\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 76, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/76\hfil Green's function]
{Green's function for two-interval Sturm-Liouville problems}

\author[A. Wang, A. Zettl \hfil EJDE-2013/76\hfilneg]
{Aiping Wang, Anton Zettl}  % in alphabetical order

\dedicatory{Dedicated to John W. Neuberger on his 80th birthday}

\address{Aiping Wang \newline
 Mathematics Department, Harbin Institute of Technology, Harbin,
150001, China}
\email{wapxf@163.com}

\address{Anton Zettl \newline
 Mathematics Deparment, Northern Illinois University, DeKalb, IL 60115, USA}
\email{zettl@math.niu.edu}

\thanks{Submitted March 3, 2013. Published March 18, 2013.}
\subjclass[2000]{34B20, 34B24, 47B25}
\keywords{Two-interval problems; Green's function; characteristic function}

\begin{abstract}
 We construct the Green's function and the characteristic function for
 two-interval regular Sturm-Liouville problems with separated and coupled,
 self-adjoint and non-self-adjoint, boundary conditions. In the self-adjoint
 case these problems may have boundary conditions requiring jump
 discontinuities of the eigenfunctions or their derivatives. Such conditions
 are known by various names including transmission and interface conditions
 and have been studied by many authors in the recent literature.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

We construct the Green's function for two-interval regular self-adjoint and
non-self-adjoint Sturm-Liouville problems. The two intervals may be
disjoint, overlap, or be identical.

In recent years Sturm-Liouville problems with boundary conditions requiring
discontinuous eigenfunctions or discontinuous derivatives of eigenfunctions
have been studied by many authors. Such conditions are known by various
names including: transmission conditions 
\cite{akdm,aosz,muya02,mukm,MKM,tumu04,wshy}, interface
conditions \cite{kowa,suwa08,zett68}, discontinuous
conditions \cite{DZ,kamu07,mukm,shyu,MT},
multi-point conditions \cite{kkkw,loud68,zett66}, point
interactions (in the Physics literature), conditions on trees, graphs or
networks \cite{scw,JMR,pobo04,popr04}, etc. For an
informative survey of such problems arising in applications including an
extensive bibliography and historical notes, see Pokornyi-Borovskikh 
\cite{pobo04} and Prokornyi-Pryadiev \cite{popr04}.

As a special case our construction applies to such problems. It is modeled
on a construction of Neuberger \cite{neub60} for the one interval case.
Neuberger's construction differs from the usual one found in textbooks and
in most of the literature, in that the discontinuity of the derivative of
the Green's function along the diagonal occurs naturally, in contrast to the
usual construction as found, for example, in Coddington and Levinson
 \cite{code55} where this discontinuity is assigned a priori as part of the
construction.

\section{Basic theory and notation}

In this section we briefly review the basic theory and make some
definitions. Although we only use the results for the second order case $n=2$
we state them for general $n$ since this does not introduce any additional
complexity or length. Let $\mathbb{N}=\{1,2,3,\dots \}$ and
denote by $L(J,\mathbb{C})$ the set of complex valued Lebesgue integrable
functions on compact interval $J$ of the real line and $AC(J)$ denotes the
absolutely continuous complex valued functions on $J$.

Let $M_{n\times m}(\mathbb{C})$ denote the set of $n\times m$ matrices with
complex entries. If $n=m$ we write 
$M_{n}(\mathbb{C})=M_{n\times n}(\mathbb{C})$. Let $M_{n}(S)$ be the $n$ by $n$ matrices with entries from an arbitrary set $S$.

We start with some definitions and preliminary lemmas.

\begin{lemma}[Existence and Uniqueness]
Let $n,m\in \mathbb{N}$. If
\begin{gather}
P\in M_{n}(L(J,\mathbb{C})), \\
F\in M_{n,m}(L(J,\mathbb{C}))
\end{gather}
then every initial value problem
\begin{gather}
Y'=PY+F,  \label{4.1}\\
Y(u)=C,\quad u\in J,\quad C\in M_{n,m}(\mathbb{C})  \label{4.2}
\end{gather}
has a unique solution defined on all of $J$. Furthermore, if $C$, $P$, $F$
are all real-valued, then there is a unique real valued solution.
\end{lemma}

Proofs of the two lemmas above can be found in  \cite{zett05}.
Let $P\in M_{n}(L(J))$. From this Lemma we know that for each point $u$
of $J $ there is exactly one matrix solution $X$ of
\begin{equation}
Y'=PY\quad  \text{on } J  \label{4.3}
\end{equation}
satisfying $X(u)=I_{n}$ where $I_{n}$ denotes the $n$ by $n$ identity matrix.

\begin{definition}[Primary fundamental matrix] \rm
For each fixed $u\in J$ let $\Phi (\cdot ,u)$
be the fundamental matrix of \eqref{4.3} satisfying $\Phi (u,u)=I_{n}$. Note
that for each fixed $u$ in $J$, $\Phi (\cdot ,u)$ belongs to $
M_{n}(AC_{loc}(J))$. Furthermore, if $J$ is compact and $P\in M_{n}(L(J,
\mathbb{C}))$ then $u$ can be an endpoint of $J$ and $\Phi (\cdot ,u)$
belongs to $M_{n}({AC(J)})$. We note that $\Phi (t,u)$ is invertible for
each $t$, $u\in J$ and $\Phi (t,u)=Y(t)Y^{-1}(u)$ for any fundamental matrix
$Y$ of \eqref{4.3}.
\end{definition}

We call $\Phi $ the primary fundamental matrix of \eqref{4.3}. Note that for
any constant $n\times m$ matrix $C$, $\Phi C$ is also a solution of $
Y'=PY$. If $C$ is a constant nonsingular $n\times n$ matrix then $
\Phi C$ is a fundamental matrix solution and every fundamental matrix
solution has this form. For these and other basic facts, notation and
terminology see Chapter 1 in \cite{zett05}.

The next lemma is fundamental in the theory of linear differential equations.

\begin{lemma}[Variation of Parameters Formula, see \protect\cite{zett05}]
Let $J$ be any compact interval, $P\in M_{n}(L(J,\mathbb{C}))$ and let
$\Phi =\Phi (\cdot ,\cdot ,P)$ be the primary fundamental matrix of $Y'=PY $ on $J$. Let $F\in M_{n,m}(L(J,\mathbb{C}))$, $u\in J$ and $C\in
M_{n,m}(\mathbb{C})$. Then
\begin{equation}
Y(t)=\Phi (t,u,P)C+\int_{u}^{t}\Phi (t,s,P)F(s)\,ds,\quad t\in J
\end{equation}
is the solution of \eqref{4.1}, \eqref{4.2}. Note that $Y\in M_{n,m}(AC(J))$.
\end{lemma}

\section{The Characteristic Function}

Next we study the two-interval characteristic function with general, not
necessarily self-adjoint, boundary conditions. Let
\begin{equation*}
J_r=(a_r,b_r),\quad -\infty <a_r<b_r<\infty ,\; r=1,2,
\end{equation*}
and assume the coefficients and weight functions satisfy
\begin{equation}
p_r^{-1}=\frac{1}{p_r},\quad q_r,w_r\in L(J_r,\mathbb{C)},\;
 r=1,2. \label{eq2.1a}
\end{equation}

Define differential expressions $M_r$ by
\begin{equation}
M_ry=-(p_ry')'+q_ry\quad \text{on }
J_r,\; r=1,2.  \label{eq2.2a}
\end{equation}

Below we use the notation with a subscipt $r$ to denote the $r-th$ interval.
The subscript $r$ is sometimes omitted when it is clear from the context. We
consider the second order scalar differential equations
\begin{equation}
-(p_ry')'+q_ry=\lambda w_ry\quad \text{on }
J_r,\quad r=1,2,\; \lambda \in \mathbb{C},  \label{eq11a}
\end{equation}
together with boundary conditions
\begin{equation}
A_1Y_1(a_1)+B_1Y_1(b_1)+A_2Y_2(a_2)+B_2Y_2(b_2)=0,\quad
Y_r=\begin{bmatrix}
y_r \\
(p_ry_r')
\end{bmatrix},\;
 r=1,2.  \label{eq12}
\end{equation}
Here $A_r,B_r\in M_{4\times 2}(\mathbb{C})$, $r=1,2$.
From \eqref{eq2.1a} and the basic theory of linear ordinary differential equations the
boundary condition \eqref{eq12} is well defined.
Next We comment on the assumption \eqref{eq2.1a}, equations \eqref{eq2.2a} and
condition \eqref{eq12}.

\begin{remark} \rm
It follows from the basic theory that, under condition \eqref{eq2.1a}, every
solution $y_r$ and its quasi-derivative $py_r'$ are
continuous on $J_r$ but, $p_r(t)$ and $y_r'(t)$ may not exist
for some $t$ in $J$ so we use the notation $(py')$ to indicate that
this is a continuous function which cannot, in general, be separated into $
p(t)y'(t)$ for all $t$ in $J$.
\end{remark}

\begin{remark} \rm
Note that each of $\frac{1}{p_r},q_r,w_r$ can be zero not only at some
points of $J$ but on subintervals and even the whole interval.  If $q_r$
is zero on $J$ then we simply have a restricted class of problems.
If $\frac{1}{p_r}=0$ or $w_r=0$ on $J,$ then we have a degenerate and
uninteresting equation. In the latter case there is no $\lambda $ dependence
and so no need for a Green's function. Kong, Wu and Zettl and Volkmer, Kong,
and Zettl \cite{kovz09} found a class of S-L problems where each of $\frac{1
}{p},q,w$ is identically zero on certain subintervals of $J$ and whose
spectrum has $n$ eigenvalues for any $n=1,2,3,\dots $. It is for
this reason that we do not want to place any unnecessary restrictions on the
coefficients. In the classical one-interval self-adjoint case the
coefficients $\frac{1}{p},q,w$ are assumed to be in $L(J,\mathbb{R)}$ with
$p,w>0$ a.e. in $J$ and the spectral properties are studied in the Hilbert
space $L^{2}(J,w)$.
\end{remark}

Below we will construct the characteristic function whose zeros are
precisely the eigenvalues of the two-interval SLP.
Let
\begin{equation}
P_r=\begin{bmatrix}
0 & \frac{1}{p_r} \\
q_r & 0
\end{bmatrix}, \quad
W_r=\begin{bmatrix}
0 & 0 \\
w_r & 0
\end{bmatrix}.  \label{eq13}
\end{equation}
Then the scalar equation \eqref{eq11a} is equivalent to the first-order
system
\begin{equation}
Y'=(P_r-\lambda W_r)Y=\begin{bmatrix}
0 & \frac{1}{p_r} \\
q_r-\lambda w_r & 0
\end{bmatrix},
\quad Y=\begin{bmatrix}
y \\
p_ry'
\end{bmatrix}.  \label{eq14a}
\end{equation}
Note that, given any scalar solution $y_r$ of $-(p_ry')'+q_ry=\lambda w_ry$ on $J_r$ the vector $Y_r$ defined by
\eqref{eq12} is a solution of the system $Y'=(P_r-\lambda W_r)Y$
on $J_r$. Conversely, given any vector solution $Y_r$ of system $
Y'=(P_r-\lambda W_r)Y$ its top component $y_r$ is a solution
of $-(p_ry')'+q_ry=\lambda w_ry$.

Let $\Phi _r(\cdot ,u_r,P_r,w_r,\lambda )$ be the primary
fundamental matrix of \eqref{eq14a} and we have
\begin{equation}
\Phi _r'=(P_r-\lambda W_r)\Phi _r\text{ on } J_r,\quad
\Phi _r(u_r,u_r,\lambda )=I,\; a_r\leq u_r\leq b_r,\; \lambda
\in \mathbb{C},  \label{eq15}
\end{equation}
where $I$ denotes 2 by 2 identity matrix. 

Here we use the notation
$\Phi _r=\Phi _r(\cdot ,u_r,P_r,w_r,\lambda )$ to indicate the
dependence of the primary fundamental matrix on these quantities. Since
$P_r,w_r$ are fixed here, we simplify it to
$\Phi _r(\cdot ,u_r,\lambda )$. By \eqref{eq2.1a}, we have 
$\Phi (b_r,a_r,\lambda )$ exists.

Define the characteristic function $\Delta $ by
\begin{equation}
\begin{split}
\Delta (\lambda )&= \Delta
(a_1,b_1,a_2,b_2,A_1,B_1,A_2,B_2,P_1,P_2,w_1,w_2,
\lambda ) \\
&= \det [(A_1+B_1\Phi _1(b_1,a_1,\lambda )\mid A_2+B_2\Phi
_2(b_2,a_2,\lambda ))],\quad \lambda \in \mathbb{C},
\end{split}
\end{equation}
where $(A_1+B_1\Phi _1(b_1,a_1,\lambda )\mid A_2+B_2\Phi
_2(b_2,a_2,\lambda ))$ denote the 4 by 4 complex matrix whose first
two columns are those of $A_1+B_1\Phi _1(b_1,a_1,\lambda )$, and
the second two columns are those of $A_2+B_2\Phi
_2(b_2,a_2,\lambda )$.

\begin{definition} \rm
By a trivial solution of equation $M_ry=\lambda w_ry$ on some interval $
I_r$ we mean a solution $y_r$ which is identical zero on $I_r$ and
whose quasi-derivative $(p_ry_r')$ is also identically zero on $
I_r$. ($I_r$ may be a subinterval of $J_r$ or it may be the whole
interval $J_r$.) Note that, under the assumptions \eqref{eq2.1a}, solution
$y_r$ might be identically zero on $I_r$ but its quasi-derivative $
(p_ry_r')$ might not be identically zero on $I_r$.
\end{definition}

\begin{definition} \rm
By a trivial solution of the two-interval Sturm-Liouville equations (\eqref
{eq11a}  we mean a solution $\mathbf{y}=\{y_1,y_2\}$ each of whose
components $y_r$ is a trivial solution of equation $M_ry=\lambda w_ry$
on $J_r\,,r=1,2$ i.e. $y_r$ and $(p_ry_r')$ both are
identically zero on $J_r,\,\ r=1,2$.
\end{definition}

\begin{definition} \rm
Let \eqref{eq2.1a} hold. A complex number $\lambda $ is called an eigenvalue
of the two-interval S-L boundary value problems (BVP) consisting of
\eqref{eq11a} and \eqref{eq12} if the two-interval S-L equations
\eqref{eq11a} have a nontrivial solution $\mathbf{y}$ satisfying the
boundary conditions \eqref{eq12}. Such a solution $\mathbf{y}$ is called an
eigenfunction of $\lambda $. Any multiple of an eigenfunction is also an
eigenfunction.
\end{definition}

\begin{theorem}
Let \eqref{eq2.1a} hold. Then a complex number $\lambda $ is an eigenvalue
of the boundary value problems \eqref{eq11a}, \eqref{eq12} if and only if $
\Delta (\lambda )=0$.
\end{theorem}

\begin{proof}
If $\lambda $ is an eigenvalue and $\mathbf{y}=\{y_1,y_2\}$ an
eigenfunction of $\lambda $, then there exist $C_r\in M_{2\times 1}(
\mathbb{C}),\,r=1,2$ and at least one of the vectors $C_1$ and $C_2$ is
nonzero, such that
\begin{equation}
Y_r(t)=\Phi _r(t,a_r,\lambda )C_r.  \label{eq16}
\end{equation}
Note that
$\Phi _r(a_r,a_r,\lambda )=I$, $r=1,2$.
Substituting \eqref{eq16} into the boundary conditions \eqref{eq12}, we
obtain
\begin{equation}
A_1C_1+B_1\Phi _1(b_1,a_1,\lambda )C_1+A_2C_2+B_2\Phi
_2(b_2,a_2,\lambda )C_2=0.  \label{eq17}
\end{equation}
Set $C=\begin{bmatrix}
C_1 \\
C_2
\end{bmatrix}$. Therefore \eqref{eq17} can be written as
\begin{equation}
(A_1+B_1\Phi _1(b_1,a_1,\lambda )\mid A_2+B_2\Phi
_2(b_2,a_2,\lambda ))C=0.  \label{eq18}
\end{equation}
Since $C\neq 0,$ and $\lambda $ is an eigenvalue of BVP \eqref{eq11a},
\eqref{eq12} by assumption, it then follows that
\begin{equation*}
\det [(A_1+B_1\Phi _1(b_1,a_1,\lambda )\mid A_2+B_2\Phi
_2(b_2,a_2,\lambda ))]=0;
\end{equation*}
i.e., $\Delta (\lambda )=0$.

Conversely, suppose $\Delta (\lambda )=0$. Then \eqref{eq18} has a
nontrivial vector $C\in M_{4\times 1}(\mathbb{C})$. 
We use the notation $C_1\in M_{2\times 1}(\mathbb{C})$ denotes the vector
 whose rows are the first two rows of $C$, and 
$C_2\in M_{2\times 1}(\mathbb{C})$ denotes the
vector whose rows are the last two rows of $C$. At least one of the vectors
 $C_1$ and $C_2$ is nontrivial. Solve the initial value problems
\begin{equation*}
Y'=(P_r-\lambda W_r)Y\text{ on } J_r,\quad
Y_r(a_r)=C_r,\; r=1,2.
\end{equation*}
Then
\begin{gather*}
Y_r(b_r)=\Phi _r(b_r,a_r,\lambda )Y_r(a_r),\\
(A_1+B_1\Phi _1(b_1,a_1,\lambda ))Y_1(a_1)+(A_2+B_2\Phi
_2(b_2,a_2,\lambda ))Y_2(a_2)=0.
\end{gather*}
Therefore, we have that $\mathbf{y}=\{y_1,y_2\}$ is an eigenfunction of
the BVP \eqref{eq11a},\eqref{eq12}, where $y_r$ is the top component of 
$Y_r$, $r=1,2$. This shows that $\lambda $ is an eigenvalue of this BVP.
\end{proof}

\section{The Green's Function}

Since, as mentioned above, our method of constructing the Green's function -
even in the one interval case - is not the standard one generally found in
the literature and in textbooks we make it self-contained by presenting the
basic theory used in the construction for the benefit of the reader.

Let $p_r^{-1}$, $q_r$, $w_r$ satisfy \eqref{eq2.1a} and 
$f_r\in L(J_r,\mathbb{C})$. We consider the two-interval boundary-value 
problem
\begin{gather}
-(p_ry')'+q_ry=\lambda w_ry+f_r\quad \text{on }
J_r=(a_r,b_r),\; r=1,2,\;\lambda \in \mathbb{C},
\label{eq19} \\
  \label{3.1}
A_1Y_1(a_1)+B_1Y_1(b_1)+A_2Y_2(a_2)+B_2Y_2(b_2)=0,\quad
 Y_r=\begin{bmatrix}
y_r \\
p_ry_r'
\end{bmatrix}, \; r=1,2.
\end{gather}
This boundary-value problem  is equivalent to the system
 boundary-value problem
\begin{equation}  \label{eq22}
Y'=(P_r-\lambda W_r)Y+F_r,\quad
A_1Y_1(a_1)+B_1Y_1(b_1)+A_2Y_2(a_2)+B_2Y_2(b_2)=0,
\end{equation}
where $P_r,W_r$ are defined by \eqref{eq13} and
\[
F_r=\begin{bmatrix}
0 \\
-f_r
\end{bmatrix}
\]

Let $\Phi_r=\Phi_r(\cdot,\cdot,\lambda)$ be the primary fundamental matrix
of the homogeneous system
\begin{equation}  \label{3.2}
Y'=(P_r-\lambda W_r)Y.
\end{equation}
Note that
\begin{equation*}
\Phi _r(t,u_r,\lambda )=\Phi _r(t,a_r,\lambda )\,\Phi
_r(a_r,u_r,\lambda )
\end{equation*}
for $a_r\leq t,u_r\leq b_r$.

The next theorem is a special case of the well known Fredholm alternative.

\begin{theorem}
Let \eqref{eq2.1a}, \eqref{eq19}--\eqref{3.2} hold. 
Let $\lambda \in \mathbb{C} $. Then the following three statements are equivalent:
\begin{itemize}
\item[(1)] when $\mathbf{f}=\{f_1,f_2\}=0$, i.e. $f_r=0$ on $J_r$,
 $r=1,2$, the two-interval BVP \eqref{eq19}-\eqref{3.1} (and consequently also
\eqref{eq22}) has only the trivial solution.

\item[(2)] The matrix $[A_1+B_1\Phi_1(b_1,a_1,\lambda)|A_2+B_2\Phi_2(b_2,a_2,
\lambda)]$ has an inverse.

\item[(3)] For every $\mathbf{f}=\{f_1,f_2\}$, $f_r\in L(J_r,\mathbb{C})$, $
r=1,2$, each of the problems \eqref{eq19}- \eqref{3.1} and \eqref{eq22} has
a unique solution.
\end{itemize}
\end{theorem}

\begin{proof}
We know that $Y_r$ is a solution of
\begin{equation}
Y'=(P_r-\lambda W_r)Y+F_r\ \ \text{on}\ \ J_r  \label{eq20}
\end{equation}
if and only if $y_r$ is a solution of
\begin{equation}
-(p_ry')'+q_ry=\lambda w_ry+f_r\quad \text{on } J_r,  \label{eq21}
\end{equation}
where
$ Y_r=\begin{bmatrix}
y_r \\
p_ry_r'
\end{bmatrix}$.
For $C_r=\begin{bmatrix}
c_{r1} \\
c_{r2}
\end{bmatrix}$,
$c_{r1},c_{r2}\in \mathbb{C}$, $r=1,2$, determine a solution 
$Y_r$ of \eqref{eq20} on $J_r$ by the initial condition
\begin{equation*}
Y_r(a_r,\lambda )=C_r.
\end{equation*}
Then $y_r$ is a solution of \eqref{eq21} determined by the initial
conditions $y_r(a_r,\lambda)=c_{r1}$, $(p_ry'_r)(a_r,\lambda)=c_{r2}$.

By the variation of parameters formula, we have
\begin{equation}
Y_r(t,\lambda )=\Phi _r(t,a_r,\lambda )C_r+\int_{a_r}^{t}\Phi
_r(t,s,\lambda )F_r(s)ds,\quad a_r\leq t\leq b_r.  \label{eq111}
\end{equation}
In particular,
\begin{equation*}
Y_r(b_r,\lambda )=\Phi _r(b_r,a_r,\lambda
)C_r+\int_{a_r}^{b_r}\Phi _r(b_r,s,\lambda )F_r(s)ds.
\end{equation*}
Let $D(\lambda )=(A_1+B_1\Phi _1(b_1,a_1,\lambda )\mid
A_2+B_2\Phi _2(b_2,a_2,\lambda ))$ and 
$C=\begin{bmatrix}
C_1 \\
C_2
\end{bmatrix}$, Then
\begin{equation}
\begin{split}
& A_1Y_1(a_1,\lambda )+B_1Y_1(b_1,\lambda
)+A_2Y_2(a_2,\lambda )+B_2Y_2(b_2,\lambda ) \\
& =D(\lambda )C+B_1\int_{a_1}^{b_1}\Phi _1(b_1,s,\lambda )F_1(s)
\,\mathrm{d}s+B_2\int_{a_2}^{b_2}\Phi _2(b_2,s,\lambda )F_2(s)
\,\mathrm{d}s.
\end{split}
\label{eq23}
\end{equation}
When $f_r=0$ on $J_r$$(r=1,2)$, $\mathbf{Y}=\{Y_1,Y_2\}$ and
 $\mathbf{y}=\{y_1,y_2\}$ are nontrivial solutions if and only if $C$ is
not the zero vector. By \eqref{eq23}, we have that when $f_r=0$ on $J_r$$
(r=1,2)$, there is a nontrivial solution $\{Y_1,Y_2\}$ (and a nontrivial
solution $\{y_1,y_2\}$ of \eqref{eq19}) satisfying the boundary
conditions
\begin{equation*}
A_1Y_1(a_1)+B_1Y_1(b_1)+A_2Y_2(a_2)+B_2Y_2(b_2)=0
\end{equation*}
if and only if $D(\lambda )$ is singular. It also follows from \eqref{eq23}
that there is a unique solution $\{Y_1,Y_2\}$ satisfying the boundary
conditions \eqref{3.1} for every $f_r\in L(J_r,\mathbb{C}),r=1,2$, if
and only if $D(\lambda )$ is nonsingular. Similarly there is a unique
solution $\mathbf{y}=\{y_1,y_2\}$ satisfying the boundary conditions
\eqref{3.1} for every $\mathbf{f}=\{f_1,f_2\}$, $f_r\in L(J_r,
\mathbb{C})$,$r=1,2$ if and only if $D(\lambda )$ is nonsingular.
\end{proof}

Next we construct the Green's function for two-interval boundary-value
problems. Assume that
\begin{equation*}
D(\lambda )=(A_1+B_1\Phi _1(b_1,a_1,\lambda )\mid A_2+B_2\Phi
_2(b_2,a_2,\lambda ))
\end{equation*}
is nonsingular. We use the notation $D_1(\lambda )$ denotes the 2 by 4
matrix whose rows are the first two rows of $D^{-1}(\lambda )$, and $
D_2(\lambda )$ denotes the 2 by 4 matrix whose rows are the last two rows
of $D^{-1}(\lambda )$. Let
\begin{gather*}
G_1(t,s,\lambda )=\begin{cases}
-\Phi _1(t,a_1,\lambda )D_1(\lambda )B_1\Phi _1(b_1,s,\lambda ),
& a_1\leq t<s\leq b_1, \\
-\Phi _1(t,a_1,\lambda )D_1(\lambda )B_1\Phi _1(b_1,s,\lambda
)+\Phi _1(t,s,\lambda ), & a_1\leq s\leq t\leq b_1,
\end{cases}\\
\widetilde {G}_1(t,s,\lambda)
=-\Phi_1(t,a_1,\lambda)D_1(\lambda)B_2
\Phi_2(b_2,s,\lambda),\quad a_1\leq t\leq b_1, \; a_2\leq s\leq b_2.
\\
{G_2}(t,s,\lambda)=-\Phi_2(t,a_2,\lambda)D_2(\lambda)
B_1\Phi_1(b_1,s,\lambda),\quad a_2\leq t\leq b_2,\; a_1\leq s\leq b_1,
\\
\widetilde G_2(t,s,\lambda)
=\begin{cases}
-\Phi_2(t,a_2,\lambda)D_2(\lambda) B_2\Phi_2(b_2,s,\lambda), & a_2\leq t<
s\leq b_2, \\
-\Phi_2(t,a_2,\lambda)D_2(\lambda)
B_2\Phi_2(b_2,s,\lambda)+\Phi_2(t,s,\lambda), & a_2\leq s\leq t\leq b_2.
\end{cases}
\end{gather*}

\begin{theorem}
Assume $D(\lambda )$ is nonsingular; i.e., $[A_1+B_1\Phi
_1(b_1,a_1,\lambda )\mid A_2+B_2\Phi _2(b_2,a_2,\lambda
)]^{-1}$ exists, then for any $\mathbf{f}=\{f_1,f_2\}$, 
$f_r\in L(J, \mathbb{C}),r=1,2$, the unique solution $\mathbf{y}=\{y_1,y_2\}$ 
of \eqref{eq19}-\eqref{3.1} and the unique solution $\mathbf{Y}=\{Y_1,Y_2\}$
of \eqref{eq22}, respectively, are given by
\begin{gather}
y_1(t)=-\int_{a_1}^{b_1}G_{1,(12)}(t,s,\lambda )f_1(s)\,\mathrm{d}
s-\int_{a_2}^{b_2}\widetilde{G}_{1,(12)}(t,s,\lambda )f_2(s)\,\mathrm{d}
s,\quad a_1\leq t\leq b_1,  \label{eq25}
\\
 \label{eq30}
y_2(t)=-\int_{a_1}^{b_1}G_{2,(12)}(t,s,\lambda )f_1(s)\,\mathrm{d}
s-\int_{a_2}^{b_2}\widetilde{G}_{2,(12)}(t,s,\lambda )f_2(s)\,\mathrm{d}
s,\quad a_2\leq t\leq b_2,
\\
Y_1(t)=\int_{a_1}^{b_1}G_1(t,s,\lambda )F_1(s)\,\mathrm{d}
s+\int_{a_2}^{b_2}\widetilde{G}_1(t,s,\lambda )F_2(s)\,\mathrm{d}
s,\quad a_1\leq t\leq b_1,  \label{eq24}
\\
Y_2(t)=\int_{a_1}^{b_1}G_2(t,s,\lambda )F_1(s)\,\mathrm{d}
s+\int_{a_2}^{b_2}\widetilde{G}_2(t,s,\lambda )F_2(s)\,\mathrm{d}
s,\quad a_2\leq t\leq b_2.  \label{eq26}
\end{gather}
Set $\mathbf{K}(t,s,\lambda )=\{K_1(t,s,\lambda ),K_2(t,s,\lambda )\}$,
where
\begin{gather*}
K_1(t,s,\lambda )=\begin{cases}
G_1(t,s,\lambda ) & a_1\leq s\leq b_1, \\
\widetilde{G}_1(t,s,\lambda ), & a_2\leq s\leq b_2,
\end{cases}
\quad  a_1\leq t\leq b_1,
\\
K_2(t,s,\lambda )=\begin{cases}
G_2(t,s,\lambda ) & a_1\leq s\leq b_1, \\
\widetilde{G}_2(t,s,\lambda ), & a_2\leq s\leq b_2,
\end{cases}
\quad  a_2\leq t\leq b_2.
\end{gather*}

We call $\mathbf{K}(t,s,\lambda )=\mathbf{K}(t,s,\lambda
,P_1,P_2,W_1,W_2,A_1,A_2,B_1,B_2)$ (Here we use the complete
notation to highlight the dependence of $\mathbf{K}$ on these quantities.)
the Green's matrix of the regular boundary value problem \eqref{eq14a},
\eqref{eq12}. And we call $\mathbf{K}_{12}=\{K_{1,(12)},K_{2(12)}\}$ the
Green's function of two-interval boundary value problem \eqref{eq11a},
\eqref{eq12}.
\end{theorem}

\begin{proof}
Let
\begin{equation*}
C=D^{-1}(\lambda )(-B_1\int_{a_1}^{b_1}\Phi _1(b_1,s,\lambda
)F_1(s)\,\mathrm{d}s-B_2\int_{a_2}^{b_2}\Phi _2(b_2,s,\lambda
)F_2(s)\,\mathrm{d}s).
\end{equation*}
By \eqref{eq23}, we have
\begin{equation*}
A_1Y_1(a_1)+B_1Y_1(b_1)+A_2Y_2(a_2)+B_2Y_2(b_2)=0.
\end{equation*}
Recall the notation $D_1(\lambda )$ and $D_2(\lambda )$, we have
\begin{gather*}
C_1=D_1(\lambda )(-B_1\int_{a_1}^{b_1}\Phi _1(b_1,s,\lambda
)F_1(s)\,\mathrm{d}s-B_2\int_{a_2}^{b_2}\Phi _2(b_2,s,\lambda
)F_2(s)\,\mathrm{d}s),
\\
C_2=D_2(\lambda )(-B_1\int_{a_1}^{b_1}\Phi _1(b_1,s,\lambda
)F_1(s)\,\mathrm{d}s-B_2\int_{a_2}^{b_2}\Phi _2(b_2,s,\lambda
)F_2(s)\,\mathrm{d}s).
\end{gather*}
From \eqref{eq111}, we obtain that
\begin{equation}  \label{eq29}
\begin{split}
Y_1(t)=&\Phi_1(t,a_1,\lambda)D_1(\lambda)(-B_1\int_{a_1}^{b_1}\Phi_1(b_1,s,
\lambda)F_1(s)\,\mathrm{d}s\\
&\quad - B_2\int_{a_2}^{b_2}\Phi_2(b_2,s,\lambda)F_2(s)\,\mathrm{d}s) 
 +\int_{a_1}^t\Phi_1(t,s,\lambda)F_1(s)\,\mathrm{d}s \\
=&\int_{a_1}^{b_1}[\Phi_1(t,a_1,\lambda)D_1(\lambda)
(-B_1\Phi_1(b_1,s,\lambda)F_1(s))]\,\mathrm{d}s+
\int_{a_1}^t\Phi_1(t,s,\lambda)F_1(s)\,\mathrm{d}s \\
& + \int_{a_2}^{b_2}[\Phi_1(t,a_1,\lambda)D_1(\lambda)
(-B_2\Phi_2(b_2,s,\lambda)F_2(s))]\,\mathrm{d}s \\
=& \int_{a_1}^{b_1}G_1(t,s,\lambda)F_1(s)\,\mathrm{d}s+ \int_{a_2}^{b_2}
\widetilde{G}_1(t,s,\lambda)F_2(s)\,\mathrm{d}s,\quad a_1\leq t\leq b_1.
\end{split}
\end{equation}
\begin{equation}  \label{eq28}
\begin{split}
Y_2(t)=&\Phi_2(t,a_2,\lambda)D_2(\lambda)(-B_1\int_{a_1}^{b_1}\Phi_1(b_1,s,
\lambda)F_1(s)\,\mathrm{d}s\\
&\quad - B_2\int_{a_2}^{b_2}\Phi_2(b_2,s,\lambda)F_2(s)\,\mathrm{d}s) 
+\int_{a_2}^t\Phi_2(t,s,\lambda)F_2(s)\,\mathrm{d}s \\
=&\int_{a_1}^{b_1}[\Phi_2(t,a_2,\lambda)D_2(\lambda)
(-B_1\Phi_1(b_1,s,\lambda)F_1(s))]\,\mathrm{d}s
+\int_{a_2}^t\Phi_2(t,s,\lambda)F_2(s)\,\mathrm{d}s \\
& + \int_{a_2}^{b_2}[\Phi_2(t,a_2,\lambda)D_2(\lambda)
(-B_2\Phi_2(b_2,s,\lambda)F_2(s))]\,\mathrm{d}s \\
=& \int_{a_1}^{b_1}G_2(t,s,\lambda)F_1(s)\,\mathrm{d}s+ \int_{a_2}^{b_2}
\widetilde{G}_2(t,s,\lambda)F_2(s)\,\mathrm{d}s,\quad a_2\leq t\leq b_2.
\end{split}
\end{equation}

Note that \eqref{eq25} and \eqref{eq30}, respectively, follow from the
identities \eqref{eq29} and \eqref{eq28} by taking the upper right
component; i.e.,
\begin{equation*}
y_1(t)=-\int_{a_1}^{b_1}G_{1,(12)}(t,s,\lambda )f_1(s)\,\mathrm{d}
s-\int_{a_2}^{b_2}\widetilde{G}_{1,(12)}(t,s,\lambda )f_2(s)\,\mathrm{d}
s,\quad a_1\leq t\leq b_1,
\end{equation*}
\begin{equation*}
y_2(t)=-\int_{a_1}^{b_1}G_{2,(12)}(t,s,\lambda )f_1(s)\,\mathrm{d}
s-\int_{a_2}^{b_2}\widetilde{G}_{2,(12)}(t,s,\lambda )f_2(s)\,\mathrm{d}
s,\quad a_2\leq t\leq b_2.
\end{equation*}
\end{proof}

\begin{remark} \rm
Note that the above construction of the Green's function and the
characteristic function does not assume any symmetry or self-adjointness of
the problem. The coefficients $p_r,q_r,w_r$ may be complex valued and
the boundary conditions need not be self-adjoint. If $w_r$ is identically
zero on the whole interval $J_r$ there is no $\lambda $ dependence and the
problem becomes degenerate. Similarly the case when $1/p_r$ is identically
zero on $J_r$ the problem can be considered degenerate.
\end{remark}

\begin{remark} \rm
If $p_r,q_r,w_r$ are real valued and $w_r>0$ on $J_r,$ $r=1,2,$
the self-adjoint operators in the separate Hilbert spaces $H_1=$ $
L^{2}(J_1,w_1),$ $H_2=$ $L^{2}(J_2,w_2)$ with their usual inner
products
\begin{equation}
(f,g)_r=\int\limits_{Jr}f\overline{g}w_r,\;r=1,2  \label{u}
\end{equation}
is well known \cite{zett05} and it is a routine exercise to show that if $
S_r$ is a self-adjoint operator in $H_r,$ $r=1,2$ then the direct sum of
$S_1$ and $S_2$ is a self-adjoint operator in the direct sum space $
H_{u}=L^{2}(J_1,w_1)\dotplus L^{2}(J_2,w_2)$ where each of $H_1$
and $H_2$ is endowed with the usual inner product \eqref{u}. Everitt and
Zettl \cite{evze86} showed that there are many self-adjoint operators in $
H_{u}$ which are not generated as direct sums in this way. These `new'
self-adjoint operators involve interactions between the the intervals $J_1$
and $J_2$. In \cite{evze86} all these interactions are characterized in
terms of boundary conditions at the endpoints. Mukhtarov and Yakubov \cite
{muya02} observed that the theory in \cite{evze86} can be significantly
extended by using different multiples of the inner usual inner products (\eqref
{u}:
\begin{equation}
(f,g)_r=h_r\int\limits_{Jr}f\overline{g}w_r,\;h_r>0,\;r=1,2.
\label{uh}
\end{equation}
Wang, Sun and Zettl \cite{asz1} exploited this observation to characterize
this enlarged set of self-adjoint operators in terms of boundary conditions
at the endpoints. Recently Wang and Zettl in \cite{waze13} further enlarged
this set by removing the positivity restriction on $h_r.$This requires a
different proof since \eqref{uh} is not an inner product if $h_r$ is
negative. In Section 5 we give some examples to illustrate these
interactions between the two intervals which generate self-adjoint
extensions including those found in \cite{muya02} and \cite{waze13} and
relate these to the comments we made in the Introduction about transmission
and interface conditions.
\end{remark}

\section{Examples}

In this section we give examples to illustrate that the construction of the
two-interval Green's function applies to problems with transmission and
interface conditions as mentioned in the Introduction.

These examples are taken from \cite{waze13}. They are for the special case
when the right endpoint of $J_1$ is the same as the left endpoint of $
J_2,$ i.e. $a_2=b_1$. In order to avoid unnecessary subscripts and to
make the notation more consistent with the literature on transmission and
interface conditions we let
\begin{equation}
J_1=(a,b),\quad J_2=(c,d),\quad b=c  \label{51}
\end{equation}
and use $c^{+}=b$ for the right endpoint of $J_1$ and $c^{-}=c$ for the
left endpoint of $J_2$. Also we let $A=A_1,$ $B=B_1,$ $C=A_2,$ $
D=B_2$ in \eqref{eq12}.

Using this notation we make the following simple but key observation.

\begin{remark} \rm
To apply the above construction of the Green's function to problems with
transmission and interface conditions a simple but important observation is
that when $b=c$ the direct sum of the Hilbert spaces from the two intervals
can be identified with the Hilbert space of the `outer' interval:
\begin{equation}
L^{2}((a,b),w_1)\dotplus L^{2}((c,d),w_2)=L^{2}((a,d),w)  \label{5.2}
\end{equation}
\end{remark}
where $w_1$ is the restriction of $w$ to $J_1$ and $w_2$ is the
restriction of $w$ to $J_2$. In each example below the given boundary
conditions generate a self-adjoint operator in the Hilbert space $
L^{2}((a,d),w)$.

The first example has separated boundary conditions: these are generally
called `transmission conditions' in the literature.

\begin{example}[Transmission Conditions]\label{ex1} \rm
Separated boundary conditions:
\begin{equation}
\begin{gathered}
A_1y(a)+A_2y^{[1]}(a) =0,\quad A_1,A_2\in \mathbb{R},\;(A_1,A_2)
\neq (0,0); \\
B_1y(b)+B_2y^{[1]}(b) =0,\quad B_1,B_2\in \mathbb{R},\;(B_1,B_2)
\neq (0,0); \\
C_1y(c)+C_2y^{[1]}(c) =0,\quad C_1,C_2\in \mathbb{R},\;(C_1,C_2)
\neq (0,0); \\
D_1y(d)+D_2y^{[1]}(d) =0,\quad D_1,D_2\in \mathbb{R},\;(D_1,D_2)
\neq (0,0).
\end{gathered}
\end{equation}
Let
\begin{equation*}
A=\begin{bmatrix}
A_1 & A_2 \\
0 & 0 \\
0 & 0 \\
0 & 0
\end{bmatrix}, \quad
B=\begin{bmatrix}
0 & 0 \\
B_1 & B_2 \\
0 & 0 \\
0 & 0
\end{bmatrix}, \quad
C=\begin{bmatrix}
0 & 0 \\
0 & 0 \\
C_1 & C_2 \\
0 & 0
\end{bmatrix}, \quad
D=\begin{bmatrix}
0 & 0 \\
0 & 0 \\
0 & 0 \\
D_1 & D_2
\end{bmatrix}.
\end{equation*}
In this case the $4\times 8$ matrix $(A,B,C,D)$ has full rank and
\begin{equation}
0=AEA^{\ast }=BEB^{\ast }=CEC^{\ast }=DED^{\ast }.  \label{eq4.3}
\end{equation}
\end{example}

Considering $(a,c]\cup \lbrack c,d)$ as one interval $(a,d)$ the next
example has transmission conditions at the outer endpoint $a,d$ and
interface conditions at $c$. This example is chosen to highlight the
(discontinuous) interface conditions at an interior point $c$. The roles of
the endpoints $a,c^{+},c^{-},d$ can be interchanged in this example (but
care must be taken regarding the signs of the matrices $A,B,C,D,$ see 
\cite{waze13}).

\begin{example} \label{ex2} \rm
 Let $h,k\in \mathbb{R},$ $h\neq 0\neq k$. Separated boundary
conditions at $a$ and at $d$ and coupled jump conditions at $c$.
\begin{gather*}
A_1y(a)+A_2(py')(a) =0,\quad A_1,A_2\in \mathbb{R},\; (A_1,A_2)\neq (0,0); \\
D_1y(d)+D_2(py')(d) =0,\quad D_1,D_2\in \mathbb{R},\;(D_1,D_2)\neq (0,0).
\end{gather*}
and
\begin{equation}
\begin{gathered}
Y(c)=e^{i\gamma }KY(b),\quad
Y=\begin{bmatrix}
y \\
y^{[1]}
\end{bmatrix}, \quad 
K=(k_{ij}),\;k_{ij}\in \mathbb{R},\;1\leq i,j\leq 2,\\
\det K\neq 0,\; -\pi <\gamma \leq \pi\,. 
\end{gathered} \label{eq4.5}
\end{equation}

Let $A,D$ be as in Example \ref{ex1}, then $\operatorname{rank}(A,D)=2$ and 
$k \,AEA^{\ast}-h\,DED^{\ast }=0$ for any $h,k$ since 
$0=AEA^{\ast }=DED^{\ast }$. Let
\begin{equation}
C=\begin{bmatrix}
0 & 0 \\
-1 & 0 \\
0 & -1 \\
0 & 0
\end{bmatrix}, \quad 
B=e^{i\gamma }\begin{bmatrix}
0 & 0 \\
k_{11} & k_{12} \\
k_{21} & k_{22} \\
0 & 0
\end{bmatrix}, \quad 
-\pi <\gamma \leq \pi .  \label{eq4.6}
\end{equation}
Then a straightforward computation shows that
\begin{equation*}
h\,CEC^{\ast }=k\,BEB^{\ast }
\end{equation*}
is equivalent to
\begin{equation*}
h\,E=k\,(\det K)\,E
\end{equation*}
which is equivalent to
\begin{equation}
h=k\det K.  \label{eq4.7}
\end{equation}
Since \eqref{eq4.7} holds for any $h,k\in \mathbb{R},$ $h\neq 0\neq k,$ it
follows from \cite[Theorem 2]{waze13} that the boundary conditions of this
example are self-adjoint for any $K\in M_2(\mathbb{R})$ with 
$\det(K)\neq 0$.
\end{example}

The next remark highlights a remarkable comparison with the well known
classical one-interval self adjoint boundary conditions, see \cite{zett05}.

\begin{remark}\label{r33} \rm
It is well known that in the one-interval theory $\det K=1$ is
required for self-adjointness of the boundary conditions. We find it
remarkable that the one-interval condition $\det K=1$ extends to $det(K)\neq
0$ in the two-interval theory. And that this generalization follows from two
simple observations: (i) The Mukhtarov-Yakubov \cite{muya02} observation
that for $h>0$ and $k>0$ using inner product multiples produces an
interaction between the two intervals yielding $det(K)>0$ and (ii) the
Wang-Zettl observation that the boundary value problem is invariant under
muliplication by $-1$ and this yields the further extension $det(K)\neq 0$.
Note that the parameters $h,k$ play no role in Example \ref{ex1}.
\end{remark}

The next example illustrates the situation when there are two sets of
coupled i.e. `jump' boundary conditions, in one case the jumps are between
the outer endpoints $a,d$ and the other between the inner `endpoints, $
b=c^{+}$ and $c=c^{-}$.

\begin{example}\label{ex3} \rm
Two pairs of coupled conditions, with $-\pi <\gamma _1,\gamma_2\leq \pi$,
\begin{equation}
\begin{gathered}
Y(d) =e^{i\gamma _1}GY(a),\quad 
G=(g_{ij}),\;g_{ij}\in \mathbb{R},\;i,j=1,2,\;\det G\neq 0, \\
Y(c) =e^{i\gamma _2}KY(b),\quad K=(k_{ij}),\;k_{ij}\in \mathbb{R}
,\;i,j=1,2,\;\det K\neq 0,\\
Y=\begin{bmatrix}
y \\
y^{[1]}
\end{bmatrix}.
\end{gathered}\label{eq4.8}
\end{equation}

Proceeding as in the previous example we obtain the equivalence of the
conditions for self-adjointness:
\begin{gather*}
k\,GEG^{\ast }=h\,E\quad\text{and}\quad k\,KEK^{\ast }=hE; \\
k\det G=h\quad\text{and}\quad k\det K=h; 
\end{gather*}
i.e.,
\begin{equation*}
\det G=\det K=\frac{h}{k}.
\end{equation*}
This shows that \eqref{eq4.8} are self-adjoint boundary conditions 
for any $h,k$ positive or negative.
\end{example}

More examples can be found in \cite{waze13} where singular analogues of the
regular self-adjoint boundary conditions are also found. We plan to
construct the Green's function for singular self-adjoint problems in a
subsequent paper.

\section{The Neuberger Construction}

The remark below is written by J. W. Neuberger and published here with his
permission. We believe it is of interest not only because we refer to  `a
construction of Neuberger' in the Introduction but also for pedagogical
reasons.   

\begin{remark}[J. W. Neuberger]\rm
In the spring of 1958, I taught my first graduate course. It was an
introduction to functional analysis by means of Sturm-Liouville problems. As
was, and still is, my custom, I didn't lecture, but rather I broke up
material for the class into a sequence of problems. The night before I was
concerned with finding problems which gave a good introduction to Green's
functions to the class. The standard `recipe' with its prescribed
discontinuity, seemed contrived. I managed to come up with the algebraic
method mentioned at the start of this paper. Problems for some simple
examples quickly led to the general case, again algebraically. To me this
remains an example of how `teaching' and `research' can impact one another,
particularly in a non lecture situation. If I had been lecturing, I would
have given the standard approach, the only one I knew the day before. The
algebraic approach to Green's functions might have never seen the light of
day and some nice mathematics would have been missed.
\end{remark}



\subsection*{Acknowledgements}
A. Wang was supported by the National Natural Science Foundation of
China (Grant No. 10901119).

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