works~\cite{BL, DM1, GG} are not fulfilled.
%
They include
non-elliptic, non-semibounded operators with infinite
deficiency indices, instances where boundary mappings
are non-surjective,
problems with the spectral parameter in boundary
conditions, etc.
%
%
%
Results of the paper make various methods
considered the systems theory
specific~\cite{CuZw,Part,Sta}
available to specialists in the area of boundary value problems.
%
%
%
For example, the analysis of BVPs with the spectral parameter in
boundary conditions can be regarded as the problem of
linear systems
with non-trivial feedback~\cite{CuZw, Part, Part}.
%
%
One more connection of the paper's topics
to other disciplines is due to the
formula~(\ref{KreinResolventFormulaForZ=0}) below.
%
%
It allows one to treat the class BVPs studied in the paper
by equating them with additive bounded perturbations of
bounded operators,
thereby greatly simplify their study.
%
%
In particular, the functional model of
nonselfadjoint perturbations of a selfadjoint operator from the
paper~\cite{Na} is directly applicable in the context
of the paper.
%
It covers
situations that are not handled by the
model from~\cite{Ryz} limited to
the case of almost solvable
extensions.
%
%
%
%
The last area where the paper's approach may
prove fruitful
is the
theory of singular perturbations of
differential operators~\cite{AlbevKur} including
its relationship with
the theory of generalized optimal control, see~\cite{Lyashko}
and references therein.
%
A detailed account of these relationships
will be given elsewhere.
%%=============================================================
\smallskip
The author would like to express his sincere gratitude to Prof.
S.~N.~Naboko for the interest to the work and continual
encouragement, and to tender thanks to Prof. M.~I.~Belishev for the
introduction to the subject and motivating discussions. The author
is grateful to Dr.~A.~V.~Kiselev for the attentive reading of the
manuscript and many useful remarks.
%%================================================
\smallskip
%
A few words regarding notation and conventions accepted in the paper
are in order.
%
For two Hilbert spaces~$H_1$ and~$H_2$ the
sign~$A : H_1 \to H_2 $ is used to denote a bounded linear
operator~$A$ defined everywhere in~$H_1$ with the range in the
space~$H_2$.
%
Symbols~$\Real$, $\Complex$, $\IM (z)$ stand for the real axis, the
complex plane, and the imaginary part of a complex number~$z\in
\Complex$, respectively.
%
Furthermore, $\Complex_\pm := \{ z \in \Complex \; | \;\pm\IM (z) >
0 \}$.
%
The domain, the range and the null set of a linear operator~$A$ are
denoted $\D(A)$, $\Ran(A)$, and $\ker (A)$; the symbol~$\rho(A)$ is
used for the resolvent set of~$A$.
%
For a Hilbert space the term \textit{subspace} denotes a closed
linear set.
%
The orthogonal complement to a linear set in Hilbert
space is always closed, i.~e. is a subspace.
%
If the operator~$A$ acting on the Hilbert space~$H$ is closed, the
null set~$\ker(A)$ is a subspace in~$H$.
%
All Hilbert spaces below are separable.
%%====================================================================
%%
%%====================================================================
\section{A boundary value problem and its Weyl-Titchmarsh \\ functions}
There exist a few ways to introduce the class of problems studied in
the paper.
%
The most straightforward approach seems to be based on the concept
of null extensions of a linear operator.
%
Apart from its simplicity, it clarifies the construction of
associated boundary value problem and underlines the close
relationship between these problems and the theory of linear open
systems.
%
Furthermore, the notion of null extensions naturally leads to the
definition of Weyl-Titchmarsh functions.
%
Intrinsic relationships among null extensions, boundary value
problems, and Weyl-Titchmarsh functions are in the main focus of
this section.
%%=========================================================================
%%
%%=========================================================================
\subsection{Null extensions}
%
The formal definition of null extensions is as follows.
\begin{defn}
Let~$T$ be a linear operator on the Hilbert space~$H$ with
domain~$\D(T)$. Linear operator~$S$ is called a \textit{null
extension} of~$T$ if $\D(T)\subset\D(S)$, $Sx = Tx$ for $x \in
\D(T)$, and~$Sx =0$ for $x\in\D(S)\setminus \D(T)$.
\end{defn}
%%========================================================================
Let~$H$ be a Hilbert space.
%
Suppose~$\HH\subset H$ is an arbitrary linear manifold of elements
in~$H$, and $A_0$ is a non-bounded linear selfadjoint operator
in~$H$ defined on the domain~$\D(A_0)$.
%
Everywhere in the paper we assume that~$A_0$ is boundedly invertible
with the inverse~$A_0^{-1} : H \to H$ and the pair~$\{A_0, \HH\}$
satisfies the following assumption.
%%=========================================================================
\begin{assump}\label{Assumption}
\begin{enumerate}
%
\item Intersection of~$\HH$ and~$\D(A_0) = A_0^{-1} H$ is trivial
, i.~e. $\D(A_0) \cap \HH = \{0\}$.
%
\item There exists a linear operator~$\gamma$
that maps~$\HH$ to some
auxiliary Hilbert space~$E$.
%
The linear set~$\gamma \HH$ is dense in~$E$
and the only solution to the equation~$\gamma h = 0$, $h\in\HH$
is the null vector~$h=0$.
%
\item The left inverse of~$\gamma$ is bounded.
%
Denote it~$\Pi : E\to H$ so that
$\Pi\gamma h = h$ for any~$h\in\HH$.
\end{enumerate}
\end{assump}
%%==============================================================
Basic objects of the paper are the null extension~$A$ of the
operator~$A_0$
to the domain~$\D(A) := \D(A_0) \dot{+}\HH$ and the
the null extension~$\Gd$ of the operator~$\gamma$ originally
defined on the set~$\HH$ to the linear manifold~$\D(A)$.
%
In other words, $A$ and~$\Gd$ are
defined on~$\D(A) := \D(A_0) \dot{+}\HH$ by
%
\[
A : x + h \mapsto A_0 x, \qquad \Gd : x + h \mapsto \gamma h,
\qquad x\in \D(A_0), \quad h \in \HH
\]
%
Since~$A_0$ is selfadjoint, its domain~$\D(A_0)$ is dense in~$H$,
therefor $A$ and $\Gd$ are densely defined.
%
At the same time they are not assumed closed on~$\D(A)$, or even
closable in~$H$.
%
By the construction, the domain~$\D(A_0)$ is the null set of~$\Gd$,
$\D(A_0) = \ker(\Gd)$ where $\ker(\Gd) = \{ u \in \D(A) \; |\; \Gd u =0\}$.
%
Vectors~$h\in\HH$ are distinguished from other elements of~$\D(A)$
by the equality~$\Pi \Gd h = h$ or by its equivalent~$\Gd \Pi
\varphi = \varphi$, where~$\varphi = \Gd h$ with some~$ h \in \HH$.
%
These observations lead to the representation for~$\D(A) = \D(A_0)
\dot{+}\HH$
\[
\D(A) = \{ A_0^{-1} f + \Pi \varphi \; |\; f\in H, \varphi \in \Gd \HH\}
\]
accompanied by following refined definitions of~$A$ and~$\Gd$
\[
A : A_0^{-1} f + \Pi \varphi \mapsto f, \qquad
\Gd : A_0^{-1} f + \Pi \varphi \mapsto \varphi,
\qquad f\in H, \quad \varphi \in \Gd \HH
\]
and equalities
\[
\begin{aligned}
\ker(\Gd) & = \{A_0^{-1} f\; |\; f\in H\}, &
\quad A|_{\ker (\Gd)} & = A_0, & \\
\ker(A) & = \{\Pi \varphi\;|\; \varphi \in \Gd \HH\}, &
\quad \Gd|_{\ker (A)} & = \gamma. &
\end{aligned}
\]
%
%
Since~$A$ and~$\Gd$ need not be closed, $\ker(\Gd)$ and $\ker(A)$
are not necessarily subspaces in~$H$.
%%============================================================
For purposes of the paper we have to introduce one more operator
in addition to~$A$ and~$\Gd$.
%
Let us fix an arbitrary symmetric map~$\Lambda$ on the space~$E$
with the dense domain~$\D(\Lambda) = \Gd\HH$ and define the linear
operator~$\Gn$ on~$\D(A)$ by
%
\begin{equation}\label{DefinitionOfGammaN}
\Gn : A_0^{-1} f + \Pi \varphi \longmapsto \Pi^*f + \Lambda\varphi,
\qquad f\in H, \quad \varphi \in \Gd \HH.
\end{equation}
%
where~$\Pi^* : H \to E$ is the adjoint to~$\Pi$.
%
From~(\ref{DefinitionOfGammaN}) with~$f=0$ follows~$\Gn \Pi\varphi =
\Lambda\varphi$, where $\varphi \in \Gd \HH$.
%
Assuming $\varphi = \Gd h$ with~$h\in \HH$ we conclude that $\Gn h =
\Lambda\Gd h$ for any~$h\in\HH$.
%
Further, $\Pi^* = \Gn A_0^{-1}$, as seen
from~(\ref{DefinitionOfGammaN}) with $\varphi = 0$.
%
Note again that~$\Gn$ and~$\Lambda$ are not assumed closed nor
closable.
%%===============================================================
The rationale behind the definition~(\ref{DefinitionOfGammaN}) is
the special role operators~$\Gd$ and~$\Gn$ play as ``boundary
operators'' that map $\D(A)$ into the ``boundary space'', the
space~$E$.
%
More precisely, for the collection $\{A, \Gd, \Gn, H, E\}$ the
following version of the Green's formula holds.
\begin{thm}
For any ~$u,v\in\D(A)$
\begin{equation} \label{GreenFormula}
(Au,v)_{H} - (u,Av)_{H} =
(\Gn u, \Gd v)_E -(\Gd u, \Gn v)_E , \qquad u,v \in \D(A),
\end{equation}
\end{thm}
%%===============================================================
\begin{proof}
Let~$u\in\D(A)$ be a vector~$u = A_0^{-1} f+ h$, where $f\in H$,
$h\in\HH$.
%
Then
\[
\begin{aligned}
(Au, u) - (u, Au)
& = (A(A_0^{-1} f+ h), A_0^{-1} f+ h) - (A_0^{-1} f+ h, A(A_0^{-1} f+ h))
\\
&= ( f, A_0^{-1} f+ h) - (A_0^{-1} f+ h, f) = (f,h) - (h,f).
\end{aligned}
\]
%
From the other side, since~$\Pi\Gd h = h$, $\Gn h = \Lambda \Gd h$,
and $\Lambda$ is symmetric,
%
\[
\begin{aligned}
(\Gn u, & \Gd v)_E -(\Gd u, \Gn v)_E
\\
& = (\Gn(A_0^{-1} f+ h), \Gd(A_0^{-1} f+ h)) - (\Gd(A_0^{-1} f+ h), \Gn(A_0^{-1} f+ h))
\\
& = (\Pi^*f +\Gn h, \Gd h) -(\Gd h, \Pi^* f + \Gn h)
\\
& = (f, \Pi \Gd h) - (\Pi\Gd h, f) + (\Lambda \Gd h, \Gd h) - ( \Gd h,\Lambda \Gd h)
= (f,h) - (h,f).
\end{aligned}
\]
The proof is complete.
\end{proof}
%%===============================================================
%%
%%===============================================================
\subsection{Boundary value problem}
Considerations above, the Green's identity~(\ref{GreenFormula}) in
particular,
reveal similarity of the introduced objects to the classical setting
of boundary value problems.
%
This analogy suggests the following definition of abstract spectral
value problem for operator~$A$,
%
\begin{equation}\label{NE_SpectralBVP}
\left\{\quad
\begin{array}{l}
(A - zI) u = f \\
\Gd u = \varphi
\end{array}
\right.
\end{equation}
%
Here $f\in H$ and $\varphi\in E$ are given elements of responding
spaces, the vector~$u\in\D(A)$ is unknown, and the complex
number~$z\in\Complex$ is the spectral parameter.
%
%
Before we proceed to the main result regarding solvability
of~(\ref{NE_SpectralBVP}) note that the condition~$\Gd u =\varphi$
im\-po\-sed on~$u\in \D(A)$ implies the inclusion~$\varphi\in\Gd
\HH$.
%
However, a weak variant of~(\ref{NE_SpectralBVP})
that allows one to extend the concept of solutions
to~(\ref{NE_SpectralBVP}) to the case of all~$\varphi\in E$
can be offered.
%%=============================================================
\begin{defn}\label{DefinitionWeakSolution}
Given~$f\in H$, $\varphi \in E$ the vector~$u\in H$ is called the
\textbf{weak solution} to the problem~(\ref{NE_SpectralBVP}) if
\begin{equation}\label{FormulaWeakSolution2}
(u, (A_0 -\bar z I)v ) = (f,v) + (\varphi, \Gn v),\quad
\text{ for any }\;
v\in
\D(A_0).
\end{equation}
\end{defn}
%%===============================================================
The definition is justified by the next observation.
%
If the vector~$u\in H$ solves the equation~(\ref{NE_SpectralBVP})
with some~$f\in H$,
$\varphi \in E$, then for any~$v\in \D(A_0)$ by virtue of the
Green's formula~(\ref{GreenFormula}),
\[
\begin{aligned}
(u, (A_0 -\bar z I)v) & = (u, A_0 v ) - (z u, v) =
(u, A_0 v ) + ( f - A u,v)
\\
& = (f,v) + (u, A v ) - (A u,v)
= (f,v) + (\Gd u, \Gn v ) - (\Gn u, \Gd v)
\\
& = (f,v ) + (\varphi, \Gn v).
\end{aligned}
\]
%
Thus, any solution to~(\ref{NE_SpectralBVP})
at the same time
solves the problem~(\ref{FormulaWeakSolution2}).
%
%
According to the established terminology,
sometimes in the sequel the problem~(\ref{FormulaWeakSolution2})
is referred to as the variational form of~(\ref{NE_SpectralBVP}).
%
%%===============================================================
Next result
concerns the solvability of~(\ref{NE_SpectralBVP}) for~$f =0$.
%
In
this case
all solutions to~(\ref{NE_SpectralBVP}) are obtained
from vectors~$h\in \HH$ by the bounded
map~$h \mapsto (I - zA_0^{-1})^{-1} h $.
%%===============================================================
\begin{lem}\label{NE_Lemma}
Suppose~$z\in\rho(A_0)$.
The map
\begin{equation*} %% \label{HtoHZMap}
h \longmapsto (I - zA_0^{-1})^{-1} h, \quad h\in \HH, \, z\in\rho(A_0)
\end{equation*}
establishes an one-to-one correspondence between $\HH = \ker (A) $
and $\ker (A - zI)$.
%
For vectors~$h\in \HH$ and $h_z := (I - zA_0^{-1})^{-1} h \in
\ker(A-zI)$ the equality $\Gd h = \Gd h_z$ holds.
%
If two arbitrary vectors~$u_1, u_2 \in \ker(A-zI)$ satisfy the
condition~$\Gd u_1 = \Gd u_2$, then $u_1 = u_2$.
\end{lem}
%%===============================================================
\begin{proof}
Since~$A$ is an extension of~$A_0$, we have
$(A - zI)(A_0 - zI)^{-1} =I$ where~$z\in\rho(A_0)$.
%
Therefore, for any~$h\in\HH$
\[
(A - zI)(I -z A_0^{-1})^{-1} h = (A - zI)[I + z ( A_0 - zI)^{-1}] h
= (A -zI) h + z h = 0. \]
%
Conversely, if $h_z \in \ker (A - zI)$ and
$h := (I - zA_0^{-1})h_z$, then
%
\[
Ah = A h_z - z A A_0^{-1}h_z = (A - zI)h_z = 0.
\]
%
The equality $\Gd h = \Gd h_z$ follows from the
relations~$h = (I - zA_0^{-1})h_z$ and $\Gd \D(A_0) =0$.
%
Finally, if $u_1, u_2\in \ker(A-zI)$ then $u_j = (I -
zA_0^{-1})^{-1}h_j $, $j=1,2$ with some~$h_1, h_2\in \HH$.
%
Assumption~$\Gd u_1 = \Gd u_2$ leads to the equality~$\Gd h_1 -\Gd
h_2 =0$.
%
Applying operator~$\Pi$ to both sides of this identity and recalling
that~$\Pi \Gd h =h$ for any~$h\in \HH$, we conclude that~$h_1 =
h_2$, hence $u_1 = u_2$.
The proof is complete.
\end{proof}
%%===============================================================
Now we can formulate the solvability criteria for the
problem~(\ref{NE_SpectralBVP}) with~$f\neq 0$.
%
As one may expect by analogy with the classical
boundary value problems theory, the solution
to~(\ref{NE_SpectralBVP}) is a sum of solutions to
two semi-homogeneous problems obtained from~(\ref{NE_SpectralBVP})
by assuming~$f=0$, $\varphi \neq 0$ and $f\neq 0$, $\varphi =0$.
%
%%============================================================
\begin{thm}\label{NE_BVPTheorem}
Suppose~$z\in\rho(A_0)$, $\varphi \in \Gd \HH$, $f\in H$.
Then the
solution
$u = u_z^{f,\varphi}$ to the problem~(\ref{NE_SpectralBVP}) exists
and is unique.
%
It is represented in the form
\begin{equation}\label{NE_BVPsolution}
u_z^{f,\varphi} = (A_0 - zI)^{-1}f + ( I - zA_0^{-1})^{-1} \Pi
\varphi.
\end{equation}
%
If~$\varphi \in E$ is arbitrary, the vector~$u_z^{f,\varphi}$
defined
by~(\ref{NE_BVPsolution}) is a weak solution
to~(\ref{NE_SpectralBVP}).
\end{thm}
%%==================================================================================
\begin{proof}
Uniqueness of the solution~(\ref{NE_BVPsolution}) is easily verified.
%
Assume that for $z\in\rho(A_0)$, $\varphi \in \Gd\HH$, and $f\in H$
there exist two solutions~$u_1, u_2 \in \D(A)$ to the
problem~(\ref{NE_SpectralBVP}).
%
Then their difference~$u_0 := u_1 - u_2$ satisfies both
equations~(\ref{NE_SpectralBVP}) with $f = 0$, $\varphi = 0$.
%
Because~$\Gd u_0 =0$, the vector~$u_0$ belongs to the domain of
operator~$A_0$.
%
Then it follows from~(\ref{NE_SpectralBVP}) than $(A - zI)u_0 = (A_0
- zI)u_0 = 0$.
%
Therefore, $u_0 = 0$ since $z \in \rho(A_0)$.
%%==================================================================================
Now consider (\ref{NE_BVPsolution}) with~$\varphi \in \Gd \HH$.
%
According to Lemma~\ref{NE_Lemma}, the second summand
in~(\ref{NE_BVPsolution}) belongs to~$\ker(A -zI)$.
%
The equalities~$(A - zI)u_z^{f,\varphi} =(A - zI)(A_0 - zI)^{-1}f = f$
follow from the
definition of~$A$.
%
Further, from Lemma~\ref{NE_Lemma} with~$h = \Pi\varphi \in \HH$
we have
\[
\Gd u_z^{f,\varphi} = \Gd (I - zA_0^{-1})^{-1}\Pi \varphi =
\Gd [I + z (A_0 - zI)^{-1}]\Pi \varphi =
\Gd \Pi
\varphi = \varphi.
\]
%
%%=============================================================
Let us now verify the last statement of Theorem.
Suppose~$\varphi \in E$ and define
the element~$u_z^{f,\varphi}\in H$ by the formula~(\ref{NE_BVPsolution}).
%
Then for~$z\in\rho(A_0)$ and~$v\in\D(A_0)$,
\[
\begin{aligned}
(u_z^{f,\varphi}, \, & (A_0 -\bar z I)v) \\
& =
((A_0 - zI)^{-1}f,(A_0 -\bar z I)v ) + (( I - zA_0^{-1})^{-1} \Pi \varphi,(A_0 -\bar z I)v )
\\
& = (f,v) + ( \varphi,\Pi^* ( I - \bar zA_0^{-1})^{-1} (A_0 -\bar z I)v
) = (f,v) + (\varphi , \Gn v),
\end{aligned}
\]
since $\Pi^* ( I - \bar zA_0^{-1})^{-1} (A_0 -\bar z I)v = \Gn
A_0^{-1} ( I - \bar zA_0^{-1})^{-1} (A_0 -\bar z I)v = \Gn v$.
%
According to Definition~\ref{DefinitionWeakSolution}, the element
$u_z^{f,\varphi}$ is the weak solution to~(\ref{NE_SpectralBVP}).
%
The proof is complete.
\end{proof}
%%=================================================
%
In the light of Theorem\ref{NE_BVPTheorem}
the decomposition~$\D(A) = \D(A_0) \dot{+}\HH$
is a reminiscence of the classical approach
of boundary value problems theory
(see~\cite{BergmanSchiffer,Weyl} for
instance).
%
We look for a solution to~(\ref{NE_SpectralBVP})
in the form
of sum of solutions to two semi-homogeneous
problems
obtained from~(\ref{NE_SpectralBVP})
by putting~$f=0$, $\varphi \neq 0$
and $f \neq 0$, $\varphi = 0$.
%
%%=================================================
The last result of this section is the direct description of
boundary value
problems~(\ref{NE_SpectralBVP}) that correspond to null extensions
satisfying Assumption~\ref{Assumption}.
%
It allows one to quickly verify whether results of the paper
are applicable to a given BVP.
%%==========================================================
%
\begin{thm} \label{TheoremA1}
Let $H$, $E$ be two Hilbert spaces and $A$, $\Gd$ are
two linear operators with the domain~$\D(A)$ dense
in~$H$ and with the ranges $\Ran(A) \subset H$, $\Ran(\Gd) \subset E$.
%
Operators~$A$, $\Gd$ define the spectral boundary value
problem
\begin{equation} \label{A_BVP1}
\left\{
\begin{array}{l}
(A - zI) u = 0 \\
\Gd u = \varphi
\end{array}
\right.
\end{equation}
where $\varphi \in E$ is given, and $u\in \D(A)$ is unknown.
%
Assume the next conditions are fulfilled:
\begin{enumerate}
\item Restriction of~$A$ to the domain~$\D(A)\cap \ker (\Gd)$
is a (necessarily unbounded)
selfadjoint operator~$A_0$ with the bounded inverse~$A_0^{-1}$
defined everywhere on~$H$.
\item Linear manifold
$\Gd \D(A)$ is dense in $E$.
\item
The Green's formula~(\ref{GreenFormula}) is valid for all $u, v \in \D(A)$
\[
(Au , v )_H - (u, Av)_H = (\Gn u, \Gd v)_E - (\Gd u, \Gn v)_E
\]
with some linear operator~$\Gn$
defined on~$\D(A)$ with the range~$\Ran(\Gn)\subset E$.
\end{enumerate}
%
Then the
domain~$\D(A)$ is represented as direct
sum~$\D(A) = \D(A_0) \dot{+}\HH$
where $\HH$ is defined as the null set of $A$.
%
The operator $A$ is the null
extension of~$A_0$ to the domain~$\D(A)$
satisfying Assumption~\ref{Assumption}
with~$\gamma = \left.\Gd\right|_{\ker (A)}$.
%
Moreover, the mapping~$\Gn A_0^{-1}$ defined on~$H$ is bounded.
%
Its
adjoint~$\Pi := (\Gn A_0^{-1})^*$
is the left inverse to~$\gamma$
and~$\Lambda := \Gn \Pi$
is symmetric on~$\D(\Lambda)= \Gd \HH$.
%
\end{thm}
%%=======================================================================
\begin{proof}
%
Since $\ker(A_0) =\{0\}$ and $A_0 \subset A$, it follows
form the invertibility of~$A_0$ that $\D(A_0)\cap \HH =\{0\}$
and $(A - zI)(A_0 - zI)^{-1} f = f $ for
any $f\in H$ and $z\in\rho(A_0)$.
%
In particular, $AA_0^{-1} f = f$.
%
Let~$u\in \D(A)$ be an arbitrary vector.
%
Represent $u$ in the form of sum~$u = f_u + h_u$ where
$f_u := A_0^{-1} A u \in \D(A_0)$
and $h_u := u - f_u = ( I - A_0^{-1} A ) u$.
%
Obviously, $h_u\in \D(A)$ and moreover $A h_u = (I - A A_0^{-1})A u
=0$, so that $h_u \in \ker(A) = \HH$.
%
Therefore, $u$ is represented as a sum of elements from~$\D(A_0)$
and~$\HH$.
%
This representation is unique because the intersection~$\D(A_0) \cap \HH$
is trivial.
%%======================================================================
Define the operator~$\gamma$ required by Assumption~\ref{Assumption}
to be the restriction of~$\Gd$ to the set~$\HH$.
%
If $\gamma h =0$ for some $h\in
\HH$, then $h\in \D(A_0)\cap \HH$, therefore $h =0$. Density of~$\gamma
\HH$ in $E$ is ensured by the assumption (ii) of the Theorem, since
$\gamma \HH = \Gd \HH = \Gd \D(A)$.
%%======================================================================
In order to verify existence and boundedness of the
left inverse of the operator~$\gamma$ consider the Green's formula
with $u = A_0^{-1}f$, $f\in H$ and $v= h\in \HH$.
%
Since $\Gd u=0$ and $A v=0$,
we obtain the equality~$(f,h) = (\Gn A_0^{-1}f, \Gd h)$.
%
According to the definition of adjoint operator, this
means that $\Gd h = \gamma h$ belongs to the domain of~$(\Gn A_0^{-1})^*$
and~$(\Gn A_0^{-1})^* : \gamma h \mapsto h$.
%
Therefore, $\Pi := (\Gn A_0^{-1})^*$ is the left inverse
of~$\gamma$.
%
Furthermore, $\Gn A_0^{-1}$ is defined on the whole space~$H$
since $\D(\Gn) \supset
\D(A_0)$ and $A_0^{-1} H = \D(A_0)$.
%
At the same time, its adjoint~$(\Gn A_0^{-1})^*$
is an operator with dense domain~$\gamma\HH = \Gd \HH = \Gd \D(A)$
due to condition~(ii).
%
Density of the domain of the adjoint implies closability;
thus, $\Gn A_0^{-1}$ is closable.
%
From the other hand,~$\Gn A_0^{-1}$ is already closed,
since its domain is the whole space~$H$.
%
By virtue of Closed Graph Theorem, the mapping~$\Gn A_0^{-1}$ is bounded, so
is its adjoint, the operator~$\Pi$.
%%================================
The last statement is easily proven by the calculations conducted
for~$h \in \HH$,
\[
(\Lambda\Gd h, \Gd h) - (\Gd h, \Lambda\Gd h)
= (\Gn \Pi \Gd h, \Gd h) - (\Gd h, \Gn \Pi\Gd h)
= (A h , h )\! -\! (h, Ah ) = 0.
\]
Two last equalities are valid due to the Green's formula since
$\Pi\Gd h =h$.
%%=================================================================
The proof is complete.
\end{proof}
%%=============================================================
%
%%=============================================================
\subsection{Weyl-Titchmarsh function}
We continue to denote~$u_z^{\varphi}$ the
solution~(\ref{NE_BVPsolution}) with~$f=0$.
%
Obviously, the map
\begin{equation}\label{NE_operatorR}
R(z) : \varphi \mapsto u_z^\varphi = (I -zA_0^{-1})^{-1}\Pi \varphi,
\qquad \varphi \in E, \quad z\in\rho(A_0)
\end{equation}
is bounded and $R(z)\Gd\HH = \ker(A - zI)$ according to
Lemma~\ref{NE_Lemma}.
%
If~$\varphi\in\Gd\HH$, then the
vector~$R(z) \varphi = u_z^{\varphi}$ belongs to the domain~$\D(A)$,
therefore~$\Gn u_z^\varphi = \Gn R(z)\varphi$ is well defined.
%
Let us calculate this vector.
%
For a given pair of~$\varphi \in \Gd\HH$ and~$z\in\rho(A_0)$ we have
%
\begin{align*}
\Gn u_z^{\varphi} & =
\Gn ( I -zA_0^{-1})^{-1}\Pi \varphi =
\Gn [ I + z (A_0 - zI)^{-1}]\Pi \varphi
\\
& =
\Gn \Pi \varphi + z \Gn ( A_0 - zI)^{-1} \Pi \varphi
= \Lambda \varphi + z \Gn A_0^{-1} ( I - z A_0^{-1})^{-1}\Pi \varphi
\\
& =\Lambda \varphi + z \Pi^* ( I - z A_0^{-1})^{-1}\Pi \varphi
=[\Lambda + z \Pi^* ( I - z A_0^{-1})^{-1}\Pi]\varphi
\end{align*}
%
Introduce the operator-function~$M(z)$, $z \in \rho(A_0)$ with
values in the set of operators defined on the dense domain~$\Gd\HH$
in~$E$ by
%
\begin{equation}\label{NE_MzDef}
M(z) \; : \;\; \varphi \longmapsto [\Lambda + z \Pi^* ( I - z
A_0^{-1})^{-1}\Pi] \varphi, \qquad \varphi \in \Gd \HH, \quad
z\in\rho(A_0)
\end{equation}
%
Since~$u_z^\varphi$ is the solution to the
problem~(\ref{NE_SpectralBVP}), the identity
$\varphi = \Gd u_z^\varphi$ holds and
the calculations conducted above show that
%
\begin{equation}\label{NE_Mz}
\Gn u_z^\varphi = M(z) \Gd u_z^\varphi, \qquad z \in \rho(A_0),
\quad \varphi \in \Gd \HH.
\end{equation}
%
The definition~$M(z) \varphi = \Gn R(z)\varphi$ yields another representation
%
\begin{equation}\label{NE_MzSecondForm}
M(z)\varphi = \Gn(I - zA_0^{-1})^{-1} \Pi \varphi,
\qquad \varphi\in \Gd \HH, \quad z \in \rho(A_0),
\end{equation}
%
and from Lemma~\ref{NE_Lemma} and Theorem~\ref{NE_BVPTheorem}
follows one more
\begin{equation}\label{NE_MzThirdForm}
M(z) \Gd h_z = \Gn h_z, \qquad h_z \in \ker (A - z I), \quad z\in
\rho(A_0).
\end{equation}
%%========================================================================
\begin{defn}
Function $M(\cdot)$ is called the \textbf{Weyl-Titchmarsh function}
of the problem~(\ref{NE_SpectralBVP}) or of the null extension~$A$
satisfying Assumption~\ref{Assumption}.
\end{defn}
A few remarks concerning this definition are in order.
%%========================================================================
\paragraph{ 1. } Analytic operator function~$m(z) :=M(z) - M(0) = M(z) -
\Lambda$ defined for $z\in\rho(A_0)$ is bounded and has a
non-negative imaginary part in the upper
half-plane~$z\in\Complex_+$.
%
In other words,~$m(z)$ is an operator-valued $R$-function.
%
This statement follows from the
equality $m(z) = z\Pi^*(I -zA_0^{-1})^{-1}\Pi $
and the formula for $\varphi, \psi \in E$ and $
z,\zeta \in\rho(A_0)$
%
\begin{equation*} %%\label{NE_MzMzetaDifference}
(m(z) \varphi, \psi)_E - ( \varphi,m(\zeta) \psi)_E = (z -
\overline{\zeta}) \left((I - z A_0^{-1})^{-1}\Pi \varphi,(I -\zeta
A_0^{-1})^{-1}\Pi \psi \right)_H,
\end{equation*}
%
obtained by direct calculations.
%
In the special case of~$\zeta = z$, $z\notin \Real$ and $\varphi =
\psi$ we have
\[
\IM (m(z)\varphi, \varphi) =
(\IM z)\cdot \|(I -zA_0^{-1})\Pi\varphi\|^2 =
(\IM z)\cdot\|R(z)\varphi\|^2, \quad \varphi\in E,
z\notin\Real.
\]
Therefore, imaginary parts of $m(z)$ are non-negative operators
for~$z\in\Complex_+$.
%
Suppose~$\ker(\Pi)$ is trivial, which is equivalent to the density
of~$\Ran(\Pi^*) = \Gn A_0^{-1} H = \Gn \D(A_0)$ in~$E$.
%
Then the imaginary part of~$m(z)$ is strictly positive
for~$z\in\Complex_+$.
%%========================================================================
\paragraph{ 2. }
As follows form the definition, the function~$M(\cdot)$ depends
on the particular choice of~$\Lambda$
in~(\ref{DefinitionOfGammaN}).
%
It is clear however, that all functions corresponding to different
values of this parameter differ from one another
by additive constant operators with the domain~$\Gd\HH$.
%%========================================================================
\paragraph{ 3. }
Since
$
[R(z)]^* = [(I - zA_0^{-1})^{-1}\Pi ]^*
= \Gn A_0^{-1}(I - \bar zA_0^{-1})^{-1} = \Gn (A_0 - \bar
zI)^{-1},
$
the function~$M(\cdot)$ can be rewritten in a compact,
but somewhat more obscure form
\begin{equation}\label{NE_MzFourthForm}
M(z) = \Gn [\Gn (A_0 - \bar zI)^{-1}]^*, \qquad z \in
\rho(A_0).
\end{equation}
%
Such representations when the operator $A_0$ is the Dirichlet or Neumann
Laplacian in a region of~$\Real^n$, $n = 2,3$ can be found in the literature
(cf.~\cite{AmP, GMZ}).
%
In comparison with~(\ref{NE_MzFourthForm}), the formula~(\ref{NE_operatorR})
separates out the singular part of~$M(\cdot)$ , that is, the
potentially unbounded term~$M(0) = \Lambda$.
%
This decomposition of~$M(\cdot)$ allows one to study properties
of~$M(\cdot) - M(0)$ by more elementary means of the bounded operators theory.
%
In addition, the summand~$M(0) = \Lambda$ ultimately is not a
characteristic of the spectral
problem~(\ref{NE_SpectralBVP}) or the extension~$A$.
%
It is merely an arbitrary parameter in the
definition of boundary operator~$\Gn$.
%
Therefore,
by studying~$M(\cdot) -\Lambda$ rather than~$M(\cdot)$
one eliminates this
arbitrariness from the analysis.
%
%%============================================
\paragraph{ 4. }
An additive representation similar to~(\ref{NE_MzDef})
in a special case of operator~$A_0$
was obtained
in the work~\cite{BM}, formula~(2.6).
%
This paper
uses another (unspecified) form of the bounded mapping from~$E$
to~$H$ whose role in our considerations is played by the
operator~$\Pi$.
%%=========================================================
\paragraph{ 5. }
Consider asymptotic behavior of~$M(\cdot)$ along the imaginary axis
in the upper half plane.
%
Since~$m(z) = M(z) - M(0)$ is an~$R$-function and $M(0) = \Lambda$
is symmetric, we expect~$M(iy)$ to possess some kind of limit
as~$y\to\infty$.
%
Analogy with the theory of bounded~$R$-functions and bounded
operators suggests that this limit is likely to be the null
operator.
%
%
For~$\varphi \in \Gd \HH $ and $h = \Pi\varphi$
represent~$ M(z)\varphi $ in the form
%
\[
M(z)\varphi = \Gn (I - z A_0^{-1})^{-1} \Pi \varphi = \Gn [ I + z (A_0 -
z I)^{-1}]h
\]
%
By the Spectral Theorem,~$z(A_0 - zI)^{-1} \to - I $ for~$z= iy$
when $y\to\infty$ in the strong operator topology.
%
Denote
$ F(h,z) := [ I + z
(A_0 -
z I)^{-1}]h$.
%
Then we have $F(h,iy) \to 0$ in~$H$ as~$y\to\infty$ for any~$h \in
\HH$ (in fact, for any~$h\in H$).
%
The vector function~$F(h,iy)$ can be seen as an approximation error
of~$h\in \HH$ by vectors~$ - iy (A_0 -
iy I)^{-1}h $ from the dense set~$\D(A_0)$.
%
If the operator~$\Gn$ is closable on its domain~$\D(A)$, then
$F(h,iy) \to 0$ implies~$M(iy)\varphi = \Gn F(h,iy) \to 0$.
%
However, if~$\Gn$ is not closable, this implication may be not valid
and there may exist vectors~$\varphi \in \Gd \HH$ such
that~$M(iy)\varphi$ does not converge when~$y\to\infty$.
%
From the other side, the requirement of closability of~$\Gn$ is too
generous for the existence of~$\lim_{y\to\infty} M(iy)\varphi$ with
$\varphi\in \Gd \HH$.
%
It is sufficient to request the convergence of~$\Gn F(h,iy)$ for any
$h\in \HH$ in order to conclude the existence of ~$\lim_{y\to\infty}
M(iy)\varphi$ for any $\varphi\in \Gd \HH$.
%
If, in addition, $\Gn F(h,iy) \to 0$ for each~$h\in\HH$, then the
Weyl-Titchmarsh function~$M(z)$ has the expected behavior along the
imaginary.
%
This condition is not as restrictive as the closability of~$\Gn$,
since the implication~$f_n\to 0 \Longrightarrow \Gn f_n \to 0$ for
$n\to\infty$ and any sequence~$\{f_n\}\in\D(\Gn)$, which is
equivalent to the closabilty of~${{\Gn}}$, is not assumed to be
fulfilled for any vectors from the domain~$\D(\Gn)$; only vectors of
the special form~$F(h,iy) = h + iy (A_0 - iyI)^{-1} h$, $h\in \HH$
are considered.
%
%
%
The obtained condition
\begin{equation}\label{MzBehaviorOnInfinity}
\Gn [ h + iy ( A_0 - iyI)^{-1}h] \to 0, \quad \text{when } y\to\infty \quad \text{for any } h\in \HH
\end{equation}
%
guarantees that $M(iy)\varphi\to 0$ for any~$\varphi \in \Gd\HH$
when $y\to\infty$.
%%=================================================================
\paragraph{ 6. }
The last remark is important in applications where
Assumption~\ref{Assumption} is not fulffiled.
%
It allows one to define the Weyl-Titchmarsh function
in cases when~$A_0$ is not boundedly invertible, but
$\rho(A_0) \cap \Real \neq \{ \emptyset\}$.
\begin{rem}\label{RemarkShift}
For a number~$t\in\Real$ denote $A_t := A + t I$ the ``shifted''
operator~$A$.
%
Then $\ker(A - zI ) = \ker(A_t - \zeta I)$ where~$\zeta = z + t$.
%
For $h_z\in\ker(A-zI)$ the Weyl-Titchmarsh function definition
$\Gn h_z = M(z)\Gd h_z$ may be rewritten in the form $\Gn
v_\zeta = M(\zeta -t)\Gd v_\zeta$ where $v_\zeta = h_{\zeta -t} \in
\ker(A_t - \zeta I)$.
%
Therefore, the operator function~$M_t (\zeta) := M(\zeta - t)$ for
$\zeta\in \rho(A_0 + t I)$ is naturally interpreted as the
Weyl-Titchmarsh function of~$A_t = A + t I$.
\end{rem}
%
%%======================================================
%%=====================================================
%%
%%=====================================================
\subsection{Minimal symmetric operator and its Krein extension}
%
As is well known, study of a
boundary value problem in many cases
can be reduced to analysis of
ceratin extensions
of a certain symmetric operator
conventionally called minimal.
%
%In particular,
% this operator-theoretic point of view helps
% one to reduce the solvability problem
% for a BVP defined on the bounded domain
% to the study of some equation on the boundary, thereby
% effectively decreasing dimensions of the problem~\cite{Ag,GG,ML}.
%
In this short section we give a brief account of this reduction
carried out in the paper's setting.
%
% and clarify how the null extension~$A$
% is interpreted in terms of extensions theory.
%
%
%%===================================================
Introduce the minimal operator~$A_{00}$ as a restriction of~$A$
to the do\-main $\D(A_{00})$ where $\D(A_{00}) := \{ u\in\D(A)\; |\;
\Gd u = \Gn u = 0\}$.
%
%
As follows from the Green's formula~(\ref{GreenFormula}), the
operator~$A_{00}$ is symmetric, but not necessarily densely defined.
%
The operator~$A_0$ can be seen as a selfadjoint
extension of~$A_{00}$ to the domain~$\D(A_0)$.
%
Another important extension of~$A_{00}$ is the operator~$A_K$
defined as a
restriction of~$A$ to the
set~$\D(A_K)$, whe\-re~$\D(A_K) := \{u\in \D(A)\;|\;(\Gn - \Lambda
\Gd)u=0\}$.
%
It is remarkable that neither of operators~$A_{00}$ or~$A_K$ depends
on the particular choice of~$\Lambda$ and can be expressed solely in
terms of the pair~$\{A_0,\HH\}$.
%
More precisely, the following Theorem is valid.
%%==================================================================
\begin{thm} \label{TheoremOnA00}
Domains of~$A_{00}$ and~$A_K$ are represented by formulae
\begin{equation*}
\begin{aligned}
\D(A_{00}) & = \{ u \in \D(A) \; |\; \Gn u = \Gd u =0 \}
&& =
A_0^{-1} \HH^\perp, \\
\D(A_{K}) & = \{ u \in \D(A) \; |\; (\Gn - \Lambda\Gd ) u =0
\} && = A_0^{-1} \HH^\perp \dot{+} \HH.
\end{aligned}
\end{equation*}
\end{thm}
%
%%==============================================================
\begin{proof}
Let us begin by noting that $\ker(\Pi^*) = {\HH}^\perp$.
%
Indeed, from identity~$\Pi \Gd h = h$, we obtain $(f,h) = (f,\Pi\Gd
h) = (\Pi^*f , \Gd h)$ for any~$h\in\HH$ and~$f\in H$.
%
Since~$\Gd \HH$ is dense in~$E$, the inclusion~$f\in \ker(\Pi^*)$ is
equivalent to the ortho\-go\-na\-li\-ty~$f\perp\HH$.
%%================================================================
Let~$u = A_0^{-1} f + h$ be an arbitrary element of~$\D(A)$ with
some~$f\in H$, $h\in \HH$.
%
Conditions~$\Gd u =0$ and $\Gn u =0$ result in the equality~$\Gn
A_0^{-1}f =0$, which is equivalent to~$f\in\ker(\Pi^*) = \HH^\perp$
since~$\Pi^* = \Gn A_0^{-1}$.
%
Therefore, $\D(A_{00})\subset A_0^{-1}\HH^\perp$.
%
The inverse inclusion holds true according to the equalities~$\Gd A_0^{-1} =
0$ and~$\Gn A_0^{-1} \HH^\perp = \Pi^* \HH^\perp =0$.
%
Further, for the domain of~$\D(A_K)$ the identity~$(\Gn -
\Lambda\Gd) h=0$ follows directly from the relations~$\Lambda =
\Gn\Pi$ and~$\Pi\Gd h =h$, $h\in\HH$. Therefore, $A_0^{-1} \HH^\perp
\dot{+} \HH \subset\D(A_K)$.
%
From the other side, if the element~$u = A_0^{-1} f + h \in\D(A)$
belongs to~$\D(A_K)$, then it is necessary that $f\in \HH^\perp$,
hence~$\D(A_K) \subset A_0^{-1}\HH^\perp \dot{+} \HH$.
\end{proof}
%%================================================================
Operator~$A_K$ is an analogue of the Krein extension of~$A_{00}$
(see~\cite{AS,Kr}), which explains the notation.
%
%
%
In this respect, the selfadjoint operator~$A_0$ can be interpreted
as the Friedrichs extension of~$A_{00}$.
%
%
Operator~$\Lambda$ was studied by
M.~Vishik in the context of elliptic
boundary value problems in~\cite{Vi}.
%
The same paper introduces
the boundary operator~$\Gn - \Lambda\Gd$
associated with the Krein extension~$A_K$
as an alternative to the more customary map~$\Gn$
equal to the trace of the normal derivative
on the domain's boundary.
%
Later $\Gn - \Lambda\Gd$
was rewritten as~$\Gn (I - \Pi \Gd)$ by
G.~Grubb~\cite{Grubb}
and independently by A.~Alonso and B.~Simon in~\cite{AS}.
%
See the note~\cite{Grubb2} for further references regarding
boundary conditions for the Krein extension.
%%=================================================================
In conclusion of the section we note that it is possible to develop
an analogue of extensions theory of
symmetric operators for
the pair of operators~$A_{00}$ and $A$.
%
%
In particular, consider an extension~$A_B$ of~$A_{00}$ defined as a
restriction of~$A$ to the
domain $\D(A_B):=\{u\in \D(A)\; |\; ( \Gn - \Lambda \Gd) u = B\Gd u\}$
with some
bounded and boundedly invertible operator~$B$.
%
Under assumption~$B^{-1} E \subset \Gd \HH$, it can be shown
that~$A_B$ is also boundedly invertible and
\begin{equation}\label{KreinResolventFormulaForZ=0}
A_B^{-1} = A_0^{-1} + \Pi B^{-1} \Pi^*.
\end{equation}
%
Formally, $B=\infty$ in~(\ref{KreinResolventFormulaForZ=0})
describes the operator~$A_0$,
and the case~$B =0 $ corresponds to the Krein extension~$A_K$, cf.~\cite{AS}.
%
Results regarding other types of extensions will be published elsewhere.
%%====================================================================
%%
%%====================================================================
\section{Associated colligation and corresponding open system}
In the previous sections we studied the null extension~$A$ of the
selfadjoint operator~$A_0$ to the set~$\D(A_0) \dot{+}\HH$ subject
to Assumption~\ref{Assumption}.
%
The mapping~$\Gn$ in~(\ref{DefinitionOfGammaN})
plays the role of the boundary
map complementary to~$\Gd$.
%
Its definition involves one parameter,
a symmetric operator~$\Lambda$ with the
dense domain~$\Gd \HH$.
%
It was shown that the extension~$A$ defines the
spectral boundary value problem~(\ref{NE_SpectralBVP}).
%
In this section we connect the problem~(\ref{NE_SpectralBVP}) with
the so called operator $d$-node, or an operator colligation.
%
In general, an operator colligation is a collection of two Hilbert
spaces and three bounded mappings.
%
Subsequently, there are many ways to incorporate objects related to
the boundary value problem into a colligation.
%
The guidance is provided by the book~\cite{Liv} where
an interpretation
of an operator colligation as a mathematical abstraction for an open
system is offered.
%
Roughly speaking, open systems are
systems coupled to the external world by
means of some kind of channels attached to it.
%
Notions of input, output, and internal state are fully applicable to
open systems.
%
In fact, internal states are represented as vectors from the
interior space, one of the Hilbert spaces comprising the
colligation, whereas inputs and outputs are modeled as elements of
the second Hilbert space, the coupling space.
%
One of three mappings of a colligation encapsulates the interior of
the corresponding system considered in isolation from the external
word, that is, with the coupling channels cut off.
%
The second operator depicts interactions of the interior of the
system with the channels, and the third operator describes the
metric nature of the channels.
%
Usually it is an involution, i.~e. a selfadjoint unitary operator
acting in the coupling space.
%
The key element of open systems theory is the system's transfer
function.
%
As one may expect, it is an analytic function that maps inputs into
outputs.
%%======================================================
Our goal in this section therefore can be stated as follows.
%
Given boundary value problem~(\ref{NE_SpectralBVP}) we are looking
for a suitable operator colligation that would correspond to an open
system effectively encapsulating principal characteristics of this
BVP.
%
One of these characteristics is undoubtedly the Weyl-Titchmarsh
function~$M(\cdot)$, and the open system constructed in this section
possesses the transfer function that coincides with the
function~$m(z) := M(z) - M(0)$, $z\in \rho(A_0)$.
%
Once this colligation (or open system) is obtained, other objects
such
as spaces~$E$, $H$ and operators~$\Pi$, $A_0$, and $R(\cdot)$ become
endowed with the clear physical meaning expressed in relative terms of this
system.
%
Connections among the null extension~$A$,
the BVP~(\ref{NE_SpectralBVP}),
and the open system
establish the relationship of the Weyl-Titchmarsh
functions theory
to the theory of open system we are looking for.
%
The link from open systems theory
to the linear systems with boundary control is the subject
of the next section where we show that the open system associated
with a given BVP essentially is the reciprocal
of the boundary control system
defined by this BVP in accordance with
the mainstream theory~\cite{CuZw,Lions,Sta}.
%
With this result in place, we accomplish the paper's promise
by connecting boundary value problems and their Weyl-Titchmarsh
functions to the open systems theory due to M.~S.~Liv\v{s}i{c}, and
then by going a bit further, to the linear boundary control systems.
%
%
%
%%=============================================================
%%
%%=============================================================
\subsection{Associated operator colligation}
The following definition of operator colligation is taken
from~\cite{Bro, Liv}.
%
%
%%===========================================================
%
%
\begin{defn}\label{NodeDefinition}
\textbf{Operator colligation} is the collection of five objects
traditionally written in the form
%
\begin{equation}\label{NodeDefMatrix}
\mathfrak{M} = \bigg(
\begin{array}{ccc}
T & \sqrt{2}\, K & J \\
H & \phantom{K} & E
\end{array}
\bigg)
\end{equation}
%
where $H$ and $E$ are two Hilbert spaces, and $T$, $K$, and $J = J^*
= J^{-1}$ are bounded linear operators:
%
\[
T : H \rightarrow H, \quad
K : E \rightarrow H, \quad
J : E \rightarrow E
\]
%
The mapping $T : H \to H $ is called the \textbf{interior} operator;
the operator~$K : E\to H$ and its adjoint~$K^* : H \to E$ are called
the \textbf{coupling operators}.
%
\end{defn}
%%====================================================================
In comparison~\cite{Bro, Liv}, we single out the multiplier~$\sqrt
2$, which is convenient for our purposes.
%
Following an alternative word usage, sometimes we shall employ the
term~\textbf{operator node} for the
colligation~(\ref{NodeDefMatrix}),
and sometimes shall
denote the colligation~(\ref{NodeDefMatrix})
as the list of its components:
\begin{equation*}
\mathfrak{M} = \big\{ T, K, H, E, J \big\}
\end{equation*}
%
According to the plan of section, below we put the null
extension~$A$ into a correspondence with a certain operator
colligation referred to as its associated colligation.
%%====================================================================
\begin{defn}\label{DefinitionDNode}
For the pair~$\{A_0,\HH\}$ satisfying Assumption~\ref{Assumption}
the \textbf{associated colligation (node)} is defined by
\begin{equation}\label{DNode}
\mathfrak{M} = \big\{ A_0^{-1}, \Pi, H, E, I_E \big\}
\end{equation}
\end{defn}
%%====================================================================
Definition~\ref{DefinitionDNode} is
a foundation for the subsequent interpretation of~(\ref{NE_SpectralBVP}),
or the null extension~$A$,
as a problem of the open systems theory.
%
In order to make the relationship~(\ref{DNode}) between BPVs and
colligations precise, a characterization of
colligations associated with boundary value problems
studied in Section~1 is needed.
%
It is given in the next theorem.
%%====================================================================
\begin{thm}\label{TheoremOnNode}
Suppose the pair~$\{A_0,\HH\}$ satisfies
Assumption~\ref{Assumption}, hence defines a
boundary value problem~(\ref{NE_SpectralBVP}).
The mapping
\[
\{A_0, \HH\} \mapsto ( \mathfrak M, \mathscr E)
\]
where~$\mathscr E := \Gd \HH \subset E$ is a linear set and~$\mathfrak M$
is defined by~(\ref{DNode}) is an one-to-one
correspondence between the pairs~$\{A_0,\HH\}$ subject to
Assumption~\ref{Assumption}
and the pairs~$(\mathfrak M, \mathscr E)$ made of a
colligation~$\mathfrak M = \big\{ T, K, H, E, J \big\}$ and
a linear set~$\mathscr E\subset E$ with following properties
\begin{enumerate}
%
\item
$\ker(T) = \{ 0\}$, $\R(T) \neq H$, $\overline{\R(T)} = H$, and
$T = T^*$, so that there exists the unbounded (selfadjoint)
operator~$T^{-1}$ with the domain~$\D(T^{-1}) = \R(T)$.
%
\item The equality
$\R(T) \cap K \mathscr E = \{ 0\}$ holds.
%
\item
The implication $K \varphi = 0,\; \varphi \in \mathscr E
\Longrightarrow \varphi = 0$ is valid.
%
\item
The linear set~$\mathscr E$ is dense in~$E$.
\item
$J = I_E$.
\end{enumerate}
\end{thm}
%%==================================================================
\begin{proof}
%
Let $\mathfrak M = \{T, K, H, E, J\}$ be the node~(\ref{DNode})
associated with the null extension~$A$ corresponding
to~$\{A_0, \HH\}$ and~$\Lambda =0$.
%
Then $T = A_0^{-1}$ is selfadjoint, $\D(A_0) = \Ran(T)$,
$K = \Pi$, and~$J= I_E$.
%
Define the set~$\mathscr E := \gamma \HH = \Gd \HH$, so that
$\HH = \Pi \mathscr E = K \mathscr E$.
%
Therefore, all four statements about the
pair~$(\mathfrak M, \mathscr E)$ hold true due to
Assumption~\ref{Assumption}.
%%==================================================================
In order to prove the inverse let $\mathfrak{M} = \big\{ T, K, H, E,
I_E \big\}$ be some colligation~(\ref{NodeDefMatrix}) with~$J= I_E$
and let $\mathscr E$ be a dense linear set in~$E$ satisfying all
conditions of the Theorem.
%
Then the pair~$(\mathfrak{M}, \mathscr E)$ uniquely defines a null
extension~$A$ of the selfadjoint operator~$A_0 := T^{-1}$ to the
set~$\Ran(T) \dot{+} \HH$ with $\HH := K \mathscr E$.
%
Let us show that Assumption~\ref{Assumption} for this extension is
valid.
%
Indeed, $\D(A_0)\cap \HH = \Ran(T)\cap K\mathscr E = \{0\}$ and $K$
is the left inverse to the linear mapping~$\Gd : T f \dot{+} K
\varphi \mapsto \varphi$, $f\in H$, $\varphi \in \mathscr E$
restricted to the set~$K\mathscr E$.
%
Thus, $\Pi = K$, $\HH =
K \mathscr E$, and~$\Gd \HH = \mathscr E$.
%%==================================================================
The proof is complete.
\end{proof}
%%==================================================================
Theorem~\ref{TheoremOnNode} implies that the
triplet~$\{A_0,\HH, \Lambda\} $ is uniquely determined
by~$(\mathfrak M,\Lambda)$, where~$\Lambda$ is
the parameter in~(\ref{DefinitionOfGammaN})
and $\mathfrak M$ is the colligation~(\ref{DNode}).
%
Therefore the pair~$(\mathfrak M, \Lambda)$ where
$\mathfrak M$ satisfies conditions of Theorem~\ref{TheoremOnNode}
and $\Lambda$ is some symmetric operator on the
domain~$\D(\Lambda) := \mathscr E$ determines the
Weyl-Titchmarsh function of BVP constructed by $\{A_0, \HH, \Lambda\}$
uniquely.
%
The inverse statement is valid only partially and under some
additional conditions imposed on the operator~$A$.
%
%%==========================================================
%
\begin{thm}\label{TheoremOfUniqueness}
In the notation introduced above,
let $\widetilde{\mathfrak M} = \{\widetilde T, \widetilde K, \widetilde H,
E, I_{ E}\}$ be an operator colligation
and $\widetilde{\mathscr E} \subset E$ be a linear
set that satisfy conditions of Theorem~\ref{TheoremOnNode}.
%
For $z\in \rho(\widetilde {A}_0)$
and $\widetilde {A}_0 := \widetilde T^{-1}$
denote~$\widetilde M(z)$
some Weyl-Titchmarsh function of the corresponding null
extension~$\widetilde A$.
%
Suppose both
linear spans~$\bigvee_{n \geqslant 0 } A_0^{-n} E$,
$\bigvee_{n \geqslant 0 } \widetilde{A}_0^{-n} E$
are dense in $H $ and $\widetilde H$, respectively.
%
Assume that for some neighborhood of the
origin~$\mathscr X \subset \rho(A_0) \cap \rho(\widetilde{A}_0) $
the identity
$M(z) - M(0)= \widetilde M(z) - \widetilde M(0)$, $z\in \mathscr X$ holds.
%
Then operators $A_0$ and~$\widetilde{A}_0$ are unitarily equivalent,
that is $U A_0 = \widetilde{A}_0 U$, where $U : H \to \widetilde H$
is an isometry. Moreover, $U K = \widetilde K$.
\end{thm}
%%===========================================================
\begin{proof}
According to~\cite{Bro}, the analytic operator function
\[
S(\lambda) = I + i [M(1/{\lambda}) - M(0)] = I - i K^* (T - \lambda
I)^{-1}K, \quad \lambda \in \rho(T)
\]
coincides with the so called
characteristic function of the operator colligation~$\mathfrak M$.
%
The required result is the known fact of the operator colligations
theory (see~\cite{Bro}, Theorem~3.2).
The proof is complete.
\end{proof}
%%===========================================================
%
Theorems~\ref{TheoremOnNode} and \ref{TheoremOfUniqueness}
show
the associated node defined by~(\ref{DNode})
has properties that are sufficient to recover
the boundary value problem associated with it
with the only exception of the
second boundary operator~$\Gn$.
%
Complemented with a symmetric densely defined map~$\Lambda$,
the associated node~$\mathfrak M$
fully represents the BVP and the operator~$\Gn$.
%
%
For convenience, let us summarize the
connections of the triplet~$\{A_0, \HH, \Lambda\}$ with
the corresponding pair~$(\mathfrak M, \Lambda)$
in a few formulae.
%
Below we assume~$f\in H$, $\varphi \in \mathscr E$.
\begin{equation}\label{BVPToNodeSummary}
\begin{aligned}
& A_0 = T^{-1}, \qquad
\Pi = K, \qquad
\HH = \mathscr E, \qquad
\D(A) = \Ran(T)\dot{+}K \mathscr E
\\
& A : T f + K \varphi \mapsto f
\\
& \Gd : Tf + K \varphi \mapsto \varphi
\\
& \Gn : Tf + K \varphi \mapsto K^* f + \Lambda \varphi
\\
& M(z) : \varphi \mapsto \Lambda \varphi + z K^*(I -z T)^{-1}K
\varphi
\end{aligned}
\end{equation}
%
It is clear that any particular choice of~$\Lambda$ affects the
operator~$\Gn$ and the Weyl-Titchmarsh function~$M(\cdot)$, whereas
the associated colligation~$\mathfrak M$ does not depend on such
choices.
%
Note as well that one can introduce a null extension and
corresponding spectral boundary value problem~(\ref{NE_SpectralBVP})
by presenting the colligation~$\mathfrak M$ and the dense
set~$\mathscr E\subset E$ that satisfy conditions of
Theorem~\ref{TheoremOnNode}.
%
Assuming a symmtric operator~$\Lambda$, $\D(\Lambda) = \mathscr E$
is given, two last formulae in~(\ref{BVPToNodeSummary}) serve as
definitions of the boundary operator~$\Gn$ and Weyl-Titchmarsh
function~$M(\cdot)$.
%
It is worth mentioning that the choice~$\Lambda =0$, which is
equivalent to~$M(0) = 0$, is always possible.
%%==================================================================
%%
%%==================================================================
\subsection{M.~S.~Liv\v{s}i{c}'s open system}
The open systems theory developed by M.~S.~Liv\v{s}i{c} in his
seminal book~\cite{Liv} states that any node~$\mathfrak{M} = \big\{
T, K, H, E, J \big\}$ corresponds to a certain open stationary
dynamic system connected with the external world via so-called
coupling channels.
%
The system's internal states are represented by vectors from the
interior space~$H$ often called the state space, whereas the
system's input and output are represented by vectors from the
coupling (exterior) space~$E$.
%
The stationarity of the system signifies that its properties do not
depend on time.
%
In the case of system corresponding to the
node~$\mathfrak{M} = \big\{ T, K, H, E, J \big\}$ the
stationarity means that the
operators~$T$, $K$, and~$J$ are constants, i.~e. are independent
on the parameter~$z\in \Complex$.
%
%
Let us set forth relevant definitions derived from~\cite{Liv}.
%%==================================================================
\begin{defn} (M.~S.~Liv\v{s}i{c}).
Let $H$, $E$ be two Hilbert spaces. An \textbf{open system}
$\mathscr F$ with \textbf{coupling space}~$E$ and \textbf{interior
space}~$H$ is comprised of two linear mappings: the
\textbf{input--interior transformation}~$R : \phi^- \mapsto\psi$ and
\textbf{input-output transformation}~$S : \phi^- \mapsto \phi^+$,
$\phi^\pm\in E$, $\psi \in H$. The vectors $\phi^-$, $\phi^+$ and
$\psi$ are called \textbf{input}, \textbf{output} and
\textbf{internal state} of the system~$\mathscr F$, respectively.
%
An open system is denoted by the symbol
%
\begin{equation} \label{GeneralSystemMatrix}
\mathscr F
\left(\vcenter{
\xymatrix@-1.6pc@M=0.6pt {
S & & \phi^+ \\
& \!\!\!\!\phi^- \ar[ur] \ar[dr] &\\
R & & {\psi}
}}
\right)
\end{equation}
\end{defn}
%%===================================================================
The book~\cite{Liv} describes various ways to relate a given
operator node to an open system.
%
%
%
One of them is commonly known; it is used
when the interior operator~$T$ of the node is nonselfadjoint and
operators~$K$ and $J$ from the definition~(\ref{NodeDefMatrix})
satisfy the
equation~$2i K J K^* = T - T^*$.
%
Another method is suitable if~$T$ is selfadjoint.
%
Then the mappings~$K$ and $J$ can be chosen arbitrary and we are
going to use this fact to define an open system corresponding
to the colligation~(\ref{DNode}) associated with a given BVP.
%
%
Let us cite the relevant definition from~\cite{Liv}.
%
%%=======================================
%
\begin{defn}\label{DefnLivsic2}
(M.~S.~Liv\v{s}i{c}).
A node~$\mathfrak M = \{T, K, H, E, J\}$ as in~(\ref{NodeDefMatrix})
is called the \textbf{$\mathbf{d}$-node of system~$\mathscr F$ with
respect to the number~$z_0\in \Complex$} if the input--output and
input--interior transformations $S:\phi^- \mapsto \phi^+$, $R :
\phi^- \mapsto \psi$ are connected with~$\mathfrak M$ by relations:
\begin{equation}\label{System}
\left.
\begin{aligned}
\left[ I - (z-z_0)T\right] \psi & = K \phi^- , \\
\phi^+ & = - i (z-z_0) J K^*\psi \\
& = - i(z -z_0) JK^*[I -(z-z_0) T]^{-1}K\phi^-
\end{aligned}
\right\}
\end{equation}
If so, it is said that the $d$-node~$\mathfrak{M}$ \textbf{belongs}
to the system~$\mathscr F = \mathscr F [\mathfrak M]$.
%
The transformation~$S : \phi^- \mapsto \phi^+$ is an operator
function defined on all input vectors~$\phi^- \in E$ and analytic
for all $z$ such that $\frac{1}{z-z_0}\in \rho(T)$.
%
It is called a \textbf{transfer function} of the system~$\mathscr
F[\mathfrak{M}]$ or its $d$-node~$\mathfrak{M}$.
%
Vectors~$\{K\phi^-\}$ are termed \textbf{channel vectors}.
\end{defn}
%%====================================================================
Now we return to the triplet~$\{A_0, \HH, \Lambda\}$
and the corresponding pair~$(\mathfrak M,\Lambda)$.
%
Assuming~$z\in\rho(A_0)$ and putting~$z_0 = 0$ in~(\ref{System}) we
see that the colligation~$\mathfrak{M}$ defined in~(\ref{DNode}) and
(\ref{BVPToNodeSummary}) is a $d$-node of
the system~$\mathscr F[\mathfrak M]$
described by relations
\begin{equation}\label{SystemForBVP}
\left.
\begin{aligned}
\psi & = ( I - zA_0^{-1})^{-1} \Pi \phi^- , \\
\phi^+ & = - i z \Pi^*\psi = - iz \Pi^*(I -z A_0^{-1})^{-1}\Pi\phi^-
\end{aligned}
\right\}
\end{equation}
%
Formulae~(\ref{SystemForBVP}) express the input--interior
transformation~$R$ and the transfer function~$S$ of this system.
%
They are
%
\begin{equation}\label{TransferFunctionAndInteriorOperatorOfF}
R (z) = (I -zA_0^{-1})^{-1} \Pi , \qquad S(z) = - iz \Pi^*(I -z
A_0^{-1})^{-1}\Pi, \quad z\in\rho(A_0)
\end{equation}
%
The input and output of system~$\mathscr F[\mathfrak M]$ are
vectors~$\phi^\pm$ of the space~$E$.
%
Now we can make the
fundamental observation, namely that the function~$R(z) : E\to H$,
$z\in \rho(A_0)$ in fact was introduced earlier by~(\ref{NE_operatorR}).
%
It coincides with the
mapping~$\varphi \mapsto u_z^\varphi$, where~$u_z^\varphi$ is a
solution to~(\ref{NE_SpectralBVP}) with~$f=0$.
%
Moreover, the transfer function~$S$
from~(\ref{TransferFunctionAndInteriorOperatorOfF})
is the $z$-dependent part of the Weyl-Titchmarsh function
multiplied by $-i$, in other words,
$S(z) = -i [ M(z) - M(0)]$, $z\in \rho (A_0)$.
%
Let us summarize these and some other observations directly obtained
by comparison of results of Section~1 with the established
relationship between the BVP~(\ref{NE_SpectralBVP}) (or the null extension~$A$)
and the
open system~$\mathscr F [\mathfrak M]$.
%
\begin{prop}\label{PropOnTheBVPandSystem}
Following statements hold true:
\begin{itemize}
\item
The internal state~$\psi = R\phi^-$ of the
system~$\mathscr F[\mathfrak M]$ defined in~(\ref{SystemForBVP}) is the
solution~$u_z^\varphi$ to the
problem~(\ref{NE_SpectralBVP}) with $f=0$ as described by
Theorem~\ref{NE_BVPTheorem}.
It corresponds to
the choice of input~$\phi^- = \varphi \in \Gd \HH$, in which case
$R\phi^-$ belongs to $\ker(A-zI)$.
\item
The set~$\HH$ consists of the channel
vectors of system~$\mathscr F[\mathfrak M]$ obtained by the
mapping~$\phi^- \mapsto \Pi \phi^-$ from the inputs~$\phi^- \in \Gd
\HH$.
\item
For an arbitrary input~$\phi^- \in E$ the internal
state~$R\phi^-$ is the weak solution to the
problem~(\ref{NE_SpectralBVP}) with $f =0$
in the sense of Definition~\ref{DefinitionWeakSolution}
and $\Pi \phi^-$ is the corresponding channel vector.
\item
The transfer function~$S$
from~(\ref{TransferFunctionAndInteriorOperatorOfF}) is an analytic
bounded operator function.
%
It does not depend on the particular choice of operator~$\Lambda$
and is related to the Weyl-Titchmarsh
function~$M$ of~$\{A_0, \HH, \Lambda\}$ by the
formula~$S = - i (M - \Lambda)$.
\end{itemize}
\end{prop}
Proposition~\ref{PropOnTheBVPandSystem} provides the link
announced in
the beginning of section between BVPs
studied in Section~1 and a class of open systems, thereby offering a
system-theoretic interpretation for BVPs that
correspond to null extensions satisfying
Assumption~\ref{Assumption}.
%
%
%
%%================================================================
%%
%%================================================================
\subsection{Remarks}
%
The
established connection between boundary value problems and open
systems is further clarified by means
of simple observations regarding their relationships.
%
%%========================================
%
\paragraph{ 1.} Comparison of~(\ref{NE_MzDef})
and~(\ref{TransferFunctionAndInteriorOperatorOfF}) leads to
the representation for Weyl-Titch\-marsh function~$M(\cdot)$ in terms of the
pair~$(\mathfrak M,\Lambda)$
%
\begin{equation}\label{MasLabmdaPlusS}
M(z) = \Lambda + i S(z), \quad z \in\rho(A_0),
\end{equation}
%
where the domain of~$M(\cdot)$ is equal to~$\D(\Lambda) = \Gd \HH$.
%
Noting that the output of system~$\mathscr F [\mathfrak M]$ is given
by the equality~$\phi^+ = S(z) \phi^-$, we can interpret the
Weyl-Titchmarsh function as a transfer function of an open system
formally written as
%
\begin{equation} \label{FullSystemMatrix}
\mathfrak F
\left(\vcenter{
\xymatrix@-1.6pc@M=0.6pt {
\Lambda + i S & & \upphi^+ \\
& \!\!\!\!\upphi^- \ar[ur] \ar[dr] &\\
R & & {\uppsi}
}}
\right), \qquad \upphi^- \in \D(\Lambda)
\end{equation}
%
This notation is formal because values of~$\Lambda + i S$ need not
be bounded operators defined everywhere on the coupling space~$E$.
%
Admissible input vectors~$\{\upphi^- \}$ of this system for which
the input-output mapping makes sense, belong to the domain~$\D(\Lambda) = \Gd \HH$.
%
The interior states of systems
$\mathfrak F$ and $\mathscr F[\mathfrak M]$ coincide and equal to
the set of solutions~$\{u_z^\varphi\}$ of
the spectral
problem~(\ref{NE_SpectralBVP}) with $f=0$,
$\varphi\in\Gd\HH$, $z\in\rho(A_0)$.
%
In other words,
the input--interior transformation of system~$\mathfrak F$ is the
restriction of the input--interior transformation~$R$
of system~$\mathscr F [\mathfrak M]$ to the set~$\Gd \HH$.
%
Finally, the output
vectors of~(\ref{FullSystemMatrix}) are
vectors~$\{\Gn u^\varphi_z\}$, that is, images of admissible
inputs under the mapping of Weyl-Titchmarsh function $M(\cdot)$.
%
For~$\upphi^- \in \D(\Lambda) = \Gd \HH$ inputs, outputs and
internal states of systems~$\mathfrak F$ and $\mathscr F[\mathfrak
M]$ from (\ref{FullSystemMatrix}) and~(\ref{GeneralSystemMatrix})
are related as follows
\begin{equation*}
\phi^- = \upphi^-, \quad \phi^+ = S \phi^- = i ( \Lambda \upphi^- - \upphi^+ ) ,
\quad \psi = \uppsi, \qquad \upphi^- \in \D(\Lambda).
\end{equation*}
%%=============================================================
\paragraph{ 2.}
For $z\in \rho(A_0)$ and input vector~$\varphi \in \Gd \HH$ the
output~$\upphi^+ = \Gn u^\varphi_z = M(z) \Gd
u^\varphi_z $ of system~$\mathfrak F$ is represented as a sum
of~$\Lambda \varphi$ and an analytic vector function~$iS(z)\varphi$
of the variable~$z\in\rho(A_0)$.
%
In applications, where the spectral parameter~$z\in\Complex$ has
the meaning of oscillation frequency, the first
summand~$\Lambda\varphi$ is interpreted as a ``static reaction'' of
the system, i.~e. the reaction at the zero frequency.
%
Following terminology of the system theory, the map~$\Lambda$ is
called a \textbf{feedthrough operator}.
%
Vector~$\Lambda\varphi$ is not defined for all inputs~$\varphi\in E$
unless~$\Lambda$ is bounded.
%
For $z\in\rho(A_0)$ in a small vicinity of the origin, the
summand~$i S(z)\varphi$ describes low-frequency oscillations of the
system~$\mathfrak F$ around its static reaction~$\Lambda\varphi$.
%
Obviously, if the static reaction is taken into account by
extraction of~$\Lambda\varphi$ from the output, the analysis of a
such modified system should be greatly simplified.
%
More accurately, according to~(\ref{MasLabmdaPlusS}), the
equality~$\Gn u^\varphi_z = M(z)\Gd u^\varphi_z$ for the
input--output mapping of the system~$\mathfrak F$ can be rewritten
in the form
%
\begin{equation}\label{KreinExtensionSystem}
\left(\Gn - \Lambda \Gd \right) u^\varphi_z =
i S(z) \Gd u^\varphi_z,
\end{equation}
%
hence the function~$iS(z)$ maps the input vector~$\varphi = \Gd
u^\varphi_z$ into~$\left(\Gn - \Lambda \Gd \right) u^\varphi_z$.
%
Thus we arrive at the system with the input--output
transformation~$\phi^- \mapsto \phi^+ - \Lambda \phi^-$, the null
feedback operator, and the transfer function~$i S(z)$ describing
small oscillations.
%
%%==================================================================
\paragraph{ 3.}
In the terminology accepted in the systems theory, if the
set of internal states is dense in the interior space then
the system~$\mathscr F[\mathfrak M]$ is
called \textbf{approximately controllable}.
%
According to~(\ref{TransferFunctionAndInteriorOperatorOfF}), this
condition means density of the linear span
\begin{equation*}
\bigvee_{e \in E,\; z \in \rho(A_0) }( I -
zA_0^{-1})^{-1}\Pi e
\end{equation*}
in the whole space~$H$.
%
The power series expansion of the resolvent combined with the
Theorem~\ref{TheoremOfUniqueness} now result in
the following observation.
\begin{rem}
Let $S$ and $\widetilde S$ be the transfer functions
of two approximately controllable systems with the same coupling
space~$E$. If $S(0) =\widetilde S(0) = 0$ and
$S(z) =\widetilde S(z)$ in some neighborhood of the origin, then
the $d$-nodes that belong to
these systems
are unitarily equivalent
in the sense of Theorem~\ref{TheoremOfUniqueness}.
\end{rem}
%%====================================================================
Another important notion of systems theory is the system's
\textbf{observability}.
%
In application to the system~$\mathscr F[\mathfrak M]$ with the
transfer function~$S(z)$, $z\in\rho(A_0)$ it seems natural to call
an internal state~$u_z \in \ker(A - zI)$ \textbf{unobservable} if
its corresponding output~$(\Gn - \Lambda \Gd) u_z$ is equal to zero,
cf.~(\ref{KreinExtensionSystem}).
%
Note that according to this
definition the set of unobservable vectors depends on the
parameter~$z\in\rho(A_0)$ and can be empty.
%
Since all internal states of~$\mathscr F[\mathfrak M]$ are elements
of~$H$ uniquely represented in the form~$u_z^\varphi =
(I-zA_0^{-1})^{-1}\Pi\varphi$ with $z\in\rho(A_0)$, $\varphi \in
\Gd\HH$, we see that for a given~$z\in\rho(A_0)$ unobservable
vectors of~$\mathscr F[\mathfrak M]$ are in fact the solutions to
the homogeneous BVP
%
\begin{equation}\label{KreinExtension}
\left\{\quad
\begin{array}{l}
(A - zI) u = 0 \\
( \Gn - \Lambda \Gd) u = 0
\end{array}
\right.
\end{equation}
%
In other words, if for $\varphi\in\Gd \HH$ and~$z\in\rho(A_0)$ the
internal state~$u_z^\varphi = (I - zA_0^{-1})^{-1}\Pi\varphi$
satisfies the ``boundary condition'' $( \Gn - \Lambda \Gd)
u_z^\varphi = 0$, then the state $u_z^\varphi$ is unobservable.
%
Another way to describe unobservable vectors can be based directly
on the definition of the transfer function~$S(\cdot)$ of the
system~$\mathscr F[\mathfrak M]$.
%
Indeed, the state~$u_z^\varphi$ is
unobservable if and only if~$S(z)\varphi = 0$.
%
%
Obviously, corresponding unobservable internal states in this case
are nothing but weak solutions to the equation~$(A - zI)u =0$.
%
Note as well that any vector from~$\HH $
satisfies~(\ref{KreinExtension}) with~$z=0$, therefore $\HH$
consists of unobservable states of the system~$\mathscr F [\mathfrak
M]$ at the zero frequency~$z=0$ (unobservable static reactions).
%%==============================================================
%%=======================================
%%
%%=======================================
\paragraph{ 4.}
The condition~(\ref{MzBehaviorOnInfinity}) of the ``regular''
behavior of the function~$M(z)$ along the imaginary axis in the
upper half plane can be used
as a definition for a class of ``regular'' systems (and
corresponding boundary value problems).
%
For these systems the summands in the decomposition~$M = \Lambda + i
S$ are not independent.
%
Equivalently, the components of the pair~$(\mathfrak M, \Lambda)$,
independent in general, in this case are related to each other.
%
Indeed, under assumption~$M(iy)\varphi \to 0$ when $y\to \infty$
for any~$\varphi \in \Gd\HH$, the operator~$\Lambda$ is reconstructed
from the operator function~$S(z)$ uniquely determined by
the colligation~$\mathfrak M$, by the limiting procedure~$\Lambda
\varphi = - \lim_{y\to\infty} i S(iy)\varphi$ where~$\varphi \in
\Gd\HH$.
%
%
%%==================================================================
%%
%%==================================================================
\section{Connection with the linear boundary control systems theory}
The preceding sections showed that the null extension~$A$
uniquely determines the open system~$\mathscr F[\mathfrak
M]$ and the spectral boundary value problem~(\ref{NE_SpectralBVP}).
%
Subsequent introduction of one more operator~$\Gn = \Pi^* A + \Lambda \Gd$
allowed us to define the Weyl-Titchmarsh function~$M(z)$, $z\in \rho(A_0)$
of the problem~(\ref{NE_SpectralBVP}) and express its various properties
in terms of the open
system~$\mathscr F[\mathfrak M]$
whose transfer function
in turn is uniquely determined by~$M(z)$.
%
%
In this section we
establish relationship between
the theory developed in the paper and the theory of
linear systems with boundary control.
%
It turns out that the open system~(\ref{DefinitionDNode})
associated with the BVP~(\ref{NE_SpectralBVP})
is the reciprocal~\cite{Cur, Sta}
of the system with boundary control as
defined by
the control theory~\cite{CuZw,Lions,Part,Sta}.
%
It will be shown that
the transfer function of this boundary control systems
coincides with
Weyl-Titchmarsh function of the problem~(\ref{NE_SpectralBVP}).
%
%%===============================================================
The research is based on the
obtained in Theorem~\ref{TheoremA1}
characteristics of
BVPs corresponding to null extensions
subject to Assumption~\ref{Assumption}.
%
%
The problem~(\ref{A_BVP1}) gives rise to the linear system described
by
the main operator~$A$ and control introduced by the
vector~$\varphi$ from the boundary space~$E$.
%
In accordance with common practice established in the control
theory,
consider $\Gn$ as the observation operator that maps
internal states of the
system into its output.
%
In other words, define the output of system
as~$y = \Gn u_z^\varphi$
where $u_z^\varphi$ is the internal
state corresponding to the input~$\varphi\in E$.
%
%%========================================================
Suppose all conditions of Theorem~\ref{TheoremA1} are satisfied
for~$A$, $\Gd$,
$\Gn$,
and the input~$\varphi$ belongs to~$\Gd\HH$.
%
Then we can seek the solution~$u$ to the problem~(\ref{A_BVP1}) in the
form $u =A_0^{-1} x + h$ with some $x\in H$ and $h\in \HH$.
%
Substitution into~(\ref{A_BVP1}) results in the
equation~$(I -zA_0^{-1}) x = z h$ for unknown vector~$x$.
%
From $\Gd u = \varphi$ we obtain $\Gd h = \varphi$,
therefore~$h = \Pi\varphi$.
%
It follows that for the solution~$u = A_0^{-1} x + h$
the controllability condition~$\Gd u = \varphi$
is fulfilled automatically if we put~$h = \Pi\varphi$.
%
At the same time the equation for~$x \in H$ takes the form $
\left(z^{-1} -A_0^{-1}\right) x = \Pi \varphi $.
%
The output~$y = \Gn u$ now can be rewritten
as~$y = \Gn (A_0^{-1}x + h) = \Pi^* x + \Lambda \varphi$.
%
Let us sum up these results in a proposition.
%
%%======================================
%
\begin{prop}
For any control~$\varphi\in \Gd \HH $
the system with boundary control
described by the problem~(\ref{A_BVP1})
with unknown $u\in \D(A)$ and
with the observation mapping
defined as~$u\mapsto y = \Gn u$
is equivalent to the system associated with the problem
\begin{equation}\label{A_BVP2}
\begin{aligned}
z^{-1} x & = A_0^{-1} x + \Pi \varphi
\\
y & = \Pi^* x + \Lambda \varphi
\end{aligned}
\end{equation}
where unknown vectors~$u\in \D(A)$ and~$x\in H$
are
related by the formula~$u = A_0^{-1} x + \Pi \varphi$.
%
\end{prop}
%%======================================================
Equalities~(\ref{A_BVP2}) describe the linear system with the bounded main
operator~$A_0^{-1}$, internal state~$x$, and the summand~$\Pi \varphi$
representing control.
%
The output of system~(\ref{A_BVP2}) is given by the
map~$ x \mapsto \Pi^* x + \Lambda \varphi$.
%
In this light,
$\Pi^*$ and~$\Lambda$ are the observation and feedthrough
operators, respectively.
%
Thus, the system~(\ref{A_BVP1}) with internal states~$u\in \D(A)$ is
equivalent to the system described by~(\ref{A_BVP2}).
%
Internal states of these two systems are related by the
equality~$u = A_0^{-1} x + \Pi \varphi$.
%
Moreover,
calculations of Section~1 show that the transfer
function of~(\ref{A_BVP1})
defined as~$\varphi \mapsto \Gn u_z^\varphi$,
where $u_z^\varphi$ is the solution to~(\ref{A_BVP1}),
is the Weyl-Titchmarsh function~$M(z)$, $z\in \rho(A_0)$
whose building blocks $A_0^{-1}$, $\Pi$, and $\Lambda$
are bounded operators defined by the system~(\ref{A_BVP2}).
%
%
The price paid for this reduction from
unbounded maps~$A$, $\Gd$ in~(\ref{A_BVP1})
to bounded~$A_0^{-1}$, $\Pi$ in~(\ref{A_BVP2})
is the possibly unbounded feedthrough
operator~$\Lambda$ of~(\ref{A_BVP2}) compared to
the null feedthrough operator
of~(\ref{A_BVP1}).
%
%%=====================================================
One remarkable detail about system~(\ref{A_BVP2}) is that
the control and
observation maps are
mutually adjoint to each other.
%
Consequently,
the matrix of this system written according to
the standard notation of linear systems
theory (see~\cite{Ar, Sta})
\[
\left(
\begin{array}{ll}
A_0^{-1} & \Pi \\
\Pi^* & \Lambda
\end{array}
\right)
\]
is selfadjoint (assuming the symmetric feedthrough map~$\Lambda$
is in fact selfadjoint).
%
Such systems are studied in the theory of electrical circuits, where
they are commonly termed \textbf{resistance
systems}.
%
The meaning of their transfer functions
is the electrical impedance that
maps ``voltage'' into ``current strength'', see~\cite{Ar}.
%%=============================================================
%%==============================================================
%%
%%==============================================================
\section{Weyl-Titchmarsh function of the Schr\"odinger operator}
Obtained results are equally applicable to the theory of boundary
value problems and to the theory of linear systems theory;
consequently, it is possible to illustrate the main points
of the paper
by examples originating in either of these two disciplines.
%
One such example has been already mentioned in the Introduction.
%
It consists of
the Dirichlet boundary value problem for the Laplacian defined
in the bounded simply
connected domain~$\Omega \subset \Real^3$ with the
smooth boundary~$\Gamma$.
%
The Dirichlet Laplacian~$A_0$ defined on functions from the usual
Sobolev class~$H^2(\Omega)$ vanishing on the boundary~$\Gamma$
gives rise to
the null extension~$A$ of~$A_0$ to the set~$\D(A) := \D(A_0) \dot{+}
\HH$ where $\HH$ denotes the subset of harmonic functions
in~$\Omega$ with smooth traces on~$\Gamma$.
%
Assumption~\ref{Assumption} is easily verified on the grounds of
well-known properties of harmonic functions in~$\Omega$ and
selfadjointness of~$A_0$ that follows from~\cite{Browder}.
%
The boundary map~$\Gd$ is the trace operator defined on~$\D(A)$.
%
Then, obviously
$\Pi$ is the operator of harmonic continuation from the
boundary~$\Gamma$ to the interior of domain~$\Omega$
defined on smooth functions given on~$\Gamma$.
%
The Green's identity for the Laplacian suggests
the second boundary operator~$\Gn$ chosen
as the trace
of normal derivative of
functions from~$\D(A)$
on the boundary~$\Gamma$.
%
%
%
With this choice
the Weyl-Titchmarsh function~$M(\cdot)$
maps a smooth function~$\varphi$
on~$\Gamma$ to the trace of normal derivative of the solution to
the problem~$(A -zI)u = 0$ satisfying boundary
condition~$u|_\Gamma = \varphi$.
%
A representation for~$M(\cdot)$ in the form of a pseudodifferential
operator acting in~$L^2(\Gamma)$ could be found for example,
in~\cite{Ag}.
%
According to the established terminology,~$M(\cdot)$ is the
Dirichlet-to-Neumann map for the Laplace operator on
domain~$\Omega$, see~\cite{SU} for more details.
%
%
At the same time, $M(\cdot)$ is the transfer function of the linear
system generated by the Laplacian in~$\Omega$ with the Dirichlet
boundary control on~$\Gamma$.
%
%%===========================================================
Of course, this example can be extended to more general situations
of
operators and domains.
%
In particular, the case of strongly elliptic operators and
systems
defined on domains of lesser regularity~(see \cite{ML}, for instance)
is the first candidate for such generalizations.
%
However, in this section we will explore another setting
where the operator~$A_0$ is not defined in terms of boundary
conditions.
%
In fact, its definition does not involve any notion of
``boundary'' at all.
%
We continue to consider the null extensions framework as
a convenient method
to introduce coupling channels into the open system
described by operator~$A_0$.
%
However,
in this section we emphasize another interpretation
of these channels
as some sort of perturbations of the operator~$A_0$.
%
%
This ``perturbative'' aspect of null extensions and
their relation to the general singular perturbations
theory~\cite{AlbevKur}
will be treated in detail elsewhere.
%%================================================================
Our underlying motive in this section is
the reconciliation of the
Weyl-Titchmarsh function~$\mathfrak M (\cdot)$ for the
three-dimensional Schr\"odinger
operator
studied by W.~O.~Amrein and D.~B.~Pe\-ar\-son in~\cite{AmP}
with the theory developed in the paper.
%
The operator~$A_0$ is defined in~$L^2(\Real^3)$ by the
differential expression~$\mathscr L = - \Delta + q(x)$
with the real-valued
bounded potential function~$q(x)$.
%
The domain of~$A_0$ is
the usual Sobolev class~$H^2(\Real^3)$.
%
As will be shown, the
function~$\mathfrak M (\cdot)$
from~\cite{AmP}
coincides with the Weyl-Titchmarsh function
of some null extension of~$A_0$.
%
%
Under addi\-ti\-o\-nal smoothness conditions on the function~$q(x)$
we
also obtain an expression for~$\mathfrak M(\cdot)$
in the form of single layer potential
operator associated with the Green's function
of~$\mathscr L = -\Delta + q(x)$ in~$L^2(\Real^3)$.
%
%%==============================================================
Let us start with some heuristic observations.
%
Assume
$H := L^2(\Real^3)$ and $A_0$ is the
selfadjoint Schr\"odinger
operator corresponding to the
expression~$\mathscr L = - \Delta + q(x)$
defined on the domain~$\D(A_0) = H^2(\Real^3)$.
%
%
Without any loss of generality we will assume that~$A_0$
is boundedly invertible in~$H$.
%
If not, a suitable constant can be added to~$A_0$ to ensure
the equality~$\ker({A_0}) =\{0\}$,
see Remark~\ref{RemarkShift}.
%
%
%
%
Suppose
the physical problem under consideration is formulated
as a certain ``perturbation'' of~$A_0$ by a smooth closed compact
surface~$\Gamma \subset \Real^3$ that splits~$\Real^3$ into
the interior domain~$\Omega$ bounded by~$\Gamma$ and
the exterior domain~$\Real^3 \setminus \overline \Omega$.
%
The surface~$\Gamma$ is their common boundary.
%
For example, the problem in hand could
be the scattering process by an obstacle~$\Omega$
with boundary~$\Gamma$.
%
The common way to introduce the ``perturbed''
operator would be to consider a restriction of~$A_0$ to the set of
smooth functions vanishing in the neighborhood of~$\Gamma$ and then
to study various extensions of thus obtained symmetric operator
corresponding to different types of boundary conditions
on~$\Gamma$.
%
The problem naturally breaks into two independent boundary
subproblems; the first one is for the interior of the scatterer, and
the second one is for the external area that includes the infinity.
%
Thus, this approach introduces two separate Hilbert spaces of
functions defined inside and outside of the scatterer whose relation
to the unperturbed operator acting in the whole space is rather
loose.
%
Indeed, by the very nature of scattering process the ``free''
operator~$A_0$ does
not depend on the scatterer at all, whereas two former operators
act in the spaces defined exclusively in terms of the scatterer.
%
Although the situation is manageable, having to deal with three
spaces instead of the initial one can be perceived as a certain
indication of inadequate modeling.
%%====================================================================
A plausible alternative can be seen
in considering null extensions~$A$ of~$A_0$
to the direct sum~$\D(A) := \D(A_0) \dot{+} \HH$ where $\HH$ is some linear
set of solutions~$h \in L^2(\Real^3)$ to the
equation~$\mathscr L h = 0$.
%
The set~$\HH$ automatically belongs to~$L^2(\Real^3)
\setminus H^2(\Real^3)$, since otherwise $A_0$ would be not boundedly
invertible.
%
Below we explore the case when~$\HH$ is composed of
single and double layer potentials associated with~$\mathscr L$
with densities
supported by~$\Gamma$.
%%===================================================================
\paragraph{ Single layer potentials. }
First, we need to make some assumptions with regard to the
function~$q(x)$
that would allow us to employ methods of layer
potentials~\cite{CK, ML, Miranda}.
%
Although cases of less regularity can be considered, we assume for
simplicity that~$q \in C^\infty(\Real^3)$ and~$\Omega$
is~$C^\infty$\nobreakdash-domain.
%
It will be clear from the sequel that in fact the only
requirements on~$q$ and~$\Omega$ are the possibility
to define
layer potentials
associated with~$\mathscr L$
and the boundary~$\Gamma$ that possess
usual properties of their acoustic counterparts
corresponding to $q=0$, see~\cite{CK, ML}.
%
Under our smoothness assumptions
the
resolvent~$(A_0 - zI)^{-1}$, $z\in \rho(A_0)$ is an
integral operator with
the kernel~$ G(x,y, z)$, $x,y \in \Real^3$.
%
The function~$G(x,y, z)$ is infinitely differentiable if~$x \neq y$
and has singularities like~$|x-y|^{-1}$ when~$|x-y| \to 0$.
%
It is symmetric in~$x$ and $y$ and real-valued
for~$z\in\rho(A_0)\cap \Real$.
%
Traditionally $G(x,y, z)$ is called the Green's function of~$A_0$.
%
%
%
%
%
%
For
smooth functions~$w$ on~$\Gamma$ the
\textbf{single-layer potential}~$\mathscr S_z w$
is defined by
\begin{equation*} %% \label{LayerPotentials}
(\mathscr S_z w)(x) := \int_\Gamma
G(x,y, z) w (y) d\sigma_y, \qquad x\in \Real^3
\end{equation*}
%
where $d \sigma_y$ is the Euclidian surface measure on~$\Gamma$.
%
For~$q(x) =0$ the operator~$\mathscr S_z$ is the usual acoustic
single layer potential for the Helmholtz equation, cf.~\cite{CK}.
%
%
%
%
Outside
the surface~$\Gamma $
functions~$\mathscr S_z w$ are infinitely differentiable and satisfy the
equation~$(\mathscr L - zI) \mathscr S_z w =0$.
%
Note that since the Lebesque measure of~$\Gamma$ in~$\Real^3$ is
zero, we can say that the layer potential~$\mathscr S_z w$ satisfy
this equation almost everywhere in~$\Real^3$, hence in $H =
L^2(\Real^3)$.
%
This makes functions~$\mathscr S_z w$ at~$z=0$ good candidates to
the role of channel vectors~$\HH = \ker(A)$ within the developed
above theory.
%
%
%
%
Denote~$\partial_\nu = \frac{\partial}{\partial \nu}$ the normal
derivative in the direction of outer normal to the domain~$\Omega$
defined everywhere on~$\Gamma$.
%
Proofs of the following properties of $\mathscr S_z$ and
$\partial_\nu \mathscr S_z$ when $x\in \Real^3\setminus\Gamma$ tends to
some~$x_0\in \Gamma$ can be found for instance in~\cite{ML, Miranda}.
%
%
Denote~$\Omega^- := \Omega$ and ~$\Omega^+ := \Real^3 \setminus
\overline{\Omega}$.
%
For~$z\notin\Real$ operators~$\mathscr S_z$ map the
space~$E := L^2(\Gamma)$ into $L^2(\Omega^\pm)$.
%
Boundary values~$(\mathscr S_z w)^\pm $, $(\partial_\nu\mathscr S_z w)^\pm $
of~$\mathscr S_z w$ and $\partial_\nu\mathscr S_z w$
on the surface $\Gamma$ from~$\Omega^\pm$ exist as
elements of~$L^2(\Gamma)$.
%
Almost everywhere on~$\Gamma$ these values satisfy the so-called
``jump relations'' (cf.~\cite{CK}):
\begin{equation}\label{JumpRelations}
(\mathscr S_z w)^\pm = \mathsf S_z w, \qquad
(\partial_\nu\mathscr S_z w)^\pm = \mathsf T_z w \mp
\frac{1}{2}w \quad
\end{equation}
%
Here~$w$ is assumed continuous on~$\Gamma$ and
for almost all $x\in\Gamma$
%
\[
(\mathsf S_z w)(x) := \int_{\Gamma}
G(x,y,z) w(y) d\sigma_y, \qquad
(\mathsf T_z w)(x) := \int_{\Gamma}
[{\partial_{\nu(x)} G(x,y,z)}] w(y) d\sigma_y
\]
%
Operators~$\mathsf S_z$ and $\mathsf T_z$ are bounded in
$L^2(\Gamma)$.
%
%
%
From~(\ref{JumpRelations}) and usual density arguments we obtain
almost everywhere on~$\Gamma$
\begin{equation}\label{JumpRelations2}
(\mathscr S_z w)^- - (\mathscr S_z w)^+ = 0,
\quad
(\partial_\nu\mathscr S_z w)^- - (\partial_\nu\mathscr S_z w)^+ =
w, \quad \text{ for } w \in
L^2(\Gamma)
\end{equation}
%
%
%%======================================================================
Following the schema explained above, we define~$\HH$ as the set of
single layer potentials~$\{\mathscr S_0 \varphi\}$ with
densities~$\varphi$ from some linear manifold~$\mathscr E$
dense in~$L^2(\Gamma)$.
%
Without loss of generality we assume that~$\mathscr E$ consists of
smooth functions.
%
Then functions from~$\HH$ are infinitely differentiable in~$\Real^3
\setminus \Gamma$, and continuous in~$\Real^3$.
%
On the surface~$\Gamma$ normal derivatives of any function
from~$\HH$ is discontinuous
according to~(\ref{JumpRelations2}).
%
Introduce~$A$ as a null extension of $A_0$ to the domain~$\D(A)
:=\D(A_0) \dot{+} \HH = H^2(\Real^3) \dot{+} \mathscr S_0 \mathscr E
$ and define the coupling operator~$\Pi : E \to H$ as~$\varphi \mapsto
\mathscr S_0 \varphi$ where $\varphi \in \mathscr E$.
%
The jump relations~(\ref{JumpRelations2}) suggest the
following
choice for the boundary map:~$\Gd : u \mapsto (\partial_\nu u)^-|_\Gamma - (\partial_\nu
u)^+|_\Gamma$, where $(\partial_\nu u)^\pm|_\Gamma$ are the traces
of normal derivatives of $u\in\D(A)$ on the surface~$\Gamma$ from~$\Omega^\pm$.
%
Then $\Gd\Pi \varphi = \Gd \mathscr S_0 \varphi = \varphi$ for
$\varphi \in \mathscr E$, and $\Pi \Gd h = h$ for $h \in \HH$, as
required.
%
Furthermore, jumps of normal derivatives of functions from $\D(A_0)$
on the surface~$\Gamma$ are equal to zero due to Sobolev imbedding
theorems, therefore~$\Gd \D(A_0) = 0$.
%
We see now that the pair~$\{A, \HH\}$ defines
a certain boundary value problem satisfying Assumption~\ref{Assumption},
and simultaneously a system with boundary control and an open system
of M.~S.~Liv\v{s}i{c}.
%
%
%
%%==================================================================
%
According to~(\ref{DefinitionOfGammaN}), the second boundary
operator has the form~$\Gn := \Pi^* A + \Lambda \Gd$, where
$\Lambda$ is the feedthrough map of the system.
%
It determines the action of~$\Gn$ on
the set~$\HH = \mathscr S_0 \mathscr E$
and always can be chosen arbitrarily as long as it is symmetric on
the domain~$\D(\Lambda) := \mathscr E$.
%
In order to make a reasonable choice let us
calculate~$\Pi^*$ and $\Pi^* A$.
%
Then we can discuss possibilities for~$\Lambda$ and~$\Gn$ more
intelligently on the grounds of obtained results.
%
%
%
For~$\varphi \in \mathscr E$ and $f \in L^2(\Real^3)$ we have
\[
\begin{aligned}
(\Pi\varphi, f )_H & = (\mathscr S_0 \varphi , f )_H
= \int_{\Real^3}
\left (
\int_\Gamma G(x,y,0) \varphi(y) d\sigma_y
\right)
\overline{f(x)}\, dx
\\
& =
\int_\Gamma \varphi (y)
\left(
\int_{\Real^3} G(x,y,0) \overline{f(x)}dx
\right) d\sigma_y
= \left(\varphi, \left. A_0^{-1}f \right|_\Gamma\right)_E
\end{aligned}
\]
by the virtue of Fubini's theorem.
%
Therefore, $\Pi^* : f \mapsto A_0^{-1} f|_\Gamma$ and $\Pi^* A : f_0
\mapsto f_0|_\Gamma$ for $f_0\in \D(A_0)$.
%
The restrictions~$\left. A_0^{-1}f\right|_\Gamma$
of functions from~$\D(A_0) = H^2(\Real^3)$ to the
surface~$\Gamma$ exist due to imbedding theorems.
%
%
Let us choose $\Gn$ on~$\D(A)$ as an operator that maps elements
from $\D(A) = H^2(\Real^3) \dot{+}\mathscr S_0\mathscr E$ to their
traces on the surface~$\Gamma$.
%
This definition is unambiguous, since single layer potentials of
continuous functions are continuous in the whole~$\Real^3$.
%
Then for $\Lambda = \Gn \Pi$ we obtain
$\Lambda \varphi = \mathscr S_0 \varphi|_\Gamma = \mathsf S_0 \varphi$,
$\varphi \in \mathscr E$.
%
Note that~$\Lambda$ is bounded in~$L^2(E)$, hence the
Weyl-Titchmarsh is also bounded.
%
The input--interior transformation~$R(z)$ has the
form~$R(z) : \varphi \mapsto \mathscr S_{z} \varphi$
for~$\mathscr S_{z} \varphi \in \ker(A- zI)$
and $\Gd \mathscr S_{z} \varphi = \varphi$.
%
Finally, for the Weyl-Titchmarsh function we have
$M(z)\varphi = \Gn R(z)\varphi = (\mathscr S_{z}\varphi) |_\Gamma =
\mathsf S_{z}\varphi$,
so that $M(z) = \mathsf S_{z}$, where $\mathsf S_{(\cdot)}$
is the operator of single layer potential on the
surface~$\Gamma$.
%
%%====================================================================
Let us summarize obtained results. For definiteness
we assume~$\mathscr E := C^\infty(\Gamma)$
\begin{equation}\label{Gathered}
\begin{gathered}
H = L^2(\Real^3), \qquad E := L^2(\Gamma),
\qquad \mathscr E := C^\infty(\Gamma)
\\
\mathscr L := -\Delta + q(x), \quad
A_0 f \mapsto \mathscr L f, \quad f \in \D(A_0) := H^2(\Real^3),
\\
A : u \mapsto \mathscr L u, \quad u \in \D(A) := \D(A_0) \dot{+}
\HH, \; \text { where }\;
\HH := \{ \mathscr S_0 \varphi \; | \; \varphi \in \mathscr E \},
\\
\Gd : u \mapsto (\partial_\nu u)^-|_\Gamma - (\partial_\nu
u)^+|_\Gamma,
\qquad \Gn : u \mapsto \left. u\right|_\Gamma,
\qquad u \in \D(A),
\\
\Pi : \varphi \mapsto \mathscr S_0 \varphi,
\qquad
\Lambda : \varphi \mapsto \left. \mathscr S_0
\varphi\right|_\Gamma = \mathsf S_0\varphi,
\qquad \varphi \in E,
\\
R(z) : \varphi \mapsto \mathscr S_{z} \varphi, \qquad
M(z) : \varphi \mapsto \mathsf S_z \varphi,
\qquad \varphi \in
E
\end{gathered}
\end{equation}
%
Vectors~$\varphi$ on the last two lines belong to~$E$
since
operators~$\Lambda$ and $M(z)$ are bounded,
therefore can be continuously extended from the
dense set~$\mathscr E$ to the
whole space~$E$.
%%================================================================
Now we can examine a question as to how
the Weyl-Titchmarsh
function $\mathfrak M(\cdot)$ of the
Schr\"o\-din\-ger operator
introduced by W.~O.~Amrein and D.~B.~Pearson in~\cite{AmP}
relates to the construction above.
%
%
We will see that for smooth potentials~$q(x)$ the
function~$\mathfrak M(\cdot)$ coincides with the Weyl-Titchmarsh
function~$M(\cdot)$ from~(\ref{Gathered}).
%%===============================================================
\begin{defn}[\cite{AmP}]\label{AmPDefinition}
%
Let $\mathscr L$ be the Schr\"odinger differential
expression~$-\Delta + q(x)$
with the real-valued potential function~$q(x)\in L^\infty(\Real^3)$
acting in~$L^2(\Real^3)$ and $S_1$ be the unit sphere
in~$\Real^3$.
%
%
For $z\in \Complex_+$, the operator~$\mathfrak M(z) : L^2(S_1) \to L^2 (S_1)
$
is defined by
\[
\mathfrak M (z) v = - w
\]
where $w = \gamma^\pm f$ and $f \in H^2(\Real^3 \setminus S_1)$
is the unique solution
of~$(\mathscr L - zI) f = 0 $ subject to conditions
\begin{equation}\label{AmPFunction}
\gamma^+ f = \gamma^- f, \qquad
\gamma^+ \frac {\partial f}{\partial \nu } -
\gamma^- \frac {\partial f}{\partial \nu } = v.
\end{equation}
Here~$\gamma^\pm $ are trace operators associated with the exterior
and the interior of the unit ball~$B_1$ in~$\Real^3$.
%
It is assumed that the boundary values in~(\ref{AmPFunction})
exists as functions from~$L^2(S_1)$.
\end{defn}
%
%%=================================================================
The equality~$M = \mathfrak M$ already can be derived by comparison
of
Definition~\ref{AmPDefinition} with the formulae~(\ref{Gathered}).
%
However, let us give a more detailed
account assuming~$q\in C^\infty(\Real^3)$.
%
Suppose~$\varphi$ is a smooth function on~$S_1$.
%
For~$\Gamma = S_1$ the formulae~(\ref{Gathered}) and the
Weyl-Titchmarsh function definition show that~$M(z)\varphi$
can be calculated as follows.
%
First, one need to solve the
problem $(\mathscr L - zI) u =0 $
inside and outside of the unit ball
subject to
the ``transmission'' conditions $u^- = u^+$, $u^- - u^+ = \varphi$
imposed on the boundary values of the solutions~$u^\pm$.
%
Here the signs~$\pm$ correspond to the domains~$\Omega^\pm$
with~$\Omega^- = B_1$ and ~$\Omega^+ = \Real^3 \setminus \overline
B_1$.
%
%
The solution to this problem in the whole space~$\Real^3$
is represented as the sum
$u_z = u^+ + u^-$.
%
The function~$u_z$ is continuous in~$\Real^3$
and its derivatives discontinue
on the support of~$\varphi$.
%
In terms of corresponding open system, $u_z$ is the interior state
obtained from the input~$\varphi$ by the operator~$R(z)$.
%
The Weyl-Titchmarsh function~$M(z)$ maps~$\varphi$ to the
trace of $u_z$ on the surface~$S_1$ defined unambiguously since $u_z$
is continuous.
%
Now it is clear
that the functions~$f$, $v$, and $w$
from Definition~\ref{AmPDefinition} in this notation
are $u_z$, $-\varphi$, and $u|_{S_1}$, respectively.
%
Therefore, $M(z)\varphi = \mathfrak M(z)\varphi$ for
continuous~$\varphi$, hence $M(z) = \mathfrak M(z)$, $z\notin\Real$
due to boundedness of~$M$ and $\mathfrak M$.
%
%
%
%%====================================================================
%
%
\paragraph{ Double layer potentials. }
As a natural extension of the argumentation above
it is possible to
consider
the case of double layer potentials
with densities supported by the surface~$\Gamma$.
%
Then
the channel vectors from~$\HH$ are discontinuous across
the surface~$\Gamma$, but their normal derivatives are
continuous everywhere in~$\Real^3$.
%
Analogously to the above, we assume that~$q$ and~$\Omega$
are such that
\textbf{double-layer potential}~$\mathscr D_z w$
defined by
\begin{equation*}
(\mathscr D_z w)(x) := \int_\Gamma
[\partial_{\nu(y)} G(x,y, z)] w (y) d\sigma_y, \qquad x\in
\Real^3\setminus\Gamma,
\end{equation*}
%
possesses all usual properties of its acoustic counterpart.
%
In particular, we assume that
the potential~$\mathscr D_z w$ with smooth~$w$
defined on
$\Gamma$
has boundary values~$(\mathscr D_z w)^\pm$
from inside and outside of~$\Omega$ and
the jump relations
are valid:
\[
(\mathscr D_z w)^+ - (\mathscr D_z w)^- = w , \qquad
( \partial_\nu \mathscr D_z w)^+ - ( \partial_\nu \mathscr D_z w)^- = 0
\]
%
%
%
We choose $\HH$ to be the set of double layer
potentials~$\{\mathscr D_0 \varphi \}$ with smooth
densities~$\varphi$ that belong to some
linear set~$\mathscr E $ dense in $E := L_2(\Gamma)$.
%
%
Noting that the jump on the surface~$\Gamma$ for any functions
from~$\D(A_0) = H^2(\Real^3)$ is always equal to zero, we define the
operator~$\Gd$ on~$\D(A) := \D(A_0) \dot + \HH = H^2(\Real^3) \dot +
\mathscr D_0 \mathscr E$ to be~$ \Gd : u \mapsto u^+|_\Gamma -
u^-|_\Gamma$, where $u^\pm|_\Gamma$ are limit values on $\Gamma$
from $\Omega^\pm$
of the function~$u\in\D(A)$.
%
%
The inverse~$\Pi = (\Gd|_{\HH})^{-1}$ is the mapping~$\Pi : \varphi
\mapsto \mathscr D_0 \varphi$ defined on~$\mathscr E = \Gd \HH$.
%
%
Calculations similar to conducted above for single layer potentials
show that the adjoint~$\Pi^*$ is given by $ \Pi^* : f
\mapsto \left.\partial_\nu A_0^{-1} f \right|_\Gamma$,
$ f \in H$.
%
This formula holds for any~$f\in H$ due to boundedness
of~$\Pi^*$ ensured by imbedding theorems.
%
Therefore, $\Pi^* A : f_0
\mapsto \left. {\partial_\nu} f_0 \right|_{\Gamma}$, $f_0 \in
\D(A_0)$.
%
Any function~$f_0 \in\D(A_0)$ has continuous derivatives, thus the
trace~$\left.{\partial_\nu} f_0 \right|_\Gamma$ is defined
unambiguously.
%
Obtained result for~$\Pi^*A $ suggests a plausible definition for
the operator~$\Gn = \Pi^* A + \Lambda \Gd$ on the
domain~$\D(A) = \D(A_0) \dot {+} \mathscr D_0 \mathscr E$ as $\Gn : u
\mapsto \left.\partial_\nu u\right|_\Gamma$, $u \in \D(A)$.
%
%
Having made this particular choice of~$\Gn$, we can calculate action
of~$\Lambda = \Gn\Pi$ on the domain~$\mathscr E$.
%
Obviously, $\Lambda = \Gn \Pi : \varphi \mapsto \left.
\partial_\nu \mathscr D_0\varphi \right|_\Gamma$,
$\varphi \in \mathscr E$.
%
Introduce \textbf{hypersingular operator}~$\mathsf R_z$ defined on
smooth functions~$w$ from~$L^2(\Gamma)$ by
\[
(\mathsf R_z w)(x) := \left. {\partial_{\nu(x)}}
\int_{\Gamma}
[{\partial_{\nu(y)} G_z(x,y,z)}] w(y)
d\sigma_y\right|_\Gamma
\]
%
Values of~$\mathsf R_z$ are unbounded operators in~$L^2(\Gamma)$ and
it can be shown that for $q\in C^\infty$ and $\Omega$ of the
$C^\infty$ class, operators~$\mathsf R(z)$ are pseudodifferential
of order~$1$, see~\cite{ML}.
%
Therefore, $\Lambda = \mathsf {R}_0 $ is unbounded
%
Following the line of
reasoning employed for the case of single layer potentials, we
conclude that the Weyl-Titchmarsh function~$M(z)$
corresponding to the problem under consideration is~$\mathsf
R_z$, $z\notin \Real$.
%
This form of the Weyl-Titchmarsh function for the three dimensional
Schr\"odinger operator can be treated
equally with the $M$-function~$\mathfrak M(\cdot)$ of
W.~O.~Amrein and D.~B.~Pearson
discussed above.
%
%%====================================================================
The boundary value problem for~$A_0$ with ~$q=0$ perturbed by double
layer potentials with boundary conditions~$ \Gd u = \lambda\Gn u$
where $\lambda \in \Complex$ is a complex parameter is closely
related to problems arising in acoustics.
%
For the detailed analysis that involves methods of
pseudodifferential operators theory see~\S~9.4 in~\cite{Ag} and
references therein.
%
%
%%====================================================================
%
%
\paragraph{ Remarks on singular perturbations. }
We conclude this section with the following observation.
%
Arguments of systems theory indicate quite clearly that by
introduction of coupling channels realized as layer
potentials into the system
governed by the operator~$A_0$ one introduces
some kind of perturbations of~$A_0$.
%
Obviously, these perturbations are not additive.
%
%
The adequate mathematical object that describes this
type of perturbations is the operator colligation
consisting in our case of two independent operators.
%
However, if one take into consideration
the relation~(\ref{KreinResolventFormulaForZ=0}), it becomes
clear that by adding a boundary condition, in other words
by
introducing a linear dependency on the inputs and outputs of the
system,
the setting is reduced to the case of an
extension of the minimal symmetric
operator.
%%==================================
To clarify this point
let~$\Gd$ and $\Gn$ be the boundary operators defined
earlier for the BPV associated with single layer potentials
and $\gamma(x)$ be a continuous function on
$\Gamma$.
%
%
It was shown in~\cite{BEKS}
that the BVP for the Laplacian~$-\Delta$
in~$L^2(\Real^3)$
with boundary
conditions~$ \Gd u = \gamma(x)\Gn u$
can be used as a mathematical model of the quantum
mechanical Schr\"odinger operator perturbed by the singular potential
supported by the surface~$\Gamma$.
%
Denote this operator~$T$.
%
Formula~(\ref{KreinResolventFormulaForZ=0}) now
describes~$T$ in perturbative terms.
%
It shows that $T$ is not
an additive perturbation
of~$A_0$,
but its inverse~$T^{-1}$ is
an additive perturbation of~$A_0^{-1}$.
%
More precisely, $T^{-1} = A_0^{-1} + \Pi (\gamma^{-1} - \mathsf S_0)^{-1}\Pi^*$
assuming the operator~$\gamma^{-1} - \mathsf S_0$ is
boundedly invertible.
%
The connection between extensions of symmetric operators and
the theory of singular perturbations
is well known and thoroughly
described in the literature~\cite{AlbevKur,Posil}.
%
Its interpretation in the terms of systems theory
captured in the paper
may prove beneficial for the analysis
of linear systems with singular control~\cite{Lyashko}.
%
%%=======================================================================
%%
%%=======================================================================
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Elliptic Boundary Problems.
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Differential Equations IX.} (Springer-Verlag, 1997).
%
%
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{\textsc S.~Albeverio and P.~Kurasov}.
\emph{Singular Perturbations of Differential Operators}.
(Cambridge University Peress, 2000).
%
%
%
%
\bibitem{AS}
{\textsc A.~Alonso \and B.~Simon}.
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semibounded operators. \emph{ J. Oper. Theory.} \textbf{4} (1980) 251-270.
%
%
%
%
\bibitem{AmP}
{\textsc W.~O.~Amrein \and D.~B.~Pearson}.
$M$-operators: a generalisation of
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\textbf{171},\textbf{ 1-2 }(2004) 1--26.
%
%
%
%
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(Russian)
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English Transl.:
\emph{Siberian Mathematical Journal.} \textbf{20} \textbf{(2)} 149--162.
%
\bibitem{At}
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Discrete and Continuous Boundary Problems}. (Academic Press, 1964).
%
%
%%%\bibitem{BadeFreeman}
%%% {\textsc W.~G.~Bade, R.~S.~Freeman}.
%%% Closed extensions of the Laplace operator
%%% determined by a general class of boundary conditions.
%%% \emph{Paific J. Math.} \emph{12} (1962),
%%% 395-410.
%
%
%
%%%\bibitem{Beals}
%%% {\textsc R.~Beals}.
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\end{thebibliography}
\end{document}
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