\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \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 $$Y'=PY\quad \text{on } J \label{4.3}$$ 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 $$Y(t)=\Phi (t,u,P)C+\int_{u}^{t}\Phi (t,s,P)F(s)\,ds,\quad t\in J$$ 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 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 $$P_r=\begin{bmatrix} 0 & \frac{1}{p_r} \\ q_r & 0 \end{bmatrix}, \quad W_r=\begin{bmatrix} 0 & 0 \\ w_r & 0 \end{bmatrix}. \label{eq13}$$ Then the scalar equation \eqref{eq11a} is equivalent to the first-order system $$Y'=(P_r-\lambda W_r)Y=\begin{bmatrix} 0 & \frac{1}{p_r} \\ q_r-\lambda w_r & 0 \end{bmatrix}, \quad Y=\begin{bmatrix} y \\ p_ry' \end{bmatrix}. \label{eq14a}$$ 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 $$\Phi _r'=(P_r-\lambda W_r)\Phi _r\text{ on } J_r,\quad \Phi _r(u_r,u_r,\lambda )=I,\; a_r\leq u_r\leq b_r,\; \lambda \in \mathbb{C}, \label{eq15}$$ 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 $$\begin{split} \Delta (\lambda )&= \Delta (a_1,b_1,a_2,b_2,A_1,B_1,A_2,B_2,P_1,P_2,w_1,w_2, \lambda ) \\ &= \det [(A_1+B_1\Phi _1(b_1,a_1,\lambda )\mid A_2+B_2\Phi _2(b_2,a_2,\lambda ))],\quad \lambda \in \mathbb{C}, \end{split}$$ 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 $$Y_r(t)=\Phi _r(t,a_r,\lambda )C_r. \label{eq16}$$ Note that$\Phi _r(a_r,a_r,\lambda )=I$,$r=1,2$. Substituting \eqref{eq16} into the boundary conditions \eqref{eq12}, we obtain $$A_1C_1+B_1\Phi _1(b_1,a_1,\lambda )C_1+A_2C_2+B_2\Phi _2(b_2,a_2,\lambda )C_2=0. \label{eq17}$$ Set$C=\begin{bmatrix} C_1 \\ C_2 \end{bmatrix}$. Therefore \eqref{eq17} can be written as $$(A_1+B_1\Phi _1(b_1,a_1,\lambda )\mid A_2+B_2\Phi _2(b_2,a_2,\lambda ))C=0. \label{eq18}$$ 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 $$\label{eq22} Y'=(P_r-\lambda W_r)Y+F_r,\quad A_1Y_1(a_1)+B_1Y_1(b_1)+A_2Y_2(a_2)+B_2Y_2(b_2)=0,$$ 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 $$\label{3.2} Y'=(P_r-\lambda W_r)Y.$$ 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 $$Y'=(P_r-\lambda W_r)Y+F_r\ \ \text{on}\ \ J_r \label{eq20}$$ if and only if$y_r$is a solution of $$-(p_ry')'+q_ry=\lambda w_ry+f_r\quad \text{on } J_r, \label{eq21}$$ 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 $$Y_r(t,\lambda )=\Phi _r(t,a_r,\lambda )C_r+\int_{a_r}^{t}\Phi _r(t,s,\lambda )F_r(s)ds,\quad a_r\leq t\leq b_r. \label{eq111}$$ 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 $$\begin{split} & A_1Y_1(a_1,\lambda )+B_1Y_1(b_1,\lambda )+A_2Y_2(a_2,\lambda )+B_2Y_2(b_2,\lambda ) \\ & =D(\lambda )C+B_1\int_{a_1}^{b_1}\Phi _1(b_1,s,\lambda )F_1(s) \,\mathrm{d}s+B_2\int_{a_2}^{b_2}\Phi _2(b_2,s,\lambda )F_2(s) \,\mathrm{d}s. \end{split} \label{eq23}$$ 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 )$. Let \begin{gather*} G_1(t,s,\lambda )=\begin{cases} -\Phi _1(t,a_1,\lambda )D_1(\lambda )B_1\Phi _1(b_1,s,\lambda ), & a_1\leq t0$ 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 $$(f,g)_r=\int\limits_{Jr}f\overline{g}w_r,\;r=1,2 \label{u}$$ 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}: $$(f,g)_r=h_r\int\limits_{Jr}f\overline{g}w_r,\;h_r>0,\;r=1,2. \label{uh}$$ 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 $$J_1=(a,b),\quad J_2=(c,d),\quad b=c \label{51}$$ 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: $$L^{2}((a,b),w_1)\dotplus L^{2}((c,d),w_2)=L^{2}((a,d),w) \label{5.2}$$ \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: $$\begin{gathered} A_1y(a)+A_2y^{[1]}(a) =0,\quad A_1,A_2\in \mathbb{R},\;(A_1,A_2) \neq (0,0); \\ B_1y(b)+B_2y^{[1]}(b) =0,\quad B_1,B_2\in \mathbb{R},\;(B_1,B_2) \neq (0,0); \\ C_1y(c)+C_2y^{[1]}(c) =0,\quad C_1,C_2\in \mathbb{R},\;(C_1,C_2) \neq (0,0); \\ D_1y(d)+D_2y^{[1]}(d) =0,\quad D_1,D_2\in \mathbb{R},\;(D_1,D_2) \neq (0,0). \end{gathered}$$ 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 $$0=AEA^{\ast }=BEB^{\ast }=CEC^{\ast }=DED^{\ast }. \label{eq4.3}$$ \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 $$\begin{gathered} Y(c)=e^{i\gamma }KY(b),\quad Y=\begin{bmatrix} y \\ y^{[1]} \end{bmatrix}, \quad K=(k_{ij}),\;k_{ij}\in \mathbb{R},\;1\leq i,j\leq 2,\\ \det K\neq 0,\; -\pi <\gamma \leq \pi\,. \end{gathered} \label{eq4.5}$$ 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 $$C=\begin{bmatrix} 0 & 0 \\ -1 & 0 \\ 0 & -1 \\ 0 & 0 \end{bmatrix}, \quad B=e^{i\gamma }\begin{bmatrix} 0 & 0 \\ k_{11} & k_{12} \\ k_{21} & k_{22} \\ 0 & 0 \end{bmatrix}, \quad -\pi <\gamma \leq \pi . \label{eq4.6}$$ 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 $$h=k\det K. \label{eq4.7}$$ 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 $-\pi <\gamma _1,\gamma_2\leq \pi$, $$\begin{gathered} Y(d) =e^{i\gamma _1}GY(a),\quad G=(g_{ij}),\;g_{ij}\in \mathbb{R},\;i,j=1,2,\;\det G\neq 0, \\ Y(c) =e^{i\gamma _2}KY(b),\quad K=(k_{ij}),\;k_{ij}\in \mathbb{R} ,\;i,j=1,2,\;\det K\neq 0,\\ Y=\begin{bmatrix} y \\ y^{[1]} \end{bmatrix}. \end{gathered}\label{eq4.8}$$ 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). \begin{thebibliography}{99} \bibitem{akdm} Z. Akdogan, M. Dimirci, O. S. Mukhtarov; \emph{Green function of discontinuous boundary value problem with transmission conditions}, Math. Meth. Appl. Sci. (2007), 1719-1738. \bibitem{aosz} J. J. Ao, J. Sun, M. Z. Zhang; \emph{The finite spectrum of Sturm-Liouville problems with transmission conditions}, Appl. Math. 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