\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 38, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/38\hfil Deficiency indices]
{Deficiency indices of a differential operator satisfying
certain matching interface conditions}

\author[P. K. Baruah, M. Venkatesulu\hfil EJDE-2005/38\hfilneg]
{Pallav Kumar Baruah, M. Venkatesulu} % in alphabetical order

\address{Pallav Kumar Baruah \hfill\break
Department of Mathematics and Computer Science\\
Sri Sathya Sai Institute of Higher Learning,  Prasanthinilayam, India}
\email{baruahpk@yahoo.com}

\address{M. Venkatesulu \hfill\break
% Senior Professor of Mathematics & Head, P.G. 
 Department of Computer Applications,
 Arulmigu Kalasalingam College of Engineering,
 Krishnankoil-626190.
 Virudhunagar (District),
 Tamil Nadu, India}
\email{venkatesulu\_m2000@yahoo.co.in}

\date{}
\thanks{Submitted October 7, 2004. Published March 29, 2005.}
\subjclass[2000]{34B10}
\keywords{Ordinary differential operators; Green's formula; deficiency index;
\hfill\break\indent
formal selfadjoint boundary-value problems; boundary form; deficiency space}

\begin{abstract}
 A pair of differential operators with matching interface
 conditions appears in many physical applications such as:
 oceanography, the study of step index fiber in optical fiber
 communication, and one  dimensional scattering in quantum theory.
 Here we initiate the study the deficiency index theory of
 such operators which precedes the study of the spectral theory.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corrolary}

\section{Introduction}

 In the study of acoustic wave guides in the ocean, and of one dimensional
time independent scattering in quantum theory, we come across of problems
of the from
$$
L_1 f_1=\sum_{k=0}^nP_k{df_1^k\over dt^k}=\lambda f_1
$$
 defined on an interval $I_1 = (a,c]$ and
$$
L_1f_1=\sum_{k=0}^nP_k{df_1^k\over dt^k}=\lambda f_1
$$
defined on an interval $I_2 = [c,b)$,  with
 $-\infty \leq a<c<b \leq +\infty$.
Here $\lambda$ is an unknown constant and the functions $f_1,f_2$ are
required to satisfy certain mixed conditions at the interface $t=c$.
In most cases, the complete set of physical conditions on the system give
rise to selfadjoint spectral problems associated with the pair $(L_1,L_2)$.

 Initial-value problem and boundary-value problems for regular and singular
cases for these equations have been discussed in publications such as
\cite{b2,v1,v2,v3,v4,v5}.
It is important to study the deficiency index theory of an operator
before one embarks on the study of the spectral theory. Here we present a
simple result on deficiency index of such operators. We take help of the results
available in \cite{d1}; however the proof of the main theorem rendered here is
new, not the same as that found in \cite{d1}. A similar study is found
in a recent work of Orochko \cite{o1}, where he has considered two arbitrary
even ordered symmetric differential expressions degenerated at the point of
interface.
The operator depends on two parameters $p,q$ and based on certain relations
between these parameters and the order of the expressions, the interface
point is classified into penetrable or impenetrable. Whereas in this work we
consider the the interface point to be regular and the functions to be
sufficiently smooth.


\subsection*{Definitions and Notation}

Let $I_1=(a,c]$ and $I_2=[c,b)$ where $-\infty\leq <a<c<b\leq +\infty$.
For any non-negative integer n, let $C^n(I_i)$ denote the space of all
complex valued $n$-times continuously differentiable functions defined on
$I_i;i=1,2$. Let $C^\infty(I_i)$ denote the space of all infinitely many
times differentiable complex valued functions defined on $I_i;i=1,2$.
Let $A^n(I_i)$ denote the space of all functions in $C^{(n-1)}(I_i)$
such that $(n-1)^{th}$derivative is absolutely continuous over each compact
subset of $I_i;i=1,2$. For a function $f$, $f^{(j)}$ denote the $j^{th}$
derivative of $f$, if it exists. For any $m\times n$ matrix A, let $A^*$
denote the adjoint of A. For a square matrix A, $A^{-1}$ denotes the inverse
of $A$, if it exists. For any two nonempty sets(topological spaces)
$V_1$ and $V_2$, let $V_1\times V_2$ denote the cartesian product
(space equipped with product topology) of  $V_1$ and $V_2$, taken in that
order. Let $L_2(I_i)$ denote the space of all measurable complex-valued
functions square integrable on $I_i,i=1,2$. Let the inner product in
$L_2(I_i)$ be denoted by $\langle .,.\rangle$, $i=1,2$.
Let $H^n(I_i)$ denote those functions $f$ in $A^n(I_i)$ such that $f^{(n)}$
belongs to  $L_2(I_i),i=1,2$. Let $H_0^n(I_i)$ denote the space of all
functions $f$ in $H^n(I_i)$ such that $f$ vanishes in a neighbourhood of
 $a$ and $f(c)=f'(c)=\dots ...=f^{(n-1)}(c)=0$. Let $H_0^n(I_2)$ denote the
space of all functions $f$ in $H^n(I_2)$ such that $f$ vanishes in a
neighbourhood of $b$ and $f(c)=f'(c)=\dots =f^{(n-1)}(c)=0$.

Let $A$ and $B$ be non singular $n\times n $  matrices with complex
entries. For $f_i\in C^n(I_i)$, let
$\tilde f_i(t)= column( f_i(t),f'_i(t),\dots ,f^{(n-1)}(t)), t\in I_i$,
$i=1,2$. Let $H^n(I_1\times I_2)$ denote the space of  all pairs
$(f_1,f_2)\in H^n(I_1)\times H^n(I_2)$ such that
$A\tilde f_1(c)=B\tilde f_2(c)$. Let $H_0^n(I_1\times I_2)$ denote the space
of all pairs $(f_1,f_2)\in H^n(I_1\times I_2)$ such that $f_1$ vanishes in
a neighbourhood of $a$ and $f_2$ vanishes in a neighbourhood of $b$.

Let $\tau _1$ and $\tau _2$ be a pair of formal ordinary differential
operators of order n defined on the intervals $I_1$ and $I_2$, respectively,
of the form
$$
 \tau _1 = \sum _{k=0}^n a_k(t)({d\over dt})^k ,\quad
 \tau _2 = \sum _{k=0}^n b_k(t)({d\over dt})^k
$$
where the coefficients $a_k\in C^\infty(I_1)$,
$b_k\in C^\infty(I_2)$ and $a_n(t)\neq 0$ and $b_n(t)\neq 0$ on $I_1$ and
$I_2$ respectively.
For $(f_1,f_2)\in A^n(I_1)\times A^n(I_2)$, let
$$
(\tau _1,\tau_2)(f_1,f_2)= (\tau _1f_1,\tau_2f_2)
$$
where
\begin{gather*}
(\tau _1f_1)(t) = \sum _{k=0}^n a_k(t)f_1^{(k)}(t),t\in I_1 \,,\\
(\tau _2f_2)(t) = \sum _{k=0}^n b_k(t)f_2^{(k)}(t),t\in I_2
\end{gather*}
We define $T_0(\tau_i)$,$T_1(\tau_i)$ in $L_2(I_i)$, $i=1,2$
and $T_0(\tau_1,\tau_2)$, $T_1(\tau_1,\tau_2)$ in $L_2(I_1)\times L_2(I_2)$
as follows:
\begin{gather*}
D(T_0(\tau_i))=H_0^n(I_i),T_0(\tau_i)f_i=\tau_if_i,f_i\in
D(T_0(\tau_i));i=1,2;\\
D(T_1(\tau_i))=H^n(I_i),T_1(\tau_i)f_i=\tau_if_i,f_i\in D(T_1(\tau_i));i=1,2;\\
D(T_0(\tau_1,\tau_2))=H_0^n(I_1\times I_2),T_0(\tau_1,\tau_2)(f_1,f_2)
=(\tau_1f_1,\tau_2f_2);\\
D(T_1(\tau_1,\tau_2))=H^n(I_1\times I_2),T_1(\tau_1,\tau_2)(f_1,f_2)
=(\tau_1f_1,\tau_2f_2).
\end{gather*}
We note that $T_0(\tau_i),T_1(\tau_i))$ are densely defined unbounded
operators in $L_2(I_i)$, $i=1,2;T_0(\tau_1,\tau_2),T_1(\tau_1,\tau_2)$
are densely defined unbounded operators in $L_2(I_1)\times L_2( I_2)$.
We also note that the matching conditions at the interface  $t=c$ , viz.
$A\tilde f_1(c)=B\tilde f_2(c)$ are introduced into the domains of
$T_0(\tau_1,\tau_2)$ and $T_1(\tau_1,\tau_2)$. It is true that $T_0(\tau_i)$,
$i=1,2$, $T_0(\tau_1,\tau_2)$ are minimal unclosed operators and  and
$T_1(\tau_i)$, $i=1,2$; $T_1(\tau_1,\tau_2)$ are the maximal closed operators
in the respective spaces.
Moreover, $T_0(\tau_i)=T_0(\tau_i^*)^*$, where $\tau_i^*$ is the formal
adjoint of $\tau_i$, $i=1,2$. Under certain assumptions on the matrices
$A,B$ and the boundary matrices for $\tau_1,\tau_2$ at $c$, we shall prove
in the next section that $T_1(\tau_1,\tau_2)=T_0(\tau_1^*,\tau_2^*)^*$.
Thus if $\tau_1$ and $\tau_2$ are formally selfadjoint, then we have
$T_1(\tau_i)=T_0(\tau_i)^*$, $i=1,2$ and
$T_1(\tau_1,\tau_2)=T_0(\tau_1,\tau_2)^*$. The positive and negative
deficiency spaces of $T_0(\tau_1),T_0(\tau_2)$ and $T_0(\tau_1,\tau_2)$
are defined as follows:
\begin{gather*}
D'_+=\lbrace f_1\in D(T_1(\tau_1))\big/ \tau_1f_1=if_1\rbrace,\\
D'_-=\lbrace f_1\in D(T_1(\tau_1))\big/ \tau_1f_1=-if_1\rbrace,\\
D"_+=\lbrace f_2\in D(T_1(\tau_2))\big/ \tau_2f_2=if_2\rbrace,\\
D"_-=\lbrace f_2\in D(T_1(\tau_2))\big/ \tau_2f_2=-if_2\rbrace,\\
D_+=\lbrace (f_1,f_2)\in D(T_1(\tau_1,\tau_2)),
\big/ (\tau_1,\tau_2)(f_1,f_2)=i(f_1,f_2)\rbrace,\\
D_-=\lbrace (f_1,f_2)\in D(T_1(\tau_1,\tau_2)),
\big/ (\tau_1,\tau_2)(f_1,f_2)=-i(f_1,f_2)\rbrace,\\
\end{gather*}
and the following quantities
\begin{gather*}
d'_+=\dim D'_+;d'_-=\dim D'_-\,,\\
d''_+=\dim D''_+;d''_-=\dim D''_-\,,\\
d_+=dim D_+;d_-=dim D_-
\end{gather*}
are called the positive and negative deficiencies of
$T_0(\tau_1),T_0(\tau_1),T_0(\tau_1,\tau_2)$, respectively.
Our main interest here is to prove the following theorem.

\begin{theorem} \label{thm1}
If $\tau_1,\tau_2$ are formally selfadjoint and
\begin{equation} \label{e1}
 (A^{-1})^*F_c(\tau_1)A^{-1}=(B^{-1})^*F_c(\tau_2)B^{-1}
\end{equation}
where $F_c(\tau_i)$ is the boundary matrix of $\tau_i$ at $t=c, i=1,2$, then
$$
d_+=d'_++d''_+-n\quad \mbox{and}\quad d_-=d'_-+d''_--n \,.
$$
If $\tau_1=\tau_2$, that is the same differential operator is
defined on $I_1$ and $I_2$, and $A=B=I$,
(where I denotes the identity matrix) then \cite[corollary (XIII).2.26]{d1}
becomes a special case of the above theorem.
The proof of Theorem \ref{thm1}, that we present here is new and more appealing
than the proof given in \cite{d1}, for the special case $\tau_1=\tau_2$,
 and $A=B=I$.
\end{theorem}

\section{Preliminary results}

In this section, we  present a few definitions and results that
are useful towards proving Theorem \ref{thm1}.

Let $g_i$ be complex valued measurable function which is integrable over
every compact subinterval of $I_i$, $i=1,2$.
Consider  the boundary-value problem (BVP)
\begin{gather}
(\tau_1,\tau_2)(f_1,f_2)=(g_1,g_2) \label{e2}\\
A\tilde f_1(c) = B\tilde f_2(c)  \label{e3}
\end{gather}
 By a solution of problem \eqref{e2}-\eqref{e3}, we mean a pair
 $(f_1,f_2)\in A^n(I_1)\times A^n(I_2)$ such that
\begin{itemize}
\item[(i)]  $(\tau_1f_1)(t)=g_1(t)$ for almost all $t\in I_1$
\item[(ii)] $(\tau_2f_2)(t)=g_2(t)$ for almost all $t\in I_2$
\item[(iii)] $A\tilde f_1(c) = B\tilde f_2(c)$.
\end{itemize}
Let $t_i\in I_i$, $i=1,2$ and $\{c_0,\dots ,c_{n-1}\},\{d_0,\dots ,d_{n-1}\}$
be arbitrary set of complex numbers. Consider the initial conditions
\begin{gather}
f_1^{(i)}(t_1)=c_i,i=0,1,\dots ,n-1, t_1\in I_1,\label{e4} \\
f_2^{(i)}(t_2)=d_i,i=0,1,\dots ,n-1, t_2\in I_2, \label{e5}
\end{gather}
The following results can be proved easily.

\begin{lemma} \label{lem1}
 The initial boundary-value problem \eqref{e2}-\eqref{e3}-\eqref{e4}
 (\eqref{e2}-\eqref{e3}-\eqref{e5})
 has a unique solution.
\end{lemma}

\begin{lemma} \label{lem2}
 If $g_i$ has k continuous derivatives in $I_i$, then the component
 $f_i$ of the solution $(f_1,f_2)$ of \eqref{e2}-\eqref{e3}-\eqref{e4}
 (\eqref{e2}-\eqref{e3}-\eqref{e5}) has $(n+k)$ continuous derivatives
 in $I_i$, $i=1,2$.
\end{lemma}

\begin{lemma} \label{lem3}
 If $(g_1,g_2)=(0,0)$ and
$0=c_0=c_1=\dots =c_{n-1}(0=d_0=d_1=\dots =d_{n-1})$, then
$(f_1,f_2)=(0,0)$ is the only solution of
\eqref{e2}-\eqref{e3}-\eqref{e4} (\eqref{e2}-\eqref{e3}-\eqref{e5}).
\end{lemma}

We say the pairs $(f_{11},f_{21}),\dots,(f_{1p},f_{2p})$ are linearly
independent on $I_1\times I_2$ if
$$
\sum_{k=1}^p\alpha_kf^{(j)}(t)=0, \quad t\in I_i,\; j=0,1,\dots ,n-1, \;
i=1,2
$$
where $\alpha_1,\dots,\alpha_p$ are scalars, then
$\alpha_1=\alpha_2=\dots =\alpha_p=0$.

The next result follows easily from Result \ref{lem1}.

\begin{lemma} \label{lem4}
 The boundary-value problem
 \begin{gather}
(\tau_1,\tau_2)(f_1,f_2)=(0,0) \label{e6}\\
 A\tilde f_1(c) = B\tilde f_2(c) \label{e7}
\end{gather}
has exactly $n$ linearly independent solutions.
\end{lemma}
We now prove the Green's formula for the pair $(\tau_1,\tau_2)$.

\begin{theorem} \label{thm2}
Let $I_1=[a,c],I_2=[c,b]$, $-\infty<a<c<b<+\infty$.
Let relation \eqref{e1} be true.
 Then for $(f_1,f_2)(g_1,g_2)\in H^n(I_1\times I_2)$,
\begin{align*}
& \int_a^c(\tau_1f_1)(t){\bar g_1(t)}dt
+ \int_c^b(\tau_2f_2)(t){\bar g_2(t)}dt\\
&=\int_a^cf_1(t){\bar{(\tau_1^*g_1)}}(t)dt
+ \int_c^bf_2(t){\bar{(\tau_2^* g_2)}}(t)dt +F_b(f_2,g_2)-F_a(f_1,g_1)
\end{align*}
 where $F_t(f_i,g_i)$ is the boundary form for $\tau_i$ at $t\in I_i$.
\end{theorem}

\begin{proof}
Being the proof routine it suffices to verify that
$$
F_c(f_1,g_1)=F_c(f_2,g_2).
$$
To show this, we consider,
\begin{align*}
 F_c(f_1,g_1)&=(\tilde g_1(c))^*F_c(\tau_1)\tilde f_1(c)\\
&= (\tilde g_1(c))^*A^*(A^{-1})^*F_c(\tau_1)A^{-1}A\tilde f_1(c)\\
& =(A\tilde g_1(c))^*(A^{-1})^*F_c(\tau_1)A^{-1}(A\tilde f_1(c))\\
&=(B\tilde g_2(c))^*(B^{-1})^*F_c(\tau_2)B^{-1}(B\tilde f_2(c))\\
&=(\tilde g_2(c))^*B^*(B^{-1})^*F_c(\tau_2)B^{-1}B\tilde f_2(c)\\
&=(\tilde g_2(c))^*F_c(\tau_2)\tilde f_2(c)\\
&=F_c(f_2,g_2)
\end{align*}
\end{proof}

The following corollary is immediate.

\begin{corollary} \label{coro1}
If $I_1$ and $I_2$ are arbitrary intervals and Relation \eqref{e1}
is true, then Green's formula is valid for
$(f_1,f_2),(g_1,g_2)\in H^n(I_1\times I_2)$
(or even $(f_1,f_2)\in H^n(I_1\times I_2),(g_1,g_2)\in
A^n(I_1)\times A^n(I_2)$ satisfying $A\tilde g_1(c)=B\tilde g_2(c)$)
provided that either $(f_1,f_2)$ or $(g_1,g_2)$ vanishes outside a
compact subcell of $I_1\times I_2$.
\end{corollary}

In the rest of the work, we assume Relation \eqref{e1} to be true.
The following results could be proved with suitable modifications
as in \cite[pp 1291-1295]{d1}.

\begin{lemma} \label{lem5}
 Let $f_i$ be a function whose square is integrable over every
compact subinterval of $I_i,i=1,2$. Suppose that
$$
\sum_{i=1}^2\int_{I_i}f_i(t){\bar{\tau_i^*g_i}}(t)dt=0,
\quad\mbox{for all }(g_1,g_2)\in H_0^n(I_1\times I_2)\,.
$$
Then (after modification on a set of measure zero)
$$
(f_1,f_2)\in C^\infty (I_1)\times C^\infty (I_2),A
\tilde f_1(c)=B\tilde f_2(c) and (\tau_1,\tau_2)(f_1,f_2)=(0,0).
$$
\end{lemma}

\begin{lemma} \label{lem6}
$T_1(\tau_1,\tau_2)=T_0(\tau_1^*,\tau_2^*)^*$.
\end{lemma}

 From  Lemma \ref{lem6} it follows that $T_1(\tau_1,\tau_2)$ is
a closed operator. Thus $T_1(\tau_1,\tau_2)$ is an extension of
$T_0(\tau_1,\tau_2)$ and hence $T_0(\tau_1,\tau_2)$ has an minimal
closed extension $\bar{T_0(\tau_1,\tau_2)}$.

\begin{lemma} \label{lem7}
If $\tau_1,\tau_2$ are formally selfadjoint then  $T_0(\tau_1,\tau_2)$
is the restriction of $T_0(\tau_1,\tau_2)^*$.
(that is $T_0(\tau_1,\tau_2)$ is symmetric).
\end{lemma}

\begin{lemma} \label{lem8}
If $\tau_1,\tau_2$ are formally selfadjoint then
$D_+',D_-';D_+'',D_-'';D_+,D_-$ consists precisely of those solutions
of the equations $(\tau_1-i)f_1=0$, $(\tau_1+i)f_1=0$;
$(\tau_2-i)f_2=0$, $(\tau_2+i)f_2=0$;
$((\tau_1,\tau_2)+i)(f_1,f_2)=(0,0)$, satisfying
$A\tilde f_1(c)=B\tilde f_2(c)$, lying in
$L_2(I_1),L_2(I_2),L_2(I_1)\times L_2(I_2)$, respectively.
\end{lemma}

\begin{lemma} \label{lem9}
Let $J_1\times J_2$ be a compact subcell of $I_1\times I_2$.
Then
\begin{itemize}
\item[(i)] The space $H^n(J_1\times J_2)$ is complete in the norm
\begin{align*}
\parallel(f_1,f_2)\parallel=&\max\Big(\sum_{i=0}^{n-1}\max_{t\in J_1}| f_1^{(i)}(t)| ,
\sum_{i=0}^{n-1}\max_{t\in J_2}| f_2^{(i)}(t)|\Big) \\
&+\Big(\sum_{i=1}^2 \int_{J_i}| f_i^{(n)}(t)| ^2dt\Big)^{1/2}.
\end{align*}

\item[(ii)] $\{(f_{1n},f_{2n})\}$ is a sequence in
$H^n(I_1 \times I_2)$ such that $\{(f_{1n},f_{2n})\}$ and \\
$(\tau_1,\tau_2)\{(f_{1n},f_{2n})\}$ converge (converge weakly)
in $L_2(I_1)\times L_2(I_2)$, then the sequence $\{(f_{1n},f_{2n})\}$
converges (converge weakly) in the topology of
$H^n(J_1 \times J_2)$ defined by the above norm.
For $(f_1,f_2),(g_1,g_2)\in L_2(I_1)\times L_2(I_2)$, the inner
product in $L_2(I_1)\times L_2(I_2)$ is given by
$$
\big\langle (f_1,f_2),(g_1,g_2)\big\rangle
= \langle f_1,f_2\rangle + \langle g_1,g_2\rangle
$$
Since $T_1(\tau_1,\tau_2)$ is closed,
$H^n(I_1 \times I_2)= D(T_1(\tau_1,\tau_2))$ becomes a Hilbert
space upon introduction of the  inner product
$$
\big\langle (f_1,f_2),(g_1,g_2)\big\rangle^*=
\big\langle (f_1,f_2),(g_1,g_2)\big\rangle + \big\langle (\tau_1,\tau_2)
(f_1,f_2),(\tau_1,\tau_2)(g_1,g_2)\big\rangle
$$
\end{itemize}
\end{lemma}

\noindent\textbf{Definition.} % 3:
A boundary value for $(\tau_1,\tau_2)$ is a continuous linear functional
$\Theta $ on $D(T_1(\tau_1,\tau_2))$ which vanishes on $D(T_0(\tau_1,\tau_2))$.
If $\Theta(f_1,f_2)=0$ for each $(f_1,f_2)\in D(T_1(\tau_1,\tau_2))$ which
vanishes in a neighbourhood of $a$, $\Theta$ is called a boundary value at $a$.
 A boundary value at $b$ is defined similarly . An equation $\Theta(f_1,f_2)=0$,
when $\Theta$ is a boundary value for $(\tau_1,\tau_2)$, is called
a boundary condition for $(\tau_1,\tau_2)$. A complete set of
boundary values is a maximal linearly independent set of boundary
values. Similarly  a complete set of boundary values at $a(b)$ is
a maximal linearly independent set of boundary values at
$a(b)$.\smallskip

Note: If $\tau_1,\tau_2$ formally selfadjoint, the boundary values for
$(\tau_1,\tau_2)$ coincides with \cite[Definition (XII)4.20]{d1} of a
 boundary value for $T_0(\tau_1,\tau_2)$.

The following results can be provided with suitable modifications as in
\cite[pp: 1298-1301]{d1}.

\begin{lemma} \label{lem10} %Result 10:
The space of boundary values for $(\tau_1,\tau_2)$ is the direct sum of
the space of boundary values for $(\tau_1,\tau_2)$ at $a$ and the space
of  boundary values for $(\tau_1,\tau_2)$ at $b$.
\end{lemma}

\begin{lemma} \label{lem11}
There exists a one to one linear mapping of the space of all boundary values
for $\tau_1(\tau_2)$ at $a(b)$ on to the space of  all boundary values for
$(\tau_1,\tau_2)$ at $a(b)$.
\end{lemma}

\begin{lemma} \label{lem12}
$\tau_1(\tau_2)$ and $(\tau_1,\tau_2)$ have the same number of linearly
independent boundary conditions at $a(b)$.
\end{lemma}

\begin{lemma} \label{lem13}
$(\tau_1,\tau_2)$ has at most $n$ linearly independent boundary values
at $a(b)$.
\end{lemma}

\begin{lemma} \label{lem14}
If $I_1=[a,c]$, $-\infty<a(I_2=[c,b],b<+\infty)$, then the functionals
$\Theta_i(f_1,f_2)= f_1^{(i)}(a)(f_2^{(i)}(b)),i=0,1,\dots n-1$ form a
complete set of boundary values for $(\tau_1,\tau_2)$ at $a(b)$.
\end{lemma}

\begin{lemma} \label{lem15}
If $\tau_1,\tau_2$ are formally selfadjoint and
$$
d=d_++d_-,\quad d' =d_+' + d_-' ,\quad  d'' =d_+'' + d_-''
$$
then
$d=d' + d''-2n$.
\end{lemma}

\section{Proof of Theorem \ref{thm1}}

\begin{proof}
Let $(f_{11},f_{21}),\dots ,(f_{1d+},f_{2d_+})$ be a basis for
$D_+$ ; $g_{11},\dots ,g_{1d_+'}$ be basis for $D_+'$;
$h_{21},\dots,h_{2d_+}''$ be a basis for $D_+''$.
Clearly, $\{(f_{1i},f_{2i})\}$, $i=1,2\dots,d_+$ are linearly independent
and belong to $L_2(I_1)\times L_2(I_2)$; $\{g_{1i}\},i=1,\dots ,d_+'$ are
linearly independent and belong to $L_2(I_1)$; $\{h_{2i}\},i=1,\dots,d_+''$
are linearly independent and belong to $L_2(I_2)$. We have
$d_+\leq d_+'', d_+\leq d_+'$.

\noindent\textbf{Claim 1:} At least $(d_+' - d_+)$ number of $g_{i1}$s are linearly
independent with respect to the set $S=\{f_{11},\dots ,f_{1d_+}\}$.
For, if possible, let this number of $g_{i1}$s be strictly less than
$(d_+' - d_+)$. Then at least $(d_++1)$ number of $g_{i1}$s shall be linearly
dependent to $S$. Without loss of generality, we may assume that
$g_{11},\dots,g_{1d_+}$ are linearly independent  to $S$. Then there exists
scalars $\alpha_{ij},i,j=1,2,dots,d_+$ and $\beta_1,\dots,\beta_{d_+}$ such that
\begin{equation} \label{e8}
\begin{gathered}
\alpha_{11}f_{11}+\dots +\alpha_{1d_+}f_{1d_+}=g_{11}\\
\alpha_{21}f_{11}+\dots +\alpha_{2d_+}f_{1d_+}=g_{12}\\
         \vdots \\
\alpha_{d_+1}f_{11}+\dots +\alpha_{d_+d_+}f_{1d_+}=g_{1d_+}
\end{gathered}
\end{equation}
and
\begin{equation} \label{e9}
\beta_1f_{11}+\dots +\beta_{d_+}f_{1d_+}=g_{1d_++1}
\end{equation}
Since $g_{11},\dots,g_{1d_+}$ are linearly independent, the matrix
$$
\begin{pmatrix}
\alpha_{11}&\dots &\alpha_{1d_+}\\
\alpha_{21}&\dots  &\alpha_{2d_+}\\
 \vdots & & \vdots\\
\alpha_{d_+1}&\dots &\alpha_{d_+d_+}
\end{pmatrix}
$$
is nonsingular and consequently system \eqref{e8}
gives that each $f_{1i},i=1,\dots ,d_+$ can be expressed as a linear combination
of $g_{11},\dots,g_{1d_+}$ and then substituting into equation \eqref{e9},
 we get $g_{1d_++1}$ is a linear combination of  $g_{11},\dots,g_{1d_+}$, a
contradiction. Hence the claim is true.

Now, let $g_{1d_++1},\dots,g_{1d_+'}$ be linearly independent with respect to $S$.
Using Lemma \ref{lem1}, we can  extend these functions to the pairs \\
$(g_{1d_++1},g_{2d_++1}), \dots, (g_{1d_+'},g_{2d_+'})$ satisfying
\begin{gather} \label{e10}
((\tau_1,\tau_2)-i)(g_{1i},g_{2i})=(0,0),  \\
A\tilde g_{1i}(c)=B\tilde g_{2i}(c),i=d_++1,\dots ,d_+' \label{e11}
\end{gather}
Clearly, $(f_{11},f_{21}),\dots ,(f_{1d_+},f_{2d_+}),(g_{1d_++1},g_{2d_++1}),
\dots,(g_{1d_+'},g_{2d_+'})$ are linearly independent and
$g_{2i}\notin L_2(I_2)$, for any $i=d_++1,\dots ,d_+'$.

Next, let $\tilde S=\{f_{21},\dots,f_{2d_+}\}$. As in claim 1, we can prove
at least $(d_+''-d_+)$ number of $h_{2i}$s must be linearly independent with
respect to $\tilde S$. Using  Lemma \ref{lem1}, we can extend these functions
to the pairs $(h_{1d_++1},h_{2d_++1}),\dots \dots,(h_{1d_+''},h_{2d_+''})$
satisfying

\begin{gather} \label{e12}
((\tau_1,\tau_2)-i)(h_{1i},h_{2i})=(0,0) \\
 A\tilde h_{1i}(c)=B\tilde h_{2i}(c)  \label{e13}
\end{gather}
Clearly, $(f_{11},f_{21}),\dots ,(f_{1d_+},f_{2d_+})$, $(h_{1d_++1},h_{2d_++1}),
\dots,(h_{1d_+''},h_{2d_+''})$ are linearly independent and
$h_{1i}\notin L_2(I_1)$, for any $i=d_++1,\dots,d_+''$. \smallskip

\noindent\textbf{Claim 2:} $(f_{11},f_{21}),\dots,(f_{1d_+},f_{2d_+}),
(g_{1d_++1},g_{2d_++1}),\dots,(g_{1d_+'},g_{2d_+'})$, \\
$(h_{1d_++1},h_{2d_++1}),\dots,(h_{1d_+''},h_{2d_+''})$ are linearly
independent solutions of
\begin{gather} \label{e14}
((\tau_1,\tau_2)-i)(f_1,f_2)=(0,0) \\
A\tilde f_1(c)=B\tilde f_2(c)\,. \label{e15}
\end{gather}
It suffices to verify the linear independency of these pairs of functions.
Again it suffices to show that $g$'s and $h$'s are mutually linear independent.
If possible for some $i$, let
$$
(g_{1i},g_{2i})=\alpha_1(h_{1d_++1},h_{2d_++1})+\dots
+\alpha_{d_+''-d_+}(h_{1d_+''},h_{2d_+''})
$$
for some scalars $\alpha_1,\dots,\alpha_{d_+''-d_+}$, not all zeros.  Then
$$
g_{2i}=\sum_{i=1}^{{d_+''-d_+}}\alpha_ih_{2i}
$$
a contradiction, since the  left-hand side is not in $L_2(I_2)$, whereas the
right-hand side is in $L_2(I_2)$. Similarly, it can be proved that no
$(h_{1i},h_{2i})$ is a linear combination of $(g_{1i},g_{2i})$,
$i= d_++1,\dots ,d_+'$. This proves claim 2.

Finally by Lemma \ref{lem4}, we have
$$
d_++(d_+' - d_+)+(d_+'' - d_+)\leq n.
$$
That is
\begin{equation} \label{e16}
d_+''+d_+' -d_+\leq n  \end{equation}
Similarly we get,
\begin{equation} \label{e17}
d_-''+d_-' -d_-\leq n
\end{equation}
Claim 3: $d_+''+d_+' -d_+\leq n$ and $d_+''+d_+' -d_+\leq n$
For if possible, let the strict inequality hold in either
\eqref{e16} or \eqref{e17}. Then, adding these two we get
$$
(d_+''+d_-'')+(d_+'+d_-')-(d_++d_-)<2n\,.
$$
That is, $d''+d'-d<2n$ which is a contradiction to Lemma \ref{lem15}.
This proves claim 3 and the proof of the theorem is complete.
\end{proof}

We remark that the operators of the form considered here occur in many
physical situations such as acoustic wave guides in oceans;
see \cite{b1,v1,v2,v3,v4,v5,o1}.

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\end{document}