\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 127, pp. 1--27.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/127\hfil Sturm-Liouville problems]
{Eigenvalues of Sturm-Liouville problems with discontinuous boundary conditions}

\author[A. Wang, A. Zettl \hfil EJDE-2017/127\hfilneg]
{Aiping Wang, Anton Zettl}

\address{Aiping Wang \newline
Department of Mathematics,
Harbin Institute of Technology,
Harbin 150001, China}
\email{aiping@hit.edu.cn}

\address{Anton Zettl \newline
Northern Illinois University, DeKalb, IL, USA}
\email{zettl@msn.com}

\dedicatory{Communicated by Jerome Goldstein}

\thanks{Submitted  March 31, 2017. Published May 10, 2017.}
\subjclass[2010]{34B20, 34B24, 47B25}
\keywords{Eigenvalue properties; discontinuous boundary conditions}

\begin{abstract}
 For classical regular two-point self-adjoint Sturm-Liouville problems (SLP)
 the dependence of the eigenvalues on the boundary conditions is well
 understood because of some surprisingly recent results.
 Recently there has been a lot of interest in problems with discontinuous
 boundary conditions. Such conditions are known by various names including
 transmission conditions, interface conditions, point interactions
 (in the physics literature), etc. Here we extend the known classical results
 to such problems.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{notation}[theorem]{Notation}
\allowdisplaybreaks

\section{Introduction}

Regular Sturm-Liouville problems (SLP) with boundary conditions requiring a
jump discontinuity at an interior point of the underlying interval are a
very active current research area. Such conditions are known by various
names including: transmission conditions \cite{muya02,mukm},
discontinuous conditions \cite{shyu,kamu07}, interface conditions
\cite{kowa,suwa08,zett68}, multi-point conditions or
multi-interval problems \cite{evze86,saz2,kkkw,asz1},
conditions on trees, point interactions, etc.

Consider the equation
\begin{equation}
My=-(py')'+qy=\lambda w\,y\\quad\text{on }
J=[a,b],\; \lambda \in \mathbb{C},\; -\infty <a<b<\infty  \label{0.1}
\end{equation}
with coefficients satisfying
\begin{equation}
\frac{1}{p},q,w\in L(J,\mathbb{R}),\quad p>0,\quad w>0,\quad  \text{a.e. on }J,
\label{0.2}
\end{equation}
where $L(J,\mathbb{R)}$ denotes the real-valued functions which are Lebesgue
integrable on $J$.

Condition \eqref{0.2} implies that all solutions $y$ and their
quasi-derivatives $y^{[1]}=(py')$ of equation \eqref{0.1} are
continuous on the whole interval $J$ \cite{zett05} and thus rules out any
boundary condition requiring a discontinuity.

We call the study of equation \eqref{0.1} and its operators, under condition
\eqref{0.2}, the 1-interval theory. Of particular interest are the
self-adjoint operator realizations $S$ of equation \eqref{0.1} and their
spectrum. These are operators $S$ from $L^2(J,w)$ to $L^2(J,w)$ which
satisfy
\begin{equation}
S_{\rm min}\subset S=S^{\ast }\subset S_{\rm max},  \label{0.3}
\end{equation}
where $S_{\rm min}$ and $S_{\rm max}$ are the minimal and maximal operators of
equation \eqref{0.1} under condition \eqref{0.2} in the space $L^2(J,w)$.
For this and other definitions and basic properties of equation \eqref{0.1}
see the book \cite{zett05}.

In this article we study equation \eqref{0.1} with boundary conditions
\begin{gather}
AY(a)+BY(b)=0,  \label{0.4}\\
Y(c^{+})=C\,Y(c^{-}),\quad  a<c<b,  \label{0.5}
\end{gather}
where $Y=\begin{pmatrix}
y \\
y^{[1]}
\end{pmatrix}$,
$y^{[1]}=(py')$, and the matrices $A,B,C$ satisfy
$A,B\in M_2(\mathbb{C)}$, $C\in M_2(\mathbb{R)}$,
 $\det (C)=1$,
\begin{equation}
AEA^{\ast }=BEB^{\ast },\quad \operatorname{rank}(A:B)=2,\quad
 E=\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}.  \label{0.6}
\end{equation}

Here $\mathbb{C}$ and $\mathbb{R}$ denote the complex and real numbers,
respectively, $(A:B)$ denotes the $2\times 4$ matrix whose first two columns
are those of $A$ and the last two are the columns of $B$, and
$M_2(\mathbb{S)}$ denotes the $2\times 2$ matrices with entries from
$\mathbb{S}$.

It is well known \cite{zett05} that the boundary value problem consisting of
equation \eqref{0.1} with coefficients satisfying \eqref{0.2} and the
boundary condition \eqref{0.4} and \eqref{0.6} generates a self-adjoint
operator $S$ satisfying \eqref{0.3} and that every operator $S$ satisfying
\eqref{0.3} is generated by a two point boundary condition \eqref{0.4} and
\eqref{0.6}. Thus every eigenfunction of every operator $S$ satisfying
\eqref{0.3} is continuous on $J$. Thus if $C$ in \eqref{0.5} is not the identity
matrix, how can we find eigenvalues whose eigenfunctions satisfy boundary
conditions \eqref{0.4} and \eqref{0.5}? The next remark discusses this
question.

\begin{remark} \label{r01} \rm
In \cite{waze15} it is shown that the boundary value problem \eqref{0.1},
\eqref{0.2}, \eqref{0.4}, \eqref{0.5}, \eqref{0.6} determines an
operator $S$ satisfying \eqref{0.3} i.e. is self-adjoint in the Hilbert
space $H=L^2(J,w)$ and its spectrum is discrete consisting of an infinite
number of eigenvalues. Thus if $C$ is not the identity matrix $I$, then the
eigenfunctions are not continuous at $c$ by \eqref{0.5}. This result is a
special case of a much more general theorem from the 2-interval theory
developed by Everitt and Zettl in \cite{evze86}. See \cite{waze15} for
details. In this theory it is convenient to identify the Hilbert space $H$
with the direct sum space $H=L^2(J_1,w_1)\dotplus L^2(J_2,w_2)$
where $J_1=(a,c)$, $J_2=(c,b)$ and $w_1,w_2$ are the restrictions of
$w$ to $J_1$, $J_2$, respectively. Strictly speaking, the 2-interval
theory applied to $J_1$, $J_2$ extends \eqref{0.3} from the Hilbert
space $L^2(J,w)$ to the direct sum space $L^2(J_1,w_1)\dotplus
L^2(J_2,w_2)$. These two spaces consist of the same functions but the
direct sum space emphasizes that these functions need not be continuous at $
c$. We believe this clarifies the meaning of a statement commonly made in
the literature when authors simply say we study the equation \eqref{0.1} on
``$(a,c)\cup (c,b)"$.  See the next remark.
\end{remark}

\begin{remark}\label{r02} \rm
We comment on the nature of the solutions of equation \eqref{0.1}
 which satisfy condition \eqref{0.5} (and not necessarily \eqref{0.4} and
\eqref{0.6}). Any initial condition at $a$ determines a unique solution $y$
and its quasi-derivative $y^{[1]}$ which are continuous on $[a,c^{-}]$.
Condition \eqref{0.5} then determines $Y(c^{+})$ and using $Y(c^{+})$ as an
initial condition $y$ and $y^{[1]}$ are uniquely determined and continuous
on $[c^{+},b]$. Here, $c^{-}$ denotes the limit from the left and $c^{+}$
the limit from the right. Therefore every initial condition at $a$
determines a unique solution $y$ on the interval $[a,b]$ which satisfies
condition \eqref{0.5} and is continuous along with its quasi-derivative $
(py')$ on the intervals $[a,c^{-}]$ and $[c^{+},b]$. We call this
solution $y$ the `extended' solution, or $C$-extended solution, on $[a,b]$
and continue to denote it by $y$. Thus for any fixed matrix $C$ with
$\det C=1$ there is a 2-dimensional space of extended solutions of equation
\eqref{0.1} on the interval $[a,b]$.
\end{remark}

In this article we develop a method for studying Sturm-Liouville problems
\eqref{0.1}, \eqref{0.2}, \eqref{0.4}, \eqref{0.5}, \eqref{0.6} by constructing
operators $C_{\rm min}$ and $C_{\rm max}$ which depend on the jump condition (
\ref{0.5}) and then prove that, for any fixed condition \eqref{0.5}, all
self-adjoint operators $S$ in $L^2(J,w)$ generated by the boundary
conditions \eqref{0.4} \eqref{0.6} are characterized by
\begin{equation}
C_{\rm min}\subset S=S^{\ast }\subset C_{\rm max}.  \label{0.7}
\end{equation}

This essentially reduces problems with boundary conditions \eqref{0.4}
\eqref{0.6} and \eqref{0.5} to the study of problems with condition \eqref{0.4}
\eqref{0.6}) only and allows us to generalize known results for boundary
conditions \eqref{0.4} \eqref{0.6} to problems \eqref{0.4} \eqref{0.6} and
\eqref{0.5}. For fixed $C$ in \eqref{0.5} the well known inequalities among
eigenvalues for different boundary conditions \eqref{0.4} \eqref{0.6}
established by Eastham, Kong, Wu, Zettl \cite{ekwz99}, the characterization
of the eigenvalues as zeros of an entire function, the continuous and
discontinuous dependence of the eigenvalues on the boundary conditions \eqref
{0.4} \eqref{0.6} are extended to \eqref{0.4} \eqref{0.6} \eqref{0.5}.  We
make no attempt to state all of these extensions here. When $C$ is not the
identity matrix then the eigenfunctions are extended solutions as described
in Remark \ref{r02}. When $C$ is the identity then the extended results
reduce to the known results for \eqref{0.4} \eqref{0.6}.

A key difference between the operators $S_{\rm min}$ and $S_{\rm max}$ in
\eqref{0.3} and $C_{\rm min}$ and $C_{\rm max}$ in \eqref{0.7} is that the former do
not depend on the boundary conditions and the latter do depend on condition
\eqref{0.5}. Because of this dependence the proof of \eqref{0.7} is rather
technical. But it can readily be extended to any finite number of interior
jump conditions \eqref{0.5} but we do not pursue this extension here. It can
also be extended to an infinite number of conditions \eqref{0.5} but this
requires some additional technical considerations.

The organization of the paper is as follows: In Section 2 we construct
$C_{\rm min}$ and $C_{\rm max}$ and establish \eqref{0.7}, in Section 3 prove the
transcendental characterization of the eigenvalues. Section 4 contains a
brief review of the canonical forms of the boundary conditions \eqref{0.4}
\eqref{0.6}, existence of eigenvalues is discussed in Section 5.  The other
sections contain `applications' of \eqref{0.7}: Inequalities in Section 6,
Continuity in Section 7, differentiability in Section 8, monotonicity in 9,
and multiplicity in 10.

\section{Minimal and maximal operators for discontinuous boundary conditions}

In this section we construct the operators $C_{\rm min}$ and $C_{\rm max}$ and
characterize the boundary conditions which generate the operators $S$ in the
Hilbert space $H=L^2(J,w)$ satisfying \eqref{0.7}. Our construction is
based on the 2-interval theory applied to the intervals
\[
J_1=(a,c),\quad J_2=(c,b).
\]

For a detailed discussion of this theory and its application to intervals
which have a common endpoint see the recent paper \cite{waze15}. In this
application the Hilbert space $H$ is identified with the direct sum space
$L^2(J_1,w_1)\dotplus L^2(J_2,w_2)$ where $w_1$, $w_2$ are
the restrictions of $w$ to the intervals $J_{1,}$ $J_2$, respectively. We
briefly summarize this two interval theory next. The 2-interval definitions
of the minimal and maximal operators and their basic properties used below.

\begin{definition} \rm
\begin{gather*}
D(S_{\rm min}(J)) = D(S_{\rm min}(J_1))\dotplus D(S_{\rm min}(J_2)), \\
D(S_{\rm max}(J)) = D(S_{\rm max}(J_1))\dotplus D(S_{\rm max}(J_2)),
\end{gather*}
and the corresponding operators $S_{\rm min}(J)$ and $S_{\rm max}(J)$ have these
domains.
\end{definition}

As in the 1-interval case the Lagrange sesquilinear form is fundamental in
the study of boundary value problems. It is defined by
\[
[f,g]=[f_1,g_1](c^{-})-[f_1,g_1](a)+[f_2,g_2](b)-[f_2,g_2](c^{+})
\]
where
\[
[ f_{r},g_{r}]=f_{r}(p_{r}\,\overline{g_{r}^{'}})-\overline{
g_{r}}(p_{r}{f_{r}'}).
\]
Here $f_{r}$, $g_{r}$, $p_{r}$ denote the the restrictions of $f,g,p$ to
$J_{r}$, $r=1,2$.

From the 2-interval theory \cite{waze15}, \cite{zett05} we have the
following two lemmas. To simplify the notation we let
$S_{\rm min}=S_{\rm min}(J) $ and $S_{\rm max}=S_{\rm max}(J)$.

\begin{lemma}
\begin{enumerate}
\item The minimal operator $S_{\rm min}$ is a closed, densely defined,
symmetric operator in the Hilbert space $H$.

\item
\begin{gather*}
S_{\rm min}^{\ast } = S_{1,\min }^{\ast }\dotplus S_{2,\min }^{\ast
}=S_{1,\max }\dotplus S_{2,\max }=S_{\rm max}; \\
S_{\rm max}^{\ast } = S_{1,\max }^{\ast }\dotplus S_{2,\max }^{\ast}
=S_{1,\min }\dotplus S_{2,\min }=S_{\rm min}.
\end{gather*}
\end{enumerate}
\end{lemma}

\begin{lemma}\label{le2}
The operators $S_{\rm min}$ and $S_{\rm max}$ have the properties:
\begin{enumerate}
\item The generalized Green's formula holds
\begin{equation}
(S_{\rm max}{f},{g})-({f},S_{\rm max}{g})=[{f},{g}]\ \ ({f},{g}\in D(S_{\rm max}));
\label{eqa.3}
\end{equation}

\item $D(S_{\rm min})$ can be characterized as
\[
D(S_{\rm min})=\{{f}\in D(S_{\rm max}):[{f},{g}]=0\ \ \text{for all }{g}\in
D(S_{\rm max})\}.
\]
\end{enumerate}
\end{lemma}

For a proof of the above lemma, see \cite{zett05}.
Next we define the operators $C_{\max}$ and $C_{\rm min}$ which depend on the
interior discontinuous condition \eqref{0.5} in the space $H$.

\begin{definition} \label{d21}\rm
 Let \eqref{0.1}, \eqref{0.2}, \eqref{0.4} and \eqref{0.5} hold.
Define the operator $C_{\rm max}$ in the Hilbert space $H$ by
\[
D(C_{\rm max})=\{y=\{y_1,y_2\}\in D(S_{\rm max}): Y(c^{+})=CY(c^{-})\}
\]
and $C_{\rm max}$ is the restriction of the 2-interval maximal operator $
S_{\rm max}$ to the domain $D(C_{\rm max})$.
\end{definition}

\begin{definition}\label{d22}\rm
 Let \eqref{0.1}, \eqref{0.2}, \eqref{0.4} and \eqref{0.5} hold.
Define the operator $C_{\rm min}$ in the Hilbert space $H$ by
\[
D(C_{\rm min})=\{y=\{y_1,y_2\}\in D(C_{\rm max}): Y(a)=0=Y(b)\}
\]
and $C_{\rm min}$ is the restriction of the 2-interval maximal operator $
S_{\rm max}$ to the domain $D(C_{\rm min})$.
\end{definition}

\begin{lemma}[Naimark Patching Lemma] \label{ylw3} \rm
Given any $c_{k}\in \mathbb{C},k=1,2,\dots ,8$ there exists a
maximal domain function $g=\{ g_1,g_2\}\in D(S_{\rm max})$ such that
\begin{gather*}
{g}_1(a) =c_1,\quad (p_1{g_1'})(a)=c_2,\quad
{g}_1(c^-)=c_{3},\quad (p_1{g_1'})(c^-)=c_{4}, \\
{g}_2(c^+) =c_{5},\quad (p_2{g_2'})(c^+)=c_{6},\quad
{g}_2(b)=c_{7},\quad (p_2{g_2'})(b)=c_{8}.
\end{gather*}
\end{lemma}

For a proof of the above lemma see the two-interval S-L theory
\cite{asz1,zett05} .
From Lemma \ref{ylw3}, one can obtain the following conclusion.

\begin{lemma} \label{ylw1}
Given any complex numbers $\alpha_i$, $i=1,2,3,4$ there exists
a function $g=\{ g_1,g_2\}\in D(C_{\rm max})$ such that
\[
{g}_1(a) =\alpha_1,\quad (p_1{g_1'})(a)=\alpha_2,\quad
{g}_2(b)=\alpha_{3},\quad (p_2{g_2'})(b)=\alpha_4.
\]
\end{lemma}

\begin{proof}
This lemma is a special case of Lemma \ref{ylw3}, where the function $g$
satisfies the interior discontinuous condition i.e.
\[
\begin{pmatrix}
g_2(c^{+}) \\
(p_2{g_2'})(c^{+})
\end{pmatrix}
=C\begin{pmatrix}
g_1(c^{-}) \\
(p_1{g_1'})(c^{-})
\end{pmatrix}.
\]
\end{proof}

The well known GKN theorem and its extensions are powerful tools for
characterizing all self-adjoint realizations $S$ of equation \eqref{0.1}
i.e. all operators $S$ satisfying \eqref{0.3}, in terms of two point
boundary conditions. The next theorems in this section, especially Theorem
\ref{th4}, establish a correspondingly powerful tool which can be used to
characterize all self-adjoint realizations $S$ satisfying \eqref{0.7} in
terms of two point boundary conditions for any fixed $C$. This new tool is
used in Sections 6 to 10 to extend the known classical results to the
boundary value problem \eqref{0.1}, \eqref{0.2}, \eqref{0.4}, \eqref{0.5},
\eqref{0.6}.

\begin{theorem}\label{t21}
Let the operators $C_{\rm min}$ and $C_{\rm max}$ be defined as
above. Then we have
\begin{enumerate}
\item $D(S_{\rm min})\subset D(C_{\rm min})\subset D(C_{\rm max})
\subset D(S_{\rm max})$ and $S_{\rm min}\subset C_{\rm min}
\subset C_{\rm max}\subset S_{\rm max}$;

\item $D(C_{\rm min})$ and $D(C_{\rm max})$ are dense in $H$;

\item For any $f,g\in D(C_{\max})$,
\begin{equation}  \label{eqa.7}
(C_{\max}f,g)-(f,C_{\max}g)=[f,g]=[f_2,g_2](b)-[f_1,g_1](a);
\end{equation}

\item For any $f,g\in D(C_{\min})$, $[f,g]=0;$

\item The operator $C_{\rm min}$ is a closed symmetric extension of the
two-interval minimal operator $S_{\rm min}$;

\item $C_{\rm min}^{\ast }=C_{\rm max}$ and $C_{\rm max}^{\ast }=C_{\rm min}$;

\item $C_{\rm max}$ is closed in $H$.
\end{enumerate}
\end{theorem}

\begin{proof}
Properties (1) and (2) follow from the definition of $C_{\rm min}$ and
$C_{\rm max}$ and the fact that $D(S_{\rm min})$ is dense in $H$.

For any $f,g\in D(C_{\max})$, functions $f$ and $g$ satisfy the interior
discontinuous condition, i.e.
\begin{gather*}
\begin{pmatrix}
f_2(c^+) \\
f_2^{[1]}(c^+)
\end{pmatrix}
=C\begin{pmatrix}
f_1(c^-) \\
f_1^{[1]}(c^-)
\end{pmatrix},\quad
\begin{pmatrix}
g_2(c^+) \\
g_2^{[1]}(c^+)
\end{pmatrix}
=C\begin{pmatrix}
g_1(c^-) \\
g_1^{[1]}(c^-)
\end{pmatrix}, \\
\begin{split}
[f_2,g_2] (c^+)
&=\big(f_2(p_2\overline{g_2'})-\overline{g_2}(p_2 {f_2'})\big)(c^+) \\
&=\det(C)\big(f_1(p_1\overline{g_1'})-\overline{g_1}(p_1{f_1'})\big)(c^-) \\
&=[f_1,g_1] (c^-).
\end{split}
\end{gather*}
It follows from the generalized Green's formula \eqref{eqa.3} that, for any
 $f,g\in D(C_{\max})\subset D(S_{\max})$,
\[
(C_{\max}f,g)-(f,C_{\max}g)=[f,g]=[f_2,g_2] (b)-[f_1,g_1] (a).
\]
Therefore, for all $f,g\in D(C_{\min})$,
\[
(C_{\min}f,g)-(f,C_{\min}g)=[f,g]=0,
\]
which shows that the densely defined operator $C_{\min}$ is symmetric.

It is obvious that
\[
(C_{\min}f,g)-(f,C_{\max}g)=[f_2,g_2] (b)- [f_1,g_1] (a)=0,\quad \forall
f\in D(C_{\min}),\; g\in D(C_{\max})
\]
Hence $C_{\max}\subset C_{\min}^*$. Next we prove $C_{\min}^*\subset
C_{\max} $.

Since $S_{\rm min}\subset C_{\rm min}\subset C_{\rm max}\subset S_{\rm max}$, we
have
\begin{equation}
S_{\rm min}=S_{\rm max}^{\ast }\subset C_{\rm max}^{\ast }\subset C_{\rm min}^{\ast
}\subset S_{\rm min}^{\ast }=S_{\rm max}.  \label{eqa.6}
\end{equation}
Let $g\in D(C_{\rm min}^{\ast })$, then for any $f\in D(C_{\rm min})$, it
follows from \eqref{eqa.3} that
\begin{equation}
\begin{split}
0& =(C_{\rm min}f,{g})-({f},C_{\rm min}^{\ast }{g}) \\
& =[f,g] \\
& =[f_1,g_1] (c^{-})- [f_1,g_1] (a)+[f_2,g_2] (b)-[f_2,g_2]
(c^{+}) \\
& =[f_1,g_1] (c^{-})-[f_2,g_2] (c^{+}) \\
& =\big(f_1(p_1\overline{g_1'})-\overline{g_1}(p_1{
f_1'})\big)(c^{-})-\big(f_2(p_2\overline{g_2'})-
\overline{g_2}(p_2{f_2'})\big)(c^{+}).
\end{split}
\label{eqa.4}
\end{equation}
Since $f\in D(C_{\rm min})$, the function $f$ satisfies
\[
\begin{pmatrix}
f_2(c^{+}) \\
(p_2f_2')(c^{+})
\end{pmatrix}
 =C\begin{pmatrix}
f_1(c^{-}) \\
(p_1f_1')(c^{-})
\end{pmatrix},
\]
 and by substituting it into equation \eqref{eqa.4}, it follows that
\begin{equation}
\begin{split}
&\Big((p_1 \overline{g_1'})(c^{-})-c_{11}(p_2\overline{
g_2'})(c^{+})+c_{21}\overline{g_2}(c^{+})\Big)f_1(c^{-}) \\
& +\Big(-\overline{g_1}(c^{-})-c_{12}(p_2\overline{g_2'}
)(c^{+})+c_{22}\overline{g_2}(c^{+})\Big)(p_1f_1')(c^{-})=0.
\end{split}
\label{eqa.5}
\end{equation}
From the arbitrariness of function $f\in D(C_{\rm min})$ and the Naimark
Patching Lemma \ref{ylw1}, it follows that
\begin{gather*}
(p_1\overline{g_1'})(c^{-})-c_{11}(p_2\overline{g_2'})(c^{+})
 +c_{21}\overline{g_2}(c^{+})=0, \\
\overline{g_1}(c^{-})+c_{12}(p_2\overline{g_2'})(c^{+})-c_{22}
\overline{g_2}(c^{+})=0.
\end{gather*}
Then
\[
\begin{pmatrix}
g_2(c^{+}) \\
{(p_2g_2'})(c^{+})
\end{pmatrix}
=C\begin{pmatrix}
g_1(c^{-}) \\
{(p_1g_1'})(c^{-})
\end{pmatrix},
\]
i.e. $g\in D(C_{\rm max})$ and $C_{\rm min}^{\ast }g=C_{\rm max}g$.
Thus $C_{\rm min}^{\ast }\subset C_{\rm max}$. Hence
$C_{\rm min}^{\ast }=C_{\rm max}$. From the
facts that the adjoint of any densely defined operator is automatically
closed and $C_{\rm min}^{\ast }=C_{\rm max}$, it follows that $C_{\rm max}$ is a
closed operator in $H$.

Since $D(C_{\rm min})$ and $D(C_{\rm max})(=D(C_{\rm min}^{\ast }))$ are dense in
$H$, we have $C_{\rm min}\subset C_{\rm min}^{\ast \ast }=C_{\rm max}^{\ast }$. In
the following we prove that $C_{\rm max}^{\ast }\subset C_{\rm min}$.

Let ${g}=\{g_1,g_2\}\in D(C_{\rm max}^{\ast })$. Then for all ${f}\in
D(C_{\rm max})$,
\[
(C_{\rm max} {f}, {g})=( {f},C_{\rm max}^{\ast } {g}).
\]
From \eqref{eqa.6}, one obtains that
 $C_{\rm max}^{\ast }\subset C_{\rm min}^{\ast }=C_{\rm max}$,
 $g\in D(C_{\max})$ and then
\[
(C_{\rm max} {f}, {g})=( {f},C_{\rm max} {g}).
\]
From \eqref{eqa.7}, one has
\[
[f_2,g_2] (b)- [f_1,g_1] (a)=0, \quad\text{for  all }
 {f}\in D(C_{\rm max}),
\]
i.e.
\begin{equation}  \label{eqa.8}
\begin{split}
&f_2(b)(p_2\overline{g_2'})(b)-\overline{g_2}(b)(p_2{f_2'})(b) \\
&= f_1(a)(p_1\overline{g_1'})(a)- \overline{g_1}(a)(p_1{
f_1'})(a), \quad\text{for  all } {f=\{f_1,f_2\}}\in D(C_{\rm max}).
\end{split}
\end{equation}

In particular, using Patching Lemma \ref{ylw1}, one can select
$f\in D(C_{\rm max})$ which satisfies $f_1(a)=(p_1f_1')(a)=0$,
$f_2(b)=1$, $(p_2f_2')(b)=0$. Then from \eqref{eqa.8}, it follows that
$(p_2g_2')(b)=0$. In the same way, one has
$g_2(b)=g_1(a)=(p_1g_1')(a)=0$. Therefore $g\in D(C_{\min})$ and
$C_{\rm max}^{\ast }g=C_{\rm max} g=C_{\rm min}{g}$. Hence
$C_{\rm max}^{\ast }\subset C_{\rm min}$, then
$C_{\rm max}^{\ast }=C_{\rm min}$ and $C_{\rm min}$ is closed in $H$.
\end{proof}

\begin{corollary}\label{coro1}
$D(C_{\rm min})$ can be characterized as
\[
D(C_{\rm min})=\{g\in D(C_{\rm max}):[{f},{g}]=0 \text{ for all } f\in
D(C_{\rm max})\}.
\]
\end{corollary}

\begin{proof}
If $g\in D(C_{\rm min})$, then from \eqref{eqa.7} it is clear that for all
 $f\in D(C_{\rm max})$,
\[
[f,g]=[f_2,g_2] (b)- [f_1,g_1] (a)=0.
\]
On the other hand, if $g\in D(C_{\rm max})$ and, for all $f\in D(C_{\rm max})$,
 $[{f},{g}]=0$, i.e.
\[
[f_2,g_2] (b)- [f_1,g_1] (a)=0,
\]
then by the last part proof of Theorem \ref{t21}, it follows that
$g\in D(C_{\min})$.
\end{proof}

\begin{remark} \rm
The operators $C_{\rm min}$ and $C_{\rm max}$ defined above are our `new'
minimal and maximal operators, they play the roles of $S_{\rm min}$ and
$S_{\rm max}$ in the `standard' GKN theory as developed in the classic book of
Naimark \cite{naim68}. Our characterization of self-adjoint realizations of
Sturm-Liouville problems with interior conditions is based on the operators
$C_{\rm min}$ and $C_{\rm max}$ rather than $S_{\rm min}$ and $S_{\rm max}$. The key
difference between $(S_{\rm min}$, $S_{\rm max})$ and $(C_{\rm min},C_{\rm max})$ is
that $S_{\rm min}$ and $S_{\rm max}$ depend only on the coefficients $1/p,q,w$
whereas $C_{\rm min}$ and $C_{\rm max}$ depend on these coefficients and on the
interior discontinuous boundary conditions. Thus the study of the
multi-point boundary conditions is reduced to the study of two point
boundary conditions, the two points being the two `outer' endpoints of the
underlying interval.
\end{remark}

Next we make some further observations. If $S$ is a symmetric extension of
$C_{\min}$, then we have
\[
S_{\rm min}\subset C_{\min}\subset S\subset S^{\ast }\subset C_{\rm max}\subset
S_{\rm max}.
\]
Thus $S$ is a self-adjoint extensions of the minimal operator $C_{\rm min}$
and of the `standard' 2-interval minimal operator $S_{\rm min}$.

Each such operator $S$ satisfies
\[
S_{\rm min}\subset C_{\rm min}\subset S=S^{\ast }\subset C_{\rm max}\subset
S_{\rm max}.
\]
and is an extension of the `new' minimal operator $C_{\rm min}$ or,
equivalently, a restriction of the `new' maximal operator $C_{\rm max}$.
The next theorem characterizes all such operators $S$.

\begin{theorem}\label{th2}
 A linear manifold $D$ of $H$ is the domain of a self-adjoint
extension of $C_{\rm min}$ if and only if
\begin{enumerate}
\item $D(C_{\rm min})\subset D\subset D(C_{\rm max})$;

\item For any ${f}, {g}\in D$, $[{f},{g}]=0$;

\item If ${f}\in D(C_{\rm max})$ and $[{f},{g}]=0$ for any ${g}\in D$, then
${f}\in D$.
\end{enumerate}
\end{theorem}

\begin{proof}
\emph{Necessity.} Let $S$ be a self-adjoint extension of $C_{\rm min}$. Let $D(S)=D$
be the domain of $S$. Obviously
$C_{\rm min}\subset S=S^{\ast }\subset C_{\rm min}^{\ast }=C_{\rm max}$, i.e.
\[
D(C_{\rm min})\subset D(S)\subset D(C_{\rm max}).
\]
For any ${f}, {g}\in D(S)$, since $S$ is a restriction of the `new' maximal
operator $C_{\rm max}$ and $S$ is self-adjoint and hence symmetric, combing
\eqref{eqa.3}, it follows that
\[
[ {f}, {g}]=(S {f}, {g})-( {f},S {g})=0.
\]

Let ${f}\in D(C_{\rm max})$. If ${g}\in D(S)\subset D(C_{\rm max})$, from
\eqref{eqa.3}, one can obtain
\[
[ {f}, {g}]=(C_{\rm max} {f}, {g})-( {f},C_{\rm max} {g})=(C_{\rm max} {f}, {
g})-( {f},S {g}).
\]
Since for any ${g}\in D(S)$, $[ {f}, {g}]=0$, i.e.
\[
(C_{\rm max} {f}, {g})-( {f},S {g})=0,\ \ \text{for all}\ {g}\in D(S),
\]
Therefore ${f}\in D(S^{\ast })=D(S)$.

\emph{Sufficiency.} Let the linear manifold $D$ satisfy conditions (1), (2) and
(3) of Theorem \ref{th2}. Since $D(C_{\rm min})$ is dense in $H$ then $D$ is
also dense in $H$. We define the operator $S$: $D(S)=D\to H$ and
$S {f}=C_{\rm max} {f}$ $( {f}\in D(S))$.

For any ${f}, {g}\in D(S)$,
\[
0=[ {f}, {g}]=(S {f}, {g})-( {f},S {g}).
\]
Therefore $S\subset S^* $.

Assume that ${f}\in D(C_{\rm max})$ and for any ${g}\in D(S)$, $[ {f}, {g
}]=0$, i.e.
\[
[f,g]=(C_{\rm max} {f}, {g})-( {f},S {g})=0,
\]
which shows that ${f}\in D(S^{\ast })$. From (3), we know ${f}\in D(S)$.
Thus $S^{\ast}\subset S$ and then $S=S^{\ast } $, i.e. $S$ is a self-adjoint
operator in $H$.
\end{proof}

Next we characterize all self-adjoint extensions of $C_{\rm min}$ in $H$ or,
equivalently, all self-adjoint restrictions of $C_{\rm max}$ in $H$. These
extensions (or restrictions) differ only by their domains. These domains are
characterized by boundary conditions. How many ? And what are they? These
two questions are answered below. The number of independent boundary
conditions depends on the deficiency index which we study next.

The deficiency subspaces $\{N_{\lambda }:\lambda \in \mathbb{C}\}$ of the
closed symmetric operator $C_{\rm min}$ are defined by
\[
N_{\lambda }=\{{f}\in D(C_{\rm max}):C_{\rm max}{f}=\lambda {f}\},
\]
where $\lambda \in \mathbb{C},\,Im\lambda \neq 0$, and recall that
$C_{\rm min}^{\ast }=C_{\rm max}$. Similar to \cite{naim68}, for any
$\lambda \in \mathbb{C}$ with $Im\lambda \neq 0$, the deficiency indices
of $C_{\rm min}$ are defined by
\[
d^{+}=\dim N_{\lambda },\quad d^{-}=\dim N_{\overline{\lambda }},
\]
and $d^{+},d^{-}$ are independent of $\lambda $. Since the differential
expression is real, it follows that $d^{+}=d^{-}=d$.

It follows from the classical Von Neumann formula that, for any fixed
$\lambda \in \mathbb{C}$ with $Im\lambda \neq 0$,
\[
D(C_{\rm max})=D(C_{\rm min})+N_{\lambda }+N_{\overline{\lambda }},
\]
where the linear manifolds $D(C_{\rm min})$, $N_{\lambda }$ and
$N_{\overline{\lambda }}$ are linearly independent.

From the general theory \cite{naim68}, we obtain that an operator $S$ is a
self-adjoint extension of $C_{\rm min}$ if and only if its domain
\begin{equation}
D(S)=\{{y}\in D(C_{\rm max}):{y}={y_{0}}+{\phi }+V{\phi }\text{ for all }
 {y_{0}}\in D(C_{\rm min}) \text{ and for all } {\phi }\in N_{\lambda }\},
\label{eqa.9}
\end{equation}
where $V$ is any unitary map with the property that
\[
V:\ N_{\lambda }\to N_{\overline{\lambda }},\quad
V^{\ast }=V^{-1}: N_{\overline{\lambda }}\to N_{\lambda },
\]
and $S{f}=C_{\rm max}{f},\,{f}\in D(S)$.

Let $\{\phi _1,\dots ,{\phi _{d}}\}$ be an orthonormal basis for
$N_{\lambda }$ in $H$, and then $\{V{\phi _1},\dots ,V{\phi _{d}}\}$ is an
orthonormal basis for $N_{\overline{\lambda }}$ in $H$
(see \cite{weid1980,evze92}).

From what has been stated above, we present the following results.

\begin{theorem}\label{theo1} \label{th11}
Let the operator $S$ be a self-adjoint extension
of $C_{\rm min}$. Then the domain of $S$ can be described as follows:
\begin{equation}  \label{eqa.10}
D(S)=\{ {y}\in D(C_{\rm max}):  {y}= {y_{0}}+\sum_{r=1}^{d}\alpha _{r}{\psi
_{r}}\},
\end{equation}
where $y_{0}\in D(C_{\rm min})$, $\alpha _{r}\in \mathbb{C} $ and
$\psi _{r}=\phi _{r}+V\phi _{r}$ $(r=1,\dots ,d)$.
\end{theorem}

\begin{proof}
We just need to prove that the two domains \eqref{eqa.9} and \eqref{eqa.10}
are identical. Let $\phi \in N_{\lambda } $ and
$\{ {\phi _1},\dots , {\phi _{d }}\}$ be an orthonormal basis for $N_{\lambda } $,
then there exist
$\alpha _1$, $\dots ,\alpha _{d }\in \mathbb{C}$ such that
${\phi } =\alpha _1 {\phi _1}+\dots +\alpha _{d } \phi _{d }$.
Therefore $V{\phi}=\alpha _1V {\phi _1}+\dots +\alpha _{d }V {\phi _{d }}$ and
\[
{\phi }+V {\phi }=\alpha _1( {\phi }_1+V {\phi _1})+\dots +\alpha _{d
}( {\phi _{d }}+V {\phi _{d }})=\sum_{r=1}^{d }\alpha _{r} {\psi _{r}}.
\]

Conversely, it follows from $\sum_{r=1}^{d}\alpha _{r}{\psi _{r}}
=\sum_{r=1}^{d}\alpha _{r}(\phi _{r}+V\phi _{r})$ that $\sum_{r=1}^{d
}\alpha _{r}{\phi }_{r}=\phi\in N_{\lambda } $ and $\sum_{r=1}^{d }\alpha
_{r}V {\phi _{r}}=V {\phi } \in N_{\overline{\lambda }} $. Therefore $
\sum_{r=1}^{d}\alpha _{r}{\psi _{r}} ={\phi }+V{\phi }$.
\end{proof}

\begin{theorem}\label{th3}
Let $S$ be a self-adjoint extension of $C_{\rm min}$ with domain
\[
D(S)=\{ {y}\in D(C_{\rm max}): {y}= {y_{0}}+\sum_{r=1}^{d }\alpha _{r} {\psi
_{r}},\ \ \alpha _{r}\in \mathbb{C}\}.
\]
Then $D(S)$ is given by
\[
\{{y}\in D(C_{\rm max}):[{y},{\psi _{r}}]=0,\; r=1,\dots ,d\}.
\]
\end{theorem}

\begin{proof}
Let $D=\{{y}\in D(C_{\rm max}): [ {y}, {\psi _{r}}]=0,\ \ r=1,\dots ,d\}$. It
is easy to see that ${\psi _1},\dots ,\psi_d\in D(S)$. For ${y}\in D(S)$,
it follows from \eqref{eqa.3} that
\[
[ {y}, {\psi _{r}}]=(S {y}, {\psi _{r}})-({y},S{\psi _{r}})=0,\quad
r=1,2,\dots,d.
\]
Therefore ${y}\in D$, and then $D(S)\subset D$.

On the other hand, let ${y}\in D\subset D(C_{\rm max}) $ and ${g}\in D(S)$
then there exist ${g_{0}} \in D(C_{\rm min})$, $\alpha_1,\dots,\alpha_d\in
\mathbb{C}$ such that ${g}={g_{0}}+\alpha _1{\psi _1}+\dots +\alpha _{d}
{\psi _{d}}$. Combining with Corollary \ref{coro1}, we deduce that
\[
[ {y},{g}]=[{y},{g_{0}}]+[{y},\alpha _1{\psi _1}+\dots +\alpha
_{d}{\psi _{d}}]=0.
\]
Hence for ${y}\in D$ and any ${g}\in D(S)$, it follows that
\[
0=[ {y}, {g}]=(C_{\rm max} {y}, {g})-( {y},C_{\rm max} {g})=(C_{\rm max} {y}, {g}
)-( {y},S {g }).
\]
Therefore ${y}\in D(S^{\ast })=D(S)$. So $D\subset D(S)$ and then $D(S)=D$.
\end{proof}

\begin{theorem}[New GKN-TYPE Theorem]\label{th4}
Let $d$ denote the deficiency index of $C_{\min}$. A linear
submanifold $D(S)$ of $D(C_{\rm max})$ is the domain of a self-adjoint
extension $S$ of $C_{\rm min}$ if and only if there exist functions ${v_1}
=\{v_{11},v_{12}\},\dots $, ${v_{d}}=\{v_{d1},v_{d2}\}\in D(C_{\rm max})$
satisfying the following conditions:

\begin{enumerate}
\item ${v_1},\dots , {v_{d}}$ are linearly independent modulo $D(C_{\rm min})$;

\item $[ {v_{i}}, {v_{j}}]=0$, $i,j=1,\dots ,d$;

\item $D(S)=\{ {y}\in D(C_{\rm max}):[ {y}, {v_{i}}]=0,i=1,\dots ,d\}$.
\end{enumerate}
\end{theorem}

\begin{proof}
\emph{Necessity.} Using Theorems \ref{theo1} and \ref{th3}, we set ${v_1}
= {\psi _1}$, $\dots , {v_{d}}= {\psi _{d}}$, then
${v_1},\dots , {v_{d}}$ satisfy the conditions (1) and (2), and
the self-adjoint domain can be denoted by (3).

\emph{Sufficiency.} Assume there exist functions ${v_1},\dots , {v_{d}}
\in D(C_{\rm max})$ satisfying the conditions (1), (2) and (3). Now we prove
that $D(S)$ is a self-adjoint domain.

Conditions $[ {y}, {v_{i}}]=0$ $(i=1,\dots ,d)$ are linearly independent.
If not, there exist constants $c_1,\dots ,c_{d}$, not all zero, such that
for all ${y}\in D(C_{\rm max})$,
\[
c_1[ {y}, {v_1}]+\dots +c_{d}[ {y}, {v_{d}}]=0,
\]
i.e. $[{y},\bar{c_1}{v_1}+\dots +\bar{c_{d}}{v_{d}}]=0$. It follows
from Corollary \ref{coro1} that $\bar{c_1}{v_1}+\dots + \bar{c_{d}}{
v_{d}}\in D(C_{\rm min})$. This contradicts the linear independence of
${v_1} ,\dots ,{v_{d}}$ modulo $D(C_{\rm min})$.

Let
\[
\widehat{D} =\big\{y:\ {y}= {y_{0}}+c_1 {v_1} +\dots +c_{d} {v_{d}}
\big\},
\]
where ${y_{0}}\in D(C_{\rm min})$ and $c_1,\dots ,c_{d}$ are any complex
constants. From condition (2) and Corollary \ref{coro1}, it follows that $
\widehat{D}\subset D(S)$. Since $D(S)$ is obtained from $D(C_{\rm max})$ by
imposing $d$ linearly independent conditions, one can deduce that $\dim\big(
D(S)/D(C_{\rm min})\big)=2d-d=d$. Moreover, $\dim\big(\widehat{D}/D(C_{\rm min})
\big)=d$. Thus $\widehat{D}=D(S)$.

Note that $D(C_{\rm min})\subset \widehat{D }\subset D(C_{\rm max})$. Since ${
v_1},\dots , {v_{d}}$ satisfy condition (2), we obtain
\[
[ {f}, {g}]=0,\ \, \text{for any } {f}, {g}\in \widehat{D}.
\]
If ${f}\in D(C_{\rm max})$ and for any ${g}\in \widehat{D} $, $[ {f}, {g}]=0$,
then for ${g}={v_{i}} (i=1,\dots ,d)$, we have $[f,v_i]=0, i=1,\dots ,d$.
Hence ${f}\in D(S)=\widehat{D}$. It follows from Theorem \ref{th2} that $
\widehat{D} (=D(S))$ is a self-adjoint domain.
\end{proof}

Note that
\[
\dim \big(D(S_{\rm max})/D(S_{\rm min})\big)=2d_0=8,
\]
where $d_0$ is the deficiency index of the two-interval minimal operator
$S_{\rm min}$,
\[
\dim \big(D(S_{\rm max})/D(C_{\rm max})\big)=2,\quad
\dim \big(D(C_{\rm min})/D(S_{\rm min})\big)=2.
\]
Therefore,
\[
\dim \big(D(C_{\rm max})/D(C_{\rm min})\big)=2d_0-4=d^{+}+d^{-}=2d
\]
and then $d =2$.

\begin{theorem}\label{t22}
An operator $S$ in $H$ satisfies \eqref{0.7} if and only if its
domain $D=D(S)$ is given as
\begin{equation}  \label{eqa.13}
D(S)=\{y=\{y_1,y_2\}\in D(C_{\rm max}): AY(a)+BY(b)= {0}\},
\end{equation}
where matrices $A,B$ satisfy \eqref{0.6} i.e. $A,B\in M_2(\mathbb{C)}$,
$\operatorname{rank}(A:B)=2 $ and $AEA^*= BEB^*$.
\end{theorem}

\begin{proof}
The deficiency index of $C_{\min}$ is $d=2$.

\emph{Necessity.} Let $D(S)$ be the domain of a self-adjoint extension $S$
of $C_{\rm min} $. By Theorem \ref{th4}, there exist functions ${w_1}
=\{w_{11},w_{12}\}$, ${w_2}=\{w_{21},w_{22}\}\in D(C_{\rm max})$ satisfying
conditions (1),(2) and (3) of Theorem \ref{th4}. For any ${\ y}
=\{y_1,y_2\}\in D(C_{\rm max})$ satisfying condition (3), we have
\[
0=\begin{pmatrix}
[ {y}, {w_1}] \\
{[ {y}, {w_2}]}
\end{pmatrix}
=\begin{pmatrix}
[ y_2,w_{12}] (b)-[y_1,w_{11}] (a) \\
{[}y_2,w_{22}{]} (b )-[y_1,w_{21}] (a )
\end{pmatrix},
\]
i.e.
\[
\begin{pmatrix}
[ y_1,w_{11}] (a ) \\
{[y_1,w_{21}]} (a )
\end{pmatrix}
=\begin{pmatrix}
[ y_2,w_{12}] (b ) \\
{[y_2,w_{22}]} (b )
\end{pmatrix}.  \label{eq14}
\]
Therefore
\[
\begin{pmatrix}
\overline{ w}_{11}(a) & \overline{ w}_{11}^{[1]}(a) \\
\overline{ w}_{21}(a) & \overline{ w}_{21}^{[1]}(a)
\end{pmatrix}
EY(a)-\begin{pmatrix}
\overline{ w}_{12}(b) & \overline{ w}_{12}^{[1]}(b) \\
\overline{ w}_{22}(b) & \overline{ w}_{22}^{[1]}(b)
\end{pmatrix} EY(b)=0.
\]
Set
\[
A=\begin{pmatrix}
\overline{ w}_{11}(a) & \overline{ w}_{11}^{[1]}(a) \\
\overline{ w}_{21}(a) & \overline{ w}_{21}^{[1]}(a)
\end{pmatrix}E,\quad
 B=-\begin{pmatrix}
\overline{ w}_{12}(b) & \overline{ w}_{12}^{[1]}(b) \\
\overline{ w}_{22}(b) & \overline{ w}_{22}^{[1]}(b)
\end{pmatrix}E.
\]
Hence boundary conditions (3) of Theorem \ref{th4} is equivalent to
$AY(a)+BY(b)=0$. Compute
\begin{gather*}
A EA^{\ast }= \begin{pmatrix}
\overline{ w}_{11}(a) & \overline{ w}_{11}^{[1]}(a) \\
\overline{ w}_{21}(a) & \overline{ w}_{21}^{[1]}(a)
\end{pmatrix}E
\begin{pmatrix}
{\ w}_{11}(a) & {\ w}_{21}(a) \\
{\ w}_{11}^{[1]}(a) & {\ w}_{21}^{[1]}(a)
\end{pmatrix},
\\
BEB^{\ast }=\begin{pmatrix}
\overline{ w}_{12}(b) & \overline{ w}_{12}^{[1]}(b) \\
\overline{ w}_{22}(b) & \overline{ w}_{22}^{[1]}(b)
\end{pmatrix}E
\begin{pmatrix}
{\ w}_{12}(b) & {\ w}_{22}(b) \\
{\ w}_{12}^{[1]}(b) & {\ w}_{22}^{[1]}(b)
\end{pmatrix}.
\end{gather*}
From
\begin{align*}
0& =\begin{pmatrix}
[ {w_1},{w_1}] & [{w_2},{w_1}] \\
{[{w_1},{w_2}]} & [{w_2},{w_2}]
\end{pmatrix} \\
& =\begin{pmatrix}
{[w_{12},w_{12}] }(b )-{[w_{11},w_{11}] }(a ) & {[w_{22},w_{12}] }(b)- {
[w_{21},w_{11}] }(a ) \\
{[w_{12},w_{22}] }(b)-{[w_{11},w_{21}] }(a ) & {[w_{22},w_{22}] }(b )- {
[w_{21},w_{21}] }(a )
\end{pmatrix} \\
& =BEB^{\ast }-AEA^{\ast },
\end{align*}
it follows that $A EA^{\ast }=BEB ^{\ast }$.

It is obvious that $\operatorname{rank}(A : B )\leq 2$.
If $\operatorname{rank}(A : B )< 2$, then there
exist constants $c$ and $d$, not all zero, such that
$\begin{pmatrix}
c & d
\end{pmatrix}
(A : B )=0$. Therefore
\[
\begin{pmatrix}
c & d
\end{pmatrix}A
=\begin{pmatrix}
c & d
\end{pmatrix}
\begin{pmatrix}
\overline{ w}_{11}(a) & \overline{ w}_{11}^{[1]}(a) \\
\overline{ w}_{21}(a) & \overline{ w}_{21}^{[1]}(a)
\end{pmatrix}
E=0,
\]
i.e.
\begin{equation}  \label{eqa.11}
c\overline{ w}_{11}^{[1]}(a)+d\overline{ w}_{21}^{[1]}(a)=0,\ \ \ \ c
\overline{ w}_{11}(a)+d\overline{ w}_{21}(a)=0.
\end{equation}
Similarly,
\[
\begin{pmatrix}
c & d
\end{pmatrix}B
=\begin{pmatrix}
c & d
\end{pmatrix}
\begin{pmatrix}
\overline{ w}_{12}(b) & \overline{ w}_{12}^{[1]}(b) \\
\overline{ w}_{22}(b) & \overline{ w}_{22}^{[1]}(b)
\end{pmatrix}
(-E) =0,
\]
i.e.
\begin{equation}  \label{eqa.12}
c\overline{ w}_{12}^{[1]}(b)+d\overline{ w}_{22}^{[1]}(b)=0,\ \ \ \ c
\overline{ w}_{12}(b)+d\overline{ w}_{22}(b)=0.
\end{equation}

Let ${g}=\{g_1,g_2\}=\overline{c} {w_1}+\overline{d} {w_2}\in D(C_{\max})$.
Therefore for any ${f}=\{f_1,f_2\}\in D(C_{\rm max})$, from \eqref{eqa.11}
and \eqref{eqa.12}, one can obtain that
\begin{align*}
[ {f}, {g}]
& =[f_2,g_2] (b)-[f_1,g_1] (a) \\
& = [f_2,\overline{c}w_{12}+\overline{d}w_{22}] (b) - [f_1,\overline{c}
w_{11}+\overline{d}w_{21}] (a)
= 0.
\end{align*}
It follows from Corollary \ref{coro1} that ${g}\in D(C_{\rm min})$. This
contradicts the fact that ${w_1}, {w_2}$ are linearly independent modulo
$D(C_{\rm min})$. Thus $\operatorname{rank}(A : B )=2$.

\emph{Sufficiency.} If there exist complex $2\times 2$ matrices $A $ and $B $
satisfy $\operatorname{rank}(A : B )=2$, $A EA^{\ast }=BEB ^{\ast }$ and \eqref{eqa.13}. We
just need to prove that $D(S)$ defined by \eqref{eqa.13} is a self-adjoint
domain.

Let $A=(a_{ij})_{2\times 2}$ and $B=(b_{ij})_{2\times 2}$. From Lemma
\ref{ylw1}, there exist functions ${w_1} =\{w_{11},w_{12}\}$, ${w_2}
=\{w_{21},w_{22}\}\in D(C_{\rm max})$ such that
\begin{gather*}
w_{11}(a)=-\overline{a}_{12},\quad w_{11}^{[1]}(a)=\overline{a}_{11}, \quad
w_{12}(b)=\overline{b}_{12},\quad w_{12}^{[1]}(b)= -\overline{b}_{11}, \\
w_{21}(a)=-\overline{a}_{22},\quad w_{21}^{[1]}(a)=\overline{a}_{21}, \quad
w_{22}(b)=\overline{b}_{22},\quad w_{22}^{[1]}(b)=-\overline{b}_{21}.
\end{gather*}
For ${y}=\{y_1,y_2\}\in D(C_{\rm max})$, we have
\begin{align*}
\begin{pmatrix}
[ {y}, {w_1}] \\
{[} {y}, {w_2}{]}
\end{pmatrix}
& =\begin{pmatrix}
{[}y_2,w_{12}{]} (b) \\
{[}y_2,w_{22}{] (b )}
\end{pmatrix}
-\begin{pmatrix}
[ {y_1},w_{11}] (a ) \\
{[}{y_1},w_{21}{] (a )}
\end{pmatrix} \\
&=\begin{pmatrix}
\overline{ w}_{12}(b) & \overline{ w}_{12}^{[1]}(b) \\
\overline{ w}_{22}(b) & \overline{ w}_{22}^{[1]}(b)
\end{pmatrix}E
\begin{pmatrix}
y_2(b) \\
y_2^{[1]}(b)
\end{pmatrix} \\
&\quad -\begin{pmatrix}
\overline{ w}_{11}(a) & \overline{ w}_{11}^{[1]}(a) \\
\overline{ w}_{21}(a) & \overline{ w}_{21}^{[1]}(a)
\end{pmatrix}
E\begin{pmatrix}
y_1(a) \\
y_1^{[1]}(a)
\end{pmatrix} \\
& =-BY(b)-AY(a)
\end{align*}
Hence the boundary conditions $AY(a)+BY(b)=0$ are equivalent to
$[ {y}, {w_{i}}]=0$, $i=1,2$.

Now we prove $[ {w_i}, {w_j}]=0$, $i,j=1,2$. Compute
\begin{align*}
&\begin{pmatrix}
[ {w_1}, {w_1}] & [ {w_2}, {w_1}] \\
{[ {w_1}, {w_2}]} & [ {w_2}, {w_2}]
\end{pmatrix} \\
& =\begin{pmatrix}
{[w_{12},w_{12}]} (b ) & {[w_{22},w_{12}]} (b ) \\
{[w_{12},w_{22}]} (b ) & {[w_{22},w_{22}]} (b )
\end{pmatrix}
-\begin{pmatrix}
{[w_{11},w_{11}]} (a ) & {[w_{21},w_{11}]} (a ) \\
{[w_{11},w_{21}]} (a ) & {[w_{21},w_{21}]} (a )
\end{pmatrix}\\
&=\begin{pmatrix}
-\overline{b}_{12}b_{11}+\overline{b}_{11}b_{12} & -\overline{b}_{22}b_{11}+
\overline{b}_{21}b_{12} \\
-\overline{b}_{12}b_{21}+\overline{b}_{11}b_{22} & -\overline{b}_{22}b_{21}+
\overline{b}_{21}b_{22}
\end{pmatrix} \\
&\quad -\begin{pmatrix}
-\overline{a}_{12}a_{11}+\overline{a}_{11}a_{12} & -\overline{a}_{22}a_{11}+
\overline{a}_{21}a_{12} \\
-\overline{a}_{12}a_{21}+\overline{a}_{11}a_{22} & -\overline{a}_{22}a_{21}+
\overline{a}_{21}a_{22}
\end{pmatrix} \\
& =B EB^{\ast }-A EA^{\ast }.
\end{align*}
Hence, it follows from $A EA^{\ast }=BEB^{\ast }$ that
$[ {w_{i}}, {w_{j}}]=0$, $i,j=1,2$.

Next we prove that ${w_1}, {w_2}$ are linearly independent modulo
$D(C_{\rm min})$. If not, there exist constants $c$ and $d$, not all zero, such
that $c {w_1}+d {w_2}\in D(C_{\rm min})$.

By the Patching Lemma \ref{ylw1}, we may construct ${f}=\{f_1,f_2\}$,
${g}=\{g_1,g_2\}\in D(C_{\max})$ such that
\begin{gather*}
f_1(a)=0,\quad f_1^{[1]}(a)=-1, \quad f_2(b)=0,\quad f_2^{[1]}(b)=1, \\
g_1(a)=1,\quad g_1^{[1]}(a)=0, \quad g_2(b)=-1,\quad g_2^{[1]}(b)=0.
\end{gather*}
Therefore
\[
[c {w_1}+d {w_2},f]=0,\quad  [c {w_1}+d {w_2},g]=0,
\]
i.e.
\begin{gather*}
[cw_{11}+dw_{21},f_1] (a)=0,\quad [cw_{11}+dw_{21},g_1] (a)=0, \\
[cw_{12}+dw_{22},f_2] (b)=0,\quad [cw_{12}+dw_{22},g_2] (b)=0.
\end{gather*}
It is seen from simple computation that
\[
\begin{pmatrix}
\overline{c} & \overline{d}
\end{pmatrix}
\begin{pmatrix}
a_{12} & -a_{11} & b_{12} & -b_{11} \\
a_{22} & -a_{21} & b_{22} & -b_{21}
\end{pmatrix} =0.
\]
Namely
\[
\begin{pmatrix}
\overline{c} & \overline{d}
\end{pmatrix}
\begin{pmatrix}
A E : B E
\end{pmatrix}
=\begin{pmatrix}
\overline{c} & \overline{d}
\end{pmatrix}
\begin{pmatrix}
A : B
\end{pmatrix}
\begin{pmatrix}
E & 0 \\
0 & E
\end{pmatrix}
 =0.
\]
Since $c$ and $d$ are not both zero and $E$ is nonsingular, we have
$\operatorname{rank}(A:B)<2$. This contradicts the fact that
$\operatorname{rank}(A:B)=2$. Therefore ${w_1}, {w_2}$ are linearly
independent modulo $D(C_{\rm min})$. From the New
GKN-TYPE Theorem \ref{th4}, it follows that $D(S)$ defined by \eqref{eqa.13}
is the domain of a self-adjoint extension of $C_{\rm min}$.
\end{proof}

\section{Transcendental characterization of the eigenvalues for self-adjoint
discontinuous boundary conditions}

In this section we extend the well known characterization of the eigenvalues
of boundary value problems consisting of equation \eqref{0.1} with boundary
condition \eqref{0.4} to problems with boundary conditions \eqref{0.4} and
\eqref{0.5}. This characterization will be used below to extend the very
general Eastham, Kong, Wu, Zettl \cite{ekwz99} inequalities for boundary
conditions \eqref{0.4} to boundary conditions \eqref{0.4} and \eqref{0.5}
for fixed $C$.

Consider the equation
\begin{equation}
My=-(py')'+qy=\lambda wy\quad\text{on }
J=[a,b],\;\lambda \in \mathbb{C},\;-\infty <a<b<\infty  \label{3.1}
\end{equation}
with coefficients satisfying
\begin{equation}
p^{-1},q,w\in L(J,\mathbb{R)},\quad p>0,\quad  w>0\quad\text{a.e.\ on } J,
\label{3.2}
\end{equation}
and boundary conditions
\begin{gather}
AY(a)+BY(b)=0,  \label{3.3} \\
Y(c^{+})=CY(c^{-}),\quad a<c<b,  \label{3.4}
\end{gather}
and the matrices $A,B,C$ satisfy
\begin{equation}
AEA^{\ast }=BEB^{\ast },\quad
\operatorname{rank}(A:B)=2, \;\det (C)=1,  \label{3.5}
\end{equation}
where
\begin{equation}
A,B\in M_2(\mathbb{C)},\quad C\in M_2(\mathbb{R)},\quad
E=\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}.  \label{3.6}
\end{equation}

Although the next result follows from the standard linear ODE theory we
state it as a theorem here since it plays a major role below.

\begin{theorem}\label{t30}
Let \eqref{3.1} to \eqref{3.4} hold and let $\lambda \in \mathbb{C}$.
Every initial condition at $a$ determines a unique solution on $[a,b]$
which satisfies the jump condition \eqref{3.4} and there are exactly two such
linearly independent solutions of equation \eqref{3.1} for every
$\lambda\in \mathbb{C}$.
\end{theorem}

\begin{proof}
See Remark \ref{r02}. The proof that there are exactly two such linearly
independent solutions is similar to the proof in the general linear ode
theory for the case when $C=I$ and hence omitted.
\end{proof}

\begin{definition}\label{d31}
A solution on $[a,b]$ satisfying \eqref{3.4} is called a $C$
jump solution or just a jump solution when $C$ remains fixed. A complex
number $\lambda $ is an eigenvalue of problem \eqref{3.1} to \eqref{3.6} if
there exists a nontrivial $C$ jump solution $y$ on $[a,b]$ which satisfies
both boundary conditions \eqref{3.3} and \eqref{3.4}.
\end{definition}

As mentioned in Section 1, condition \eqref{3.2} implies that all
solutions are continuous on $[a,b]$. So if $C\neq I$, the identity matrix,
how can we get an eigenfunction satisfying both conditions \eqref{3.3} and
\eqref{3.4})? The next theorem answers this question.

\subsection*{Notation}
Below, for a fixed boundary condition \eqref{3.4}, we extend
solutions $y$ from $[a,c]$ to $[c,b]$ as in Remark \ref{r02}, and continue
to use the same notation $y$ for the extended solution. Thus if $y$ is an
eigenfunction satisfying \eqref{3.3} then it is such an extended solution.

Let
\begin{equation}
P=
\begin{pmatrix}
0 & 1/p \\
q & 0
\end{pmatrix},\quad
W=\begin{pmatrix}
0 & 0 \\
w & 0
\end{pmatrix}.  \label{3.7}
\end{equation}
Then the scalar equation \eqref{3.1} is equivalent to the first order system
\begin{equation}
Y'=(P-\lambda W)Y
=\begin{pmatrix}
0 & 1/p \\
q-\lambda w & 0
\end{pmatrix}
Y, \quad
 Y=\begin{pmatrix}
y \\
(py')
\end{pmatrix}.  \label{3.8}
\end{equation}

For fixed boundary condition \eqref{3.4} let $u,v$ be the extended solutions
of \eqref{3.1} on $[a,b]$ determined by the initial conditions:
\[
u(a)=1=v^{[1]}(a),\;v(a)=0=u^{[1]}(a)
\]
Let
\[
\Phi =\begin{pmatrix}
u & v \\
u^{[1]} & v^{[1]}
\end{pmatrix} .
\]
Then
\[
\Phi '=(P-\lambda W)\,\Phi \quad \text{on } J,\quad
\Phi (a,\lambda )=I,\; \lambda \in \mathbb{C}.
\]
Define the characteristic function $\delta $ by
\begin{equation}
\delta (\lambda )=\det [A+B\,\Phi (b,a,\lambda )],\quad
 \lambda \in \mathbb{C}.  \label{3.12}
\end{equation}
This function $\delta $ is a transcendental function whose zeros
characterize the eigenvalues as we will see below.

\begin{lemma}\label{l32}
The characteristic function $\delta $ is well defined and is an
entire function of $\lambda $ for fixed $(a,b,A,B,C,P,W)$.
\end{lemma}

The proof of the above lemma is similar to the case when $C=I$,
see \cite[Chapter 2]{zett05}.

\begin{lemma} \label{l33}
For fixed boundary condition \eqref{3.4} and $\delta (\lambda )$
defined as above in \eqref{3.12} we have:
\begin{enumerate}
\item A complex number $\lambda $ is an eigenvalue of the boundary value
problem \eqref{3.1} to \eqref{3.6} if and only if $\delta (\lambda )=0$.

\item The geometric multiplicity of an eigenvalue $\lambda $ is equal to the
number of linearly independent vector solutions $C=Y(a)$ of the linear
algebra system
\begin{equation}
[ A+B \Phi (b,a,\lambda )] C=0.  \label{3.13}
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
Suppose $\delta (\lambda )=0$. Then \eqref{3.13} has a nontrivial vector
solution for $C$. Solve the IVP
\[
Y'=(P-\lambda W)Y\quad \text{on } J,\;Y(a)=C.
\]
Then
\[
Y(b)=\Phi(b,a,\lambda)\,Y(a)\quad\text{and}\quad [A+B\Phi(b,a,\lambda)]Y(a)=0.
\]

From this it follows that the top component of $Y$, say, $y$ is an
eigenfunction of \eqref{3.1} to \eqref{3.6} and $\lambda $ is an eigenvalue
of this BVP. (Recall that the eigenfunctions are extended solutions on $
[a,b]$.

Conversely, if $\lambda $ is an eigenvalue and $y$ an eigenvector of $
\lambda $, then $Y=\begin{pmatrix}
y \\
py'
\end{pmatrix} $ satisfies $Y(b)=\Phi (b,a,\lambda )Y(a)$ and consequently
$[A+B\Phi (b,a,\lambda )]Y(a)=0$. Since $Y(a)=0$ would imply that $y$ is
the trivial solution in contradiction to it being an eigenfunction, we have
that $\det [A+B\Phi (b,a,\lambda )]=0$. If \eqref{3.13} has two linearly
independent solutions for $C$, say $C_1,C_2$, then solve the IVP with
the initial conditions $Y(a)=C_1$, $Y(a)=C_2$ to obtain solutions $Y_1$,
 $Y_2$. Then $Y_1,Y_2$ are linearly independent vector
solutions of \eqref{3.8} and their top components $y_1,y_2$ are linearly
independent solutions of \eqref{3.1}. Conversely, if $y_{1},y_2$ are
linearly dependent solutions of \eqref{3.1} we can reverse the steps above
to obtain two linearly independent vector solutions of the algebraic system
\eqref{3.13}.
\end{proof}

It is convenient to classify the boundary conditions (BC) \eqref{3.3},
\eqref{3.5} into two mutually exclusive classes: separated and coupled. Note
that, since the BC are homogeneous, multiplication on the left by a nonzero
constant or a nonsingular matrix leads to equivalent boundary conditions.

\begin{lemma}[Separated boundary conditions]\label{l34}
Assume
\[
A=\begin{pmatrix}
A_1 & A_2 \\
0 & 0
\end{pmatrix},
B=\begin{pmatrix}
0 & 0 \\
B_1 & B_2
\end{pmatrix}.
\]
Then for $\lambda \in \mathbb{C}$,
\[
\delta (\lambda )=-A_{2\,}B_1\phi _{11}(b,a,\lambda )-A_2B_2\phi
_{21}(b,a,\lambda )+A_1B_1\phi _{12}(b,a,\lambda )+A_1B_2\phi
_{22}(b,a,\lambda ).
\]
\end{lemma}

The proof of the above lemma follows from the definition of $\delta$
and a direct computation.
The characterization of the eigenvalues as zeros of an entire function given
by Lemma \ref{l33} reduces to a simpler and more informative form when the
boundary conditions are self-adjoint and coupled. This reduction is given by
the next lemma.

\begin{theorem}\label{t31}
Let \eqref{3.1} to \eqref{3.8} hold and fix \eqref{3.4} and
$P,W,J $. Define $\Phi =(\phi _{ij})$ as above and suppose that
\begin{equation}
B=-I,\quad A=e^{i\gamma }\,K,\quad 0\leq \gamma \leq \pi ,\quad
K\in M_2(\mathbb{R)},\quad \det K=1.  \label{3.16}
\end{equation}
\end{theorem}

Let $K=(k_{ij})$ and define
\begin{equation}
D(\lambda ,K)=k_{11}\,\phi _{22}(b,a,\lambda )-k_{12}\,\phi
_{21}(b,a,\lambda )-k_{21}\,\phi _{12}(b,a,\lambda )+k_{22}\,\phi
_{11}(b,a,\lambda ),  \label{3.17}
\end{equation}
for $\lambda \in \mathbb{C}$.
Then
\begin{enumerate}
\item The complex number $\lambda $ is an eigenvalue of BVP \eqref{3.1} to
\eqref{3.6} if and only if
\begin{equation}
D(\lambda ,K)=2\cos \gamma ,\quad 0\leq \gamma \leq \pi .  \label{3.18}
\end{equation}

\item If $\lambda $ is an eigenvalue for $A=$ $e^{i\gamma }K$,
$B=-I$, $0<\gamma <\pi $, with eigenfunction $u$, then $\lambda $ is also an
eigenvalue for $A=e^{-i\gamma }K$, $B=-I$, but with eigenfunction
$\overline{u}$.
\end{enumerate}

\begin{proof}[Proof of Theorem \ref{t31}]
From the basic theory of linear ordinary differential equations, see
\cite{zett05}, we have $\det \Phi (b,a,\lambda )=1$.
We abbreviate $(\phi_{ij}(b,a,\lambda ))$ to $\phi _{ij}$ and
$D(\lambda ,K)$ to $D$ for simplicity of exposition.
By \eqref{3.12} and \eqref{3.16} and recalling that $\det K=1$ we get
\begin{equation}
\begin{aligned}
\delta (\lambda )
& =\det (e^{i\gamma }\,K-\Phi )
=\begin{pmatrix}
e^{i\gamma }k_{11}-\phi _{11} & e^{i\gamma }k_{12}-\phi _{12} \\
e^{i\gamma }k_{21}-\phi _{21} & e^{i\gamma }k_{22}-\phi _{22}
\end{pmatrix}
 \\
& =(e^{i\gamma }k_{11}-\phi _{11})(e^{i\gamma }k_{22}-\phi
_{22})-(e^{i\gamma }k_{12}-\phi _{12})(e^{i\gamma }k_{21}-\phi _{21})
\\
& =e^{2i\gamma }(k_{11}k_{22}-k_{12}k_{21})-e^{i\gamma }D+\det \Phi .
\end{aligned}\label{3.19}
\end{equation}
By Lemma \ref{l33}, $\lambda $ is an eigenvalue if and only if $\delta
(\lambda )=0$. Therefore $\lambda $ is an eigenvalue if and only if
\begin{align*}
D(\lambda ) &=(1+e^{2i\gamma })/e^{i\gamma }=e^{-i\gamma }+e^{i\gamma } \\
&=\cos (-\gamma )+i\sin (-\lambda )+\cos (\gamma )+i\sin (\gamma ) \\
&=2\cos(\gamma ).
\end{align*}
This proves part (1). Part (2) follows from \eqref{3.19} and by taking
conjugates of equation \eqref{3.1}.
\end{proof}

\begin{corollary}\label{c21}
Let the hypotheses and notation of Theorem \ref{t31} hold.
If $\lambda $ is any eigenvalue and $D(\lambda ,K)$ is given by \eqref{3.17}
then
\begin{equation}
-2\leq D(\lambda ,K)\leq 2.  \label{3.20}
\end{equation}
\end{corollary}

The above corollary follows directly from \eqref{3.18}.

\begin{corollary}\label{c22}
Let the hypotheses and notation of Theorem \ref{t31} and let $I$
denote the identity matrix. Then
\begin{enumerate}
\item A complex number $\lambda $ is an eigenvalue of the periodic boundary
condition
\[
Y(b)=Y(a)
\]
if and only if $D(\lambda ,I)=2$.

\item A complex number $\lambda $ is an eigenvalue of the semi-periodic
boundary condition
\[
Y(b)=-Y(a)
\]
if and only if $D(\lambda ,-I)=-2$.

\item A complex number $\lambda $ is an eigenvalue of the complex
self-adjoint boundary condition
\[
Y(b)=e^{i\gamma }\,Y(a),\;0<\gamma <\pi
\]
if and only if $D(\lambda ,I)=2\cos (\gamma )$.
\end{enumerate}
\end{corollary}

The above corollary follows directly from \eqref{3.18}.
Next we comment on the remarkable characterization \eqref{3.18}.

\begin{remark}\label{r26} \rm
Note that in \eqref{3.18} $D(\lambda ,K)$ on the left is defined
for any on $K\in SL_2(\mathbb{R})$ and the right side depends only on
$\gamma \in [0,\pi]$. Recall the canonical form of the coupled
boundary conditions with $A$, $B$ given by \eqref{3.16}. When $\gamma =0$,
$D(\lambda ,K)=2$ characterizes the eigenvalues when $A=K$;
 when $\gamma =\pi $, $D(\lambda ,K)=-2$ characterizes the eigenvalues when
$A=-K$; when $\gamma \in (0,\pi )$ we have the complex coupled boundary condition:
$A=e^{i\gamma }\,K$. Thus the characterization $D(\lambda ,K)=2\,\cos \gamma $
suggests a close relationship between the eigenvalues of the complex coupled
condition with $A=e^{i\gamma }K$ and the eigenvalues of the two real
coupled conditions with $A=K$ and $A=-K$. Below we explore this relationship
in some detail for the special case when $K=I$, the identity matrix. Another
project we plan to pursue is to study this relationship for other
$K\in SL_2( \mathbb{R})$ using the special features of this well known special
linear group of order $2$ over the reals, i.e. $SL_2( \mathbb{R})$.
\end{remark}

\section{Canonical forms of self-adjoint boundary conditions}

The boundary condition \eqref{0.4}, \eqref{0.6} is homogeneous and thus
clearly invariant under multiplication by a nonsingular matrix or nonzero
constant. This is a serious obstacle to studying the dependence of the
eigenvalues on this condition. The conditions \eqref{0.4}, \eqref{0.6} can
be divided into three mutually exclusive classes: separated, real coupled
and complex coupled. We refer to all nonseparated conditions as coupled.
These three classes are:

\subsection*{Separated self-adjoint BC} These are
\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),\\
B_1y(b)+B_2(py')(b)=0,\quad B_1,B_2\in \mathbb{R},\; (B_1,B_2)\neq (0,0).
\end{gather*}
These separated conditions can be parameterized as follows:
\begin{gather}
\cos \alpha y(a)-\sin \alpha (py')(a)=0,\quad  0\leq\alpha <\pi ,  \label{4.3} \\
\cos \beta  y(b)-\sin \beta (py')(b)=0,\quad  0<\beta \leq\pi ,  \label{4.4}
\end{gather}
choose $\alpha \in [ 0,\pi )$ such that
\[
\tan \alpha =
\frac{-A_2}{A_1} \text{ if } A_1\neq 0,\quad
\text{and}\quad
\alpha =\pi /2 \text{ if }A_1=0,
\]
similarly, choose $\beta \in (0,\pi ]$ such that
\[
\tan \beta =\frac{-B_2}{B_1} \text{ if } B_1\neq 0,\quad
\text{and}\quad \beta =\pi /2\text{ if }B_1=0.
\]

Note the different normalization in \eqref{4.4} for $\beta $ than that used
for $\alpha $ in \eqref{4.3}. This is for convenience in using the
Pr\"{u}fer transformation which is widely used for the theoretical studies of
eigenvalues and their eigenfunction and for the numerical computation of
these. For example the FORTRAN code SLEIGN2
\cite{baez01,baze12,bgkz91,bgkz91a,bewz93} uses this normalization.

\subsection*{All real coupled self-adjoint BC} These can be
formulated as follows:
\[
Y(b)=K\,Y(a),\quad Y
=\begin{pmatrix}
y \\
(py')
\end{pmatrix},
\]
where $K\in SL_2(\mathbb{R})$, i.e. $K$ satisfies
\begin{equation}
K=\begin{pmatrix}
k_{11} & k_{12} \\
k_{21} & k_{22}
\end{pmatrix}, \quad
 k_{ij}\in \mathbb{R},\quad  \det K=1.  \label{4.8}
\end{equation}

\subsection*{All complex coupled self-adjoint BC} These are:
\[
Y(b)=e^{i\gamma } K Y(a),
\]
where $K$ satisfies \eqref{4.8} and $-\pi <\gamma <0$, or $0<\gamma <\pi $.


\begin{lemma} \label{l41}
Given a boundary condition \eqref{0.4}, \eqref{0.6} it is
equivalent to exactly one of the separated, real coupled, or complex coupled
boundary conditions defined above and each of these conditions can be
written in the form \eqref{0.4}, \eqref{0.6}.
\end{lemma}

For a proof of the above lemma, see \cite{zett05}.


\begin{notation} \label{n41} \rm
For fixed coefficients $p,q,w$, fixed endpoints $a,b$ and a fixed
jump condition \eqref{0.5} we use the following notation for the eigenvalues
of the boundary conditions \eqref{0.4}, \eqref{0.6}:
\begin{equation}
\lambda _n(\alpha ,\beta ),\quad \lambda _n(K),\quad
\lambda _n(\gamma,K),\quad n\in \mathbb{N}_{0}.  \label{4.10}
\end{equation}
Here and below $\mathbb{N}_{0}=\{0,1,2,3,\cdots \}$. Note that
$\lambda _n$ is uniquely defined although its eigenfunction may not be
unique and this notation covers all self-adjoint boundary conditions
\eqref{0.4}, \eqref{0.6}. Since each of these has a unique representation as a
separated, real coupled, or complex coupled condition we can study how the
eigenvalues change when this boundary condition changes. The existence of
eigenvalues is discussed in the next section.
\end{notation}

\section{Existence of eigenvalues}

\begin{theorem}\label{t41}
Let \eqref{0.1} to \eqref{0.6} hold and let $S$ satisfy \eqref{0.7}.
Then the spectrum of $S$ is real, discrete, bounded below and not bounded
above. We have
\begin{enumerate}
\item There are an infinite but countable number of eigenvalues with no
finite accumulation point.

\item The eigenvalues can be ordered to satisfy
\begin{equation}
-\infty <\lambda _{0}\leq \lambda _1\leq \lambda _2\leq \ldots ;\ \
\lambda _n\to +\infty ,\quad \text{as } n\to \infty .
\label{4.11}
\end{equation}
Each eigenvalue may be simple or double but there cannot be two consecutive
equalities in \eqref{4.11} since, as pointed out in Theorem \ref{t30}, for
any value of $\lambda $, the equation \eqref{0.1} has exactly two linearly
independent extended solutions. Note that $\lambda _n$ is well defined for
each $n\in \mathbb{N}_{0}$ but there is some arbitrariness in the indexing
of the eigenfunctions corresponding to a double eigenvalue since every
nontrivial extended solution of the equation for such an eigenvalue is an
eigenfunction. Let $\sigma (S)=\{\lambda _n:n\in \mathbb{N}_{0}\}$ where
the eigenvlaues are ordered to satisfy \eqref{4.11}.

\item If the boundary condition is separated then strict inequality holds
everywhere in \eqref{4.11}. Furthermore, if $u_n$ is an eigenfunction of $
\lambda _n$, then $u_n$ is unique up to constant multiples and has
exactly $n$ zeros in the open interval $(a,b)$ for each $n\in \mathbb{N}_{0}$.

\item Let $S$ be determined by a real coupled boundary condition matrix $K$
and $u_n$ be a real-valued eigenfunction of $\lambda _n(K)$. Then the
number of zeros of $u_n$ in the open interval $(a,b)$ is $0$ or $1$, if $
n=0$, and $n-1$ or $n$ or $n+1$ if $n\geq 1$.

\item Let $S$ be determined by a complex coupled boundary condition $
(K,\gamma )$ and let $\sigma (S)=\{\lambda _n:n\in \mathbb{N}_{0}\}$. Then
all eigenvalues are simple and strict inequality holds everywhere in
\eqref{4.11}. Moreover, if $u_n$ is an eigenfunction of $\lambda _n$ then the
number of zeros of $\operatorname{Re}u_n$ on $[a,b)$ is $0$ or $1$ if $n=0$, and
$n-1$ or $n$ or $n+1$ if $n\geq 1$. The same conclusion holds for
$\operatorname{Im}u_n$. Moreover, $u_n$ has no zero in $[a,b]$,
$n\in \mathbb{N}_0$.
\end{enumerate}
\end{theorem}

See \cite{zett05} for a proof or a reference to a proof and note that these
proofs can be generalized to the boundary conditions used here.


\begin{remark}\label{r41} \rm
Note that Theorem \ref{t41} justifies the notation \ref{n41}.
Thus for each $S$ satisfy \eqref{0.7} we have that the spectrum $\sigma (S)$
of $S$ is given by
\begin{enumerate}
\item $\sigma (S)=\{\lambda _n(\alpha ,\beta )$, $n\in \mathbb{N}_{0}\}$
if the boundary condition of $S$ is separated and determined by the
parameters $\alpha ,\beta ;$

\item $\sigma (S)=\{\lambda _n(K)$, $n\in \mathbb{N}_{0}\}$ if the
boundary condition of $S$ is real coupled with coupling constant $K;$

\item $\sigma (S)=\{\,\lambda _n(\gamma ,K)$, $n\in \mathbb{N}_{0}\}$ if
the boundary condition of $S$ is complex coupled with coupling constants $
K,\gamma $.
\end{enumerate}
\end{remark}

\begin{remark} \label{r42} \rm
It is the canonical forms of the boundary conditions which make
it possible to introduce the notation of Remark \ref{r41}. This notation
identifies $\lambda _n$ uniquely and makes it possible to study the
dependence of the eigenvalues on the boundary conditions and on the
equations. No comparable canonical representation of all self-adjoint
boundary conditions is known for higher order ordinary differential
equations. There are some recent results \cite{hasz12,hasz12a} but
these are much more complicated and thus more difficult to use for the study
of the dependence of the eigenvalues on the problem. But note that the jump
condition \eqref{0.5} determined by $C$ at the point $c$ remains fixed as $A
$ and $B$ vary.
\end{remark}

\section{Eigenvalue inequalities}

In this section we give a complete description of how, for a fixed equation
and fixed matrix $C$, the eigenvalues change as the boundary conditions
\eqref{0.4} determined the matrices $A,B$ vary. Since the Dirichlet and Neumann
boundary conditions play a special role we introduce the notation
\begin{equation}
\lambda _n^{D}=\lambda _n(0,\pi ),\quad
 \lambda _n^{N}=\lambda_n(\pi /2,\pi /2),\quad
 n\in \mathbb{N}_{0}.  \label{6.1}
\end{equation}

\begin{theorem}\label{t61}
Let \eqref{0.1} to \eqref{0.6} hold, let $S$ satisfy \eqref{0.7}
and let $\lambda _n^{D}$ be defined by \eqref{6.1}. Then for all $(A,B)$
satisfying \eqref{0.4} we have
\begin{enumerate}
\item
\begin{equation}
\lambda _n(A,B)\leq \lambda _n^{D},\quad n\in \mathbb{N}_{0}.
\label{6.2}
\end{equation}
Equality can hold in \eqref{6.2} for non Dirichlet eigenvalues.

\item For all $(A,B)$ satisfying \eqref{0.4} we have
\[
\lambda _n^{D}\leq \lambda _{n+2}(A,B),\quad n\in \mathbb{N}_{0}.
\]

\item The range of $\lambda _{0}(A,B)$ is $(-\infty ,\lambda _{0}^{D}]$.

\item The range of $\lambda _1(A,B)$ is $(-\infty ,\lambda _{0}^{D}]$.

\item The range of $\lambda _n(A,B)$ is $(\lambda _{n-2}^{D},\lambda
_n^{D}]$ for $n\geq 2$.

Moreover, (3), (4), (5) still hold when $A,B$ are restricted to be real.
\end{enumerate}
\end{theorem}

For a proof of the above theorem, see \cite{zett05}.
Next we investigate how the eigenvalues change when the boundary conditions
change more closely.

According to a well-known classical result (see \cite{east73} and
\cite{cole55} for the case of smooth coefficients and \cite{weid87} for the
general case), we have the following inequalities for $K=I$, the identity
matrix:
\begin{equation}
\begin{aligned}
\lambda _{0}^{N}
& \leq \lambda _{0}(I)<\lambda _{0}(e^{i\gamma }I)<\lambda_{0}(-I)
 \leq \{\lambda _{0}^{D},\lambda _1^{N}\} \\
& \leq \lambda _1(-I)<\lambda _1(e^{i\gamma }I)<\lambda _1(I)\leq
\{\lambda _1^{D},\lambda _2^{N}\} \\
& \leq \lambda _2(I)<\lambda _2(e^{i\gamma }I)<\lambda _2(-I)\leq
\{\lambda _2^{D},\lambda _{3}^{N}\} \\
& \leq \lambda _{3}(-I)<\lambda _{3}(e^{i\gamma }I)<\lambda _{3}(I)\leq
\{\lambda _{3}^{D},\lambda _{4}^{N}\}\leq \dots ,
\end{aligned} \label{6.6}
\end{equation}
where $\gamma \in (-\pi ,\pi )$ and $\gamma \neq 0$. In \eqref{6.6} notation
$\{\lambda _n^{D},\lambda _{n+1}^{N}\}$ means either of $\lambda _n^{D}$
and $\lambda _{n+1}^{N}$ and there is no comparison made between these two.
These inequalities are well known in Flochet theory.

Eastham, Kong, Wu and Zettl \cite{ekwz99} extended these inequalities to
general $K\in {{SL}_2}(\mathbb{R})$. A key feature of this extension is the
identification of separated boundary conditions which play the role of the
Dirichlet and Neumann conditions in \eqref{6.6}. These are given next.

For $K\in SL_2(\mathbb{R})$, $K=\begin{pmatrix}
k_{11} & k_{12} \\
k_{21} & k_{22}
\end{pmatrix}$, denote by $\mu _n=\mu _n(K)$ and $\nu _n=\nu _n(K)$,
$n\in \mathbb{N}_{0}$, the eigenvalues for the separated boundary conditions
\begin{gather}
y(a)=0,\quad k_{22}y(b)-k_{12}y^{[1]}(b)=0;  \label{6.7} \\
y^{[1]}(a)=0,\quad k_{21}y(b)-k_{11}y^{[1]}(b)=0;  \label{6.8}
\end{gather}
respectively. Note that $(k_{22},k_{12})\neq (0,0)\neq (k_{21},k_{11})$
since $\det K=1$. Therefore each of these is a self-adjoint separated
boundary condition with a countably infinite number of only real eigenvalues.

\begin{theorem}\label{t63}
Let \eqref{0.1} to \eqref{0.7} hold. Let $\mu _n$ and $\nu_n$,
 $n\in \mathbb{N}_{0}$ be the eigenvalues for \eqref{6.7}, and
\eqref{6.8}, respectively.  Then we have
\begin{itemize}
\item Suppose that $\;k_{12}<0\;and\;$ $k_{11}\leq0$. Then
\end{itemize}
\begin{enumerate}
\item $\lambda_{0}(K)$ is simple;

\item $\lambda _{0}(K)$ $<\lambda _{0}(-K)$;

\item The following inequalities hold for $-\pi <\gamma $ $<0$ and $0<\gamma
<\pi$:
\begin{align*}
-\infty & <\lambda _{0}(K)<\lambda _{0}(\gamma ,K)<\lambda _{0}(-K)\leq
\{\mu _{0},\nu _{0}\}   \\
& \leq \lambda _1(-K)<\lambda _1(\gamma ,K)<\lambda _1(K)\leq \{\mu
_1,\nu _1\}   \\
& \leq \lambda _2(K)<\lambda _2(\gamma ,K)<\lambda _2(-K)\leq \{\mu
_2,\nu _2\}   \\
& \leq \lambda _{3}(-K)<\lambda _{3}(\gamma ,K)<\lambda _{3}(-K)\leq \{\mu
_{3},\nu _{3}\}\leq \ldots
\end{align*}
\end{enumerate}

\begin{itemize}
\item  Suppose that $k_{12}\leq0$ and  $k_{11}>0$. Then
\end{itemize}

\begin{enumerate}
\item $\lambda_{0}(K)$ is simple;

\item $\lambda _{0}(K)$ $<\lambda _{0}(-K)$

\item The following inequalities hold for $-\pi <\gamma $ $<0$ and $0<\gamma
<\pi $:
\begin{align*}
\nu _{0}& \leq \lambda _{0}(K)<\lambda _{0}(\gamma ,K)<\lambda _{0}(-K)\leq
\{\mu _{0},\nu _1\}   \\
\quad \quad & <\lambda _1(-K)<\lambda _1(\gamma ,K)<\lambda _1(K)\leq
\{\mu _1,\nu _2\}   \\
& \leq \lambda _2(K)<\lambda _2(\gamma ,K)<\lambda _2(-K)\leq \{\mu
_2,\nu _{3}\}   \\
& \leq \lambda _{3}(-K)<\lambda _{3}(\gamma ,K)<\lambda _{3}(K)\leq \{\mu
_{3},\nu _{4}\}\leq \ldots
\end{align*}
\end{enumerate}

\begin{itemize}
\item Furthermore,  for $0<$ $\alpha<\beta<\pi$ we have
\begin{align*}
\lambda_{0}(\beta,K)
& <\lambda_{0}(\alpha,K)<\lambda_1(\alpha ,K)<\lambda_1(\beta,K)
 <\lambda_2(\beta,K)<\lambda_2(\alpha,K) \\
& <\lambda_{3}(\alpha,K)<\lambda_{3}(\beta,K)<\ldots
\end{align*}

\item If neither of the above cases holds for $K$ then one of them must hold
for $-K$. The notation $\{\mu_n,\nu_{m}\}$ is used to indicate either $
\upsilon_n$ or $\nu_{m}$ but no comparison is made between $\mu_n$ and
$\nu_{m}$.
\end{itemize}
\end{theorem}

\begin{proof}
For $K$ a diagonal matrix these inequalities were established in Weidmann
\cite{weid87}. The general result is proven in Eastham, Kong, Wu and Zettl
\cite{ekwz99}.
\end{proof}

Next we mention some interesting consequences of Theorem \ref{t63}.

\begin{remark} \label{r62} \rm
For separated boundary conditions the Pr\"{u}fer transformation
is a powerful tool for proving the existence of eigenvalues, studying their
properties and computing them numerically. There is no comparable tool for
coupled conditions. For coupled conditions the standard existence proof for
the eigenvalues is based on operator theory in Hilbert space; the Green's
function is constructed and used as a kernel in the definition of an
integral operator whose eigenvalues are those of the problem or their
reciprocals, see Coddington and Levinson \cite{cole55} or Weidmann
\cite{weid87}.

A proof based on Theorem \ref{t63} was given in \cite{ekwz99} and goes as
follows:
Starting with the eigenvalues $\mu _n$ and $\nu _n$,
 $n\in \mathbb{N}_{0}$, of the separated BC \eqref{6.7}, \eqref{6.8} the proof
of \cite[Theorem 4.8.1]{ekwz99} (although this is not explicitly pointed
out there) actually shows that there is one and only one eigenvalue of the
coupled condition determined by $K$ in the interval $(-\infty ,\mu _{0}]$
and it is $\lambda _{0}(\gamma ,K);$ there is exactly one eigenvalue in the
interval $[\mu _{0},\mu _1]$ and it is $\lambda _1(\gamma ,K)$; there is
exactly one eigenvalue in the interval $[\mu _n,\mu _{n+1}]$ and it is
$\lambda _{n+1}(\gamma ,K)$, for $n\in \mathbb{N}_{0}$. This not only proves
the existence of the eigenvalues of $K$ but can be used to construct an
algorithm to compute them. Such an algorithm is used by SLEIGN2, see
\cite{baez01}, see also \cite{bgkz91,bgkz91a}. This is the first
existence proof for coupled eigenvalues which does not use the self-adjoint
operator in Hilbert space and thus can be described as the first
`elementary' existence proof.
\end{remark}

\begin{remark} \label{r63} \rm
By Theorem \ref{t63} for \textit{any} $K\in {{SL}_2}(\mathbb{R})$
either $\lambda _{0}(K)$ or $\lambda _{0}(-K)$ is simple. This extends the
classical result that the lowest periodic eigenvalue is simple, to the
general case of arbitrary coupled self-adjoint BC's. Here simple refers to
both the algebraic and geometric multiplicities, since these are equal.
\end{remark}

\begin{theorem} \label{t64}
Let \eqref{0.1} to \eqref{0.7} hold. Let $\mu _n$ and
$\nu_n$, $n\in \mathbb{N}_{0}$ be the eigenvalues for \eqref{6.7}, and
\eqref{6.8}, respectively. Then
\begin{enumerate}
\item An eigenvalue $\lambda _n(K)$ is double if and only if there exist
$k,m\in \mathbb{N}_{0}$ such that
\[
\lambda _n(K)=\mu _{k}=\nu _{m};
\]

\item Given eigenvalues $\lambda _n(K)$ and $\lambda _{n+1}(K)$ of $K$,
distinct or not, there exist eigenvalues $\upsilon _{k},\nu _{m}$ of the
separated boundary conditions \eqref{6.7}, \eqref{6.8} such that
\[
\lambda _n(K)\leq \{\mu _{k},\nu _{m}\}\leq \lambda _{n+1}(K).
\]
\end{enumerate}
\end{theorem}

For a proof of the above theorem, see
\cite[Theorem 4.3 and Corollary 4.2]{kowz99}.

\section{Continuity of eigenvalues}

In this section we study the continuity of the eigenvalues as functions of
each parameter of the problem. We extend the notation \eqref{4.10} for the
eigenvalues to include the coefficients and the endpoints
\begin{equation}
\lambda _n(a,b,\alpha ,\beta ,p,q,w),\quad
\lambda_n(a,b,K,p,q,w),\quad  \lambda _n(a,b,\gamma ,K,p,q,w),\quad
 n\in \mathbb{N}_{0}.  \label{5.0}
\end{equation}

When we study the dependence on one parameter $x$ with the others fixed we
abbreviate the notation to $\lambda _n(x);$ thus $\lambda _n(q)$
indicates that we are studying $\lambda _n$ as a function of
$q\in L (J, \mathbb{R})$ with all other parameters of the problem fixed,
$\lambda _n(a) $ indicates that we are studying $\lambda _n$ as a function of the
left endpoint with all other parameters fixed, etc. Since $C$ is fixed in
our results below we do not include it in the notation \eqref{5.0}.

The eigenvalues are continuous functions of each of $\frac{1}{p},q,w,a,b$;
they are not continuous functions of the boundary conditions, in general.
The continuity on the coefficients $\frac{1}{p},q,w$ is with respect to the
$L(J,\mathbb{R})$ norm; the continuity on $K$ is with respect to any matrix
norm and the continuity with respect to $a,b,\alpha ,\beta ,\gamma $ is in
the reals $\mathbb{R}$. We will see below that even though, in general,
$\lambda _n$ is not a continuous function of the boundary conditions for
fixed $n$, it can always be embedded in a ``continuous branch''
 of eigenvalues by varying the index $n$. For
separated boundary conditions there is a jump discontinuity when either $
y(a)=0$ or $y(b)=0$. We also characterize the coupled boundary conditions at
which the eigenvalues are not continuous and show that all discontinuities
are finite or infinite jumps. The set of boundary conditions at which the
eigenvalues have discontinuities we call ``the jump set''
 since all discontinuities are of the jump type.

We start with the continuous dependence on the coefficients and the
endpoints. For all results in this section $C$ is fixed.

\begin{theorem}\label{t51}
Let \eqref{0.1} to \eqref{0.7} hold and let $n\in \mathbb{N}_{0}$.
Then
\begin{enumerate}
\item $\lambda _n(1/p)$ is a continuous function of $1/p\in L (J,\mathbb{R})$;

\item $\lambda _n(q)$ is a continuous function of $q\in L (J,\mathbb{R})$;

\item $\lambda _n(w)$ is a continuous function of $w\in L (J,\mathbb{R})$;

\item $\lambda _n(a)$ is a continuous function of $a$.

\item $\lambda _n(b)$ is a continuous function of $b$.
\end{enumerate}
\end{theorem}

For a proof of the above theorem, see
 Kong, Wu and Zettl \cite[Section 2]{kowz99}.
Next we characterize the boundary conditions at which $\lambda _n$ is not
continuous, we call this set the ``jump''
set since all discontinuities are of jump type.

\begin{definition}[Jump set of boundary conditions] \label{d51}\rm
The jump set of boundary conditions $\mathbb{J}$
 is the union of
\begin{enumerate}
\item the (real and complex) coupled conditions
\[
Y(b)=e^{i\gamma }\,K\,Y(a),\quad Y=\begin{pmatrix}
y \\
(py')
\end{pmatrix}, \quad
-\pi <\gamma \leq \pi ,
\]
where the $2\times 2$ matrix $K=(k_{ij})\in SL_2( \mathbb{R})$ satisfies
$k_{12}=0$, and

\item the separated boundary conditions
\begin{equation} \label{5.2}
\begin{gathered}
A_1y(a)+A_2(py')(a) =0,\quad A_1,A_2\in \mathbb{R},\; (A_1,A_2)\neq (0,0)   \\
B_1y(b)+B_2(py')(b) =0,\quad B_1,B_2\in \mathbb{R},\;(B_1,B_2)\neq (0,0)
\end{gathered}
\end{equation}
satisfying $A_2B_2=0$. Note that these are precisely the conditions
where either $\alpha =0$ or $\beta =\pi $ or both $\alpha =0$ and
$\beta=\pi $.
\end{enumerate}
\end{definition}

\begin{theorem} \label{t52}
Let \eqref{0.1} to \eqref{0.7} hold and let $n\in \mathbb{N}_{0}$.
Let $\mathbb{J}$ be given by Definition\ref{d51}. Then
\begin{enumerate}
\item If the boundary condition is not on the jump set $\mathbb{J}$, then
$\lambda _n$ is a continuous function of the boundary condition.

\item If $n\in \mathbb{N}=\{1,2,3,\cdot \cdot \cdot \}$, $k_{12}=0$ and
$\lambda _n=\lambda _{n-1}$, then $\lambda _n$ is continuous at $K$.

\item The lowest eigenvalue $\lambda _{0}$ has an infinite jump
discontinuity at each separated or (real or complex) coupled boundary
condition in $\mathbb{J}$.

\item Let $n\in \mathbb{N}$. If the boundary condition is in $\mathbb{J}$ \
and $\lambda _n$ is simple, then $\lambda _n$ has a finite jump
discontinuity at this boundary condition.
\end{enumerate}
\end{theorem}

For a proof of the above theorem, see \cite[Section 3]{kowz99}.
For the important special case of separated boundary conditions in canonical
form \eqref{4.3}, \eqref{4.4} there is a stronger result.

\begin{lemma}\label{l51}
For any $n\in \mathbb{N}_{0}$, $\lambda _n(\alpha ,\beta )$ is
jointly continuous on $[0,\alpha )\times (0,\pi ]$ and strictly decreasing
in $\alpha $ for each fixed $\beta $ and strictly increasing in $\beta $ for
each fixed $\alpha $.
\end{lemma}

The proof of the above lemma can be found in \cite{kowz99}.
The next theorem gives more detailed information about separated boundary
conditions \eqref{5.2} not in canonical form, in particular for the
separated jump boundary conditions.

\begin{theorem}[Everitt-M\"{o}ller-Zettl] \label{t53}
Fix $a,b,p,q,w$ and consider the conditions \eqref{5.2}.
\begin{itemize}
\item Fix $B_1,B_2$ and let $A_1=1$. Consider
$\lambda_n=\lambda_n(A_2)$ as a function of $A_2\in\mathbb{R}$.
Then for each $n\in \mathbb{N}_{0}$, $\lambda_n(A_2)$ is continuous at $A_2$
for $A_2>0$ and $A_2<0$ but has a jump discontinuity at $A_2=0$. More precisely we
have
\begin{enumerate}
\item $\lambda_n(A_2)\to\lambda_n(0)$ as $A_2\to0^{-}$, $n\in \mathbb{N}_{0}$.

\item $\lambda_{0}(A_2)\to-\infty$ as $A_2\to0^{+}$.

\item $\lambda_{n+1}(A_2)\to\lambda_n(0)$ as $A_2\to0^{+}$.
\end{enumerate}

\item Fix $A_1,A_2$ and let $B_1=1$. Consider
$\lambda_n=\lambda _n(B_2)$ as a function of $B_2\in\mathbb{R}$.
Then for each $n\in \mathbb{N}_{0}$, $\lambda_n(B_2)$ is continuous at $B_2$
for $B_2>0$ and $B_2<0$ but has a jump discontinuity at $B_2=0$.
More precisely we have:

\begin{enumerate}
\item $\lambda_n(B_2)\to\lambda_n(0)$ as $B_2\to 0^{+}$, $n\in \mathbb{N}_{0}$.

\item $\lambda_{0}(B_2)\to-\infty$ as $B_2\to0^{-}$.

\item $\lambda_{n+1}(B_2)\to\lambda_n(0)$ as $B_2\to 0^{-}$.
\end{enumerate}
\end{itemize}
\end{theorem}

For a proof of the above theorem, see Everitt, M\"{o}ller and Zettl
\cite{evmz97,evmz99,evmz95}.


\begin{remark}\label{r51} \rm
Note that $\lambda _{0}(A_2)$ has an infinite jump
discontinuity at $A_2=0$, but for all $n\geq 1$, $\lambda _n(A_2)$ has
a finite jump discontinuity at $A_2=0$, $\lambda _n(A_2)$ is left but
not right continuous at $0$. Similarly, $\lambda _{0}(B_2)$ has an
infinite jump discontinuity at $B_2=0$, but for all $n\geq 1$,
$\lambda _n(B_2)$ has a finite jump discontinuity at $B_2=0$
$\lambda_n(B_2)$ is right but not left continuous at $0$.
In all cases $\lambda_n(0)$ is embedded in a continuous branch of
eigenvalues as $A_2$ or $ B_2$ passes through zero but this branch
is not given by a fixed index $n$; in order to preserve continuity
the index ``jumps'' from $n$ to $n+1$ as $A_2$ or $B_2$ pass
through zero from the appropriate direction.
\end{remark}

\begin{remark} \label{r52} \rm
This forced ``index jumping'' in
order to stay on a continuous branch of eigenvalues plays an important role
in some of the algorithms and their numerical implementations used in the
code SLEIGN2 \cite{baez01} for the numerical approximation of the spectrum
of regular and singular SLP.
\end{remark}

\begin{remark}\label{r53} \rm
This ``index jumping'' phenomenon in order to stay on a
``continuous eigenvalue branch'' is quite general:
It applies to all simple eigenvalues for all boundary conditions on the
jump set $\mathbb{J}$, separated, real coupled, or complex coupled.
For details the reader is referred to
\cite[Theorems 3.39, 3.73, 3.76, Propositions 3.71, 3.72]{kowz99}.
\end{remark}

\begin{remark} \label{r54} \rm
Kong and Zettl \cite{koze96a} have shown that each continuous
eigenvalue branch is in fact differentiable everywhere including the point
$A_2=0$ (or $B_2=0$) where the index jumps. This also follows from
M\"{o}ller and Zettl \cite{moze96}.
\end{remark}

\begin{remark} \label{r55} \rm
Remarkably, if the boundary condition is in $\mathbb{J}$ and
$\lambda _n$ is simple then it can be embedded in a continuous eigenvalue
branch and this branch is differentiable. M\"{o}ller-Zettl \cite{moze96}
extended this result to abstract operators in Banach space.
\end{remark}

\section{Differentiability of eigenvalues}

Now that the continuities of $\lambda _n$ have been characterized it is
natural to investigate the differentiability of $\lambda _n$ as a function
of the parameters of the problem. This we embark upon next.
Here for each $n\in \mathbb{N}_{0}$, $u_n$ denotes a normalized eigenfunction
of $\lambda_n$. For all cases except when $\gamma \neq 0$ we choose $u_n$ to be
real valued. Again $C$ is fixed in this section.

\begin{theorem}\label{t54}
Let \eqref{0.1} to \eqref{0.7} hold. Let $n\in \mathbb{N}_{0}$.
\begin{enumerate}
\item Assume that $p,q,w$ are continuous at $a$ and $p(a)\neq 0$, then $
\lambda _n(a)$ is differentiable at $a$ and
\[
\lambda _n'(a)=\frac{1}{p(a)}|pu_n'|^2(a)-|u_n|^2(a)[q(a)-\lambda _n(a)w(a)].
\]

\item Assume that $p,q,w$ are continuous at $b$ and $p(b)\neq 0$, then $
\lambda _n(b)$ is differentiable at $b$ and
\[
\lambda _n'(b)=-\frac{1}{p(b)}|pu_n'|^2(b)+|u_n|^2(b)\,[q(b)-\lambda _n(b)w(b)].
\]

\item Let $-\pi <\gamma <0$ or $0<\gamma <\pi $. Then $\lambda _n(\gamma )$
is differentiable at $\gamma $ and
\[
\lambda _n'(\gamma )=-2\,\operatorname{Im}[u_n(b)\;(pu_n')(b)],
\]
where $\operatorname{Im}[z]$ denotes the imaginary part of $z$.

\item Let $\alpha \in (0,\pi )$. Then $\lambda _n(\alpha )$ is
differentiable and its derivative is given by
\[
\mathbb{\lambda }_n'(\alpha )=-u^2(a)-(pu')^2(a).
\]

\item Let $\beta \in (0,\pi )$. Then $\lambda _n(\beta )$ is
differentiable and its derivative is given by
\[
\mathbb{\lambda }_n'(\beta )=u^2(b)+(pu')^2(b).
\]
\end{enumerate}
\end{theorem}

For a proof of the above theorem, see \cite{koze96a}.
Next we study the differentiability of the eigenvalues with respect to the
remaining parameters: $\frac{1}{p},q,w$ and $K$.

\begin{theorem}
Let \eqref{0.1} to \eqref{0.7} hold. Let $n\in \mathbb{N}_{0}$.
\begin{enumerate}
\item Assume that $\lambda _n(q)$ is a simple eigenvalue with real valued
normalized eigenfunction $u_n(\cdot ,q)$. Then $\lambda _n(\cdot ,q)$ is
differentiable in $L(J,\mathbb{R})$ and its Frechet derivative is given by
\begin{equation}
\lambda _n'(q)h=\int_{a\,}^{b\,\;}|u_n(\cdot,q)|^{2} h,\quad
 h\in L (J,\mathbb{R}).  \label{5.8}
\end{equation}

\item Assume that $\lambda _n(1/p)$ is a simple eigenvalue with real
valued normalized eigenfunction $u_n(\cdot ,\frac{1}{p})$.
Then $\lambda_n(\cdot ,1/p)$ is differentiable in
$L (J,\mathbb{R})$ and its Frechet
derivative is given by
\[
\lambda _n'(1/p)\,h=-\int_{a}^{b\,\,\,}|u_n^{[1]}(\cdot,1/p)|^{2}h,\quad
 h\in L (J,\mathbb{R}).
\]

\item Assume that $\lambda _n(w)$ is a simple eigenvalue with real valued
normalized eigenfunction $u_n(\cdot ,w)$. Then $\lambda _n(\cdot ,w)$ is
differentiable in $L (J,\mathbb{R})$ and its Frechet derivative is given by
\[
\lambda _n'(w)h=-\lambda _n(w)\int_{a}^{b}|u_n(\cdot,w)|^{2\,}h,\quad
h\in L (J,\mathbb{R}).
\]

\item Assume that $\lambda _n(K)$ is a simple eigenvalue with real valued
normalized eigenfunction $u_n(\cdot ,K)$. Then $\lambda _n(\cdot ,K)$ is
differentiable and its Frechet derivative is given by the bounded linear
transformation defined by
\[
\lambda _n'(K)\,H = [p\overline{u_n}'(b),-\overline{u}_n(b)]HK^{-1}
\begin{pmatrix}
u_n(b) \\
(pu_n')(b)
\end{pmatrix},\quad
 H\in M_{2,2}(\mathbb{C}).
\]
\end{enumerate}
\end{theorem}

For the proof of (1), (2), (3), see \cite{koze96a},
and for (4) see \cite{moze96}.

\section{Monotonicity of eigenvalues}

In this section we fix a boundary condition and study how the eigenvalues
change when coefficient changes monotonically.

\begin{theorem}\label{t71}
Let \eqref{0.1} to \eqref{0.7} hold, let $n\in \mathbb{N}_{0}$.
\begin{enumerate}
\item Fix $p,w$. Suppose $Q\in L([a,b],\mathbb{R}$ $)$ and assume that
$Q\geq q$ a.e. on $[a,b]$.
Then $\lambda _n(Q)\geq \lambda _n(q)$. If $Q>q$ on a subset of $[a,b]$
having positive Lebesgue measure, then $\lambda _n(Q)>\lambda _n(q)$.

\item Fix $q,w$. Suppose $1/P\in L([a,b],\mathbb{R})$ and $0<P\leq p$ a.e.
on $[a,b]$.
Then $\lambda _n(1/P)\geq \lambda _n(1/p);$ if $1/P<1/p$ on a subset of $
[a,b]$ having positive Lebesgue measure, then $\lambda _n(1/P)<\lambda
_n(1/p)$.

\item Fix $p,q$. Suppose $W\in L([a,b],\mathbb{R})$ and $W\geq w>0$ a.e. on $
[a,b]$.
Then $\lambda _n(W)\geq \lambda _n(w)$ if $\lambda _n(W)<0$ and $
\lambda _n(w)<0;$ but $\lambda _n(W)\leq \lambda _n(w)$ if $\lambda
_n(W)>0$ and $\lambda _n(w)>0$. Furthermore, if strict inequality holds
in the hypothesis on a set of positive Lebesgue measure, then strict
inequality holds in the conclusion.
\end{enumerate}
\end{theorem}

\begin{proof}
We give the proof for (1), the proofs of (2) and (3) are similar. Define a
function $f:\mathbb{R}$ $\to \mathbb{R}$ by
\[
f(t)=\lambda _n(s(t)),\;s(t)=q+t(Q-q),\quad t\in [ 0,1].
\]
Then $s(t)\in L ((a,b),\mathbb{R})$ for each $t\in [ 0,1]$. From the
chain rule in Banach space and formula \eqref{5.8} for $\lambda _n'(q)$ we have
\[
f'(t)=\lambda _n'((s(t))\,s'(t)=\int_{a}^{b}|u^2(r,s(t))|\,(Q(r)-q(r))\,dr\geq 0,\;t\in [ 0,1].
\]
Hence $f$ is nondecreasing on $[0,1]$ and $f(1)=\lambda _n(Q)\geq \lambda
_n(q)=f(0)$. The strict inequality part of the theorem also follows from
this argument.
\end{proof}

\subsection*{Acknowledgements}

The first author was supported by the China Postdoctoral Science Foundation
(project 2014M561336).

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\end{document}