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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 101, pp. 1--16.\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-2013/101\hfil Formally self-adjoint quasi-differential operators]
{Formally self-adjoint quasi-differential operators and
 boundary-value problems}

\author[A. Goriunov, V. Mikhailets, K. Pankrashkin \hfil EJDE-2013/101\hfilneg]
{Andrii Goriunov, Vladimir Mikhailets, Konstantin Pankrashkin}  % in alphabetical order

\address{Andrii Goriunov \newline
Institute of Mathematics, National Academy of Sciences of Ukraine,
 Kyiv, Ukraine}
\email{goriunov@imath.kiev.ua}

\address{Vladimir Mikhailets \newline
Institute of Mathematics,  National Academy of Sciences of Ukraine, 
Kyiv, Ukraine}
\email{mikhailets@imath.kiev.ua}

\address{Konstantin Pankrashkin \newline
Laboratory of mathematics, University Paris-Sud 11, Orsay, France}
\email{konstantin.pankrashkin@math.u-psud.fr}

\thanks{Submitted March 6, 2013. Published April 19, 2013.}

\subjclass[2000]{34B05, 34L40, 47N20, 34B37}
\keywords{Quasi-differential operator; distributional coefficients;
\hfill\break\indent
self-adjoint extension; maximal dissipative extension; generalized resolvent}

\begin{abstract}
 We develop the machinery of boundary triplets for one-dimen\-sional operators
 generated by formally self-adjoint quasi-differential expression of arbitrary
 order on a finite interval. The technique is then used to describe
 all maximal dissipative, accumulative and self-adjoint extensions of the
 associated minimal operator and its generalized resolvents in terms
 of the boundary conditions. Some specific classes are considered in greater
 detail.
\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}

Many problems of the modern mathematical physics and the  quantum mechanics
lead to the study of differential operators with strongly singular coefficients 
such as Radon measures or even more singular distributions,
see the monographs \cite{Albeverio, AlbeKur} and the very recent papers 
\cite{Teschl-12-unpubl,EGNT,EGNT2,Teschl-arXiv1105.3755E}
and the references therein. In such situations one is faced with the problem 
of a correct definition of such operators as the classical methods of the 
theory of differential operators cannot be applied anymore.
It was observed in the recent years that a large class of one-dimensional 
operators can be handled
in a rather efficient way with the help of the so-called Shin-Zettl 
quasi-derivatives \cite{AtkEvZ-88, SavShkal-99}.
The class of such operators includes, for example,
the Sturm-Liouville operators  acting on $L_2([a,b], \mathbb{C})$ by the rule
\begin{equation}\label{St-L expr}
l(y) = - (py')' + qy,
\end{equation}
where the coefficients $p$ and $q$ satisfy the conditions
\[
\frac{1}{p}, \frac{Q}{p}, \frac{Q^2}{p} \in L_1([a,b], \mathbb{C}),
\]
where $Q$ is the antiderivative of the distribution $q$, and $[a,b]$ 
is a finite interval.
The condition $1/p \in L_1([a,b], \mathbb{C})$ implies that the potential 
function $q$ may be
a finite measure on $[a,b]$, see~\cite{GMSt-L}.

For the two-term formal differential expression
\begin{equation}\label{m expr}
l(y) = i^m y^{(m)} + qy,  \quad m \geq 3,
\end{equation}
where $q = Q'$ and $Q \in L_1([a,b], \mathbb{C})$,
the regularisation with quasi-derivatives
was constructed in \cite{GM_high}.
Similarly one can study the case
\[ 
q = Q^{(k)}, \quad k \leq [\frac{m}{2}],
\]
where $Q \in L_2([a,b], \mathbb{C})$ if $m$ is even and $k = m/2$,
and $Q \in L_1([a,b], \mathbb{C})$ otherwise,
and all the derivatives of $Q$ are understood in the sense of distributions.

In the present paper we consider one-dimensional operators generated 
by the most general formally self-adjoint quasi-differential expression 
of an arbitrary order on the Hilbert space $L_2([a,b], \mathbb{C})$, 
and the main result consists in an explicit construction of a boundary
 triplet for the associated symmetric minimal  quasi-differential operator.
The machinery of boundary triplets \cite{Gorb-book-eng}
is a useful tool in the description and the analysis of various 
boundary-value problems arising in mathematical physics, see e.g.
 \cite{BMN02,BGP08,DM}, and we expect that
the constructions of the present paper will be useful, in particular, 
in the study of higher order differential operators on metric graphs \cite{AGA}.

The quasi-differential operators were introduced first by Shin \cite{Shin-43-eng}
and then essentially developed by Zettl \cite{Zettl-75},  see also the 
monograph \cite{EverMark-book} and references therein.
The paper \cite{Zettl-75} provides the description of all self-adjoint 
extensions of the minimal symmetric quasi-differential operator of even 
order with real-valued coefficients.
It is based on the so-called  Glasman-Krein-Naimark theory and is rather 
implicit.
The approach of the present work gives an explicit description of the 
self-adjoint extensions as well as of all maximal dissipative/accumulative 
extensions in terms of easily checkable boundary conditions.

The paper is organised as follows. In Section 1 we recall basic definitions 
and known facts concerning the Shin-Zettl quasi-differential operators.
Section 2 presents the regularization of the formal differential expressions
\eqref{St-L expr} and \eqref{m expr}
using the quasi-derivatives, and some specific examples are considered.
In Section 3 the boundary triplets for the minimal symmetric operators are 
constructed.
All maximal dissipative, maximal accumulative and self-adjoint extensions 
of these operators are explicitly described in terms of boundary conditions.
Section 4 deals with the formally self-adjoint quasi-differential operators 
with real-valued coefficients.
We prove that every maximal dissipative/accumulative extension of the minimal
 operator in this case is self-adjoint and describe all such extensions.
In Section 5 we give an explicit description
of all maximal dissipative/accumulative and self-adjoint extensions
with separated boundary conditions for a special case.
In Section 6 we describe all generalized resolvents of the minimal operator.
Some results of this paper for some particular classes of quasi-differential 
expressions were announced without proof in \cite{GM_even, GM_odd}.
These results were used in papers \cite{BadKo-arXiv1112.4587, GM_MN}.

\section{Quasi-differential expressions}

In this section we recall the definition and the basic facts concerning
the Shin-Zettl quasi-derivatives and the quasi-differential operators 
on a finite interval,
see \cite{EverMark-book, Zettl-75}  for a more detailed discussion.

Let $m \in \mathbb{N}$ and a finite interval $[a, b]$ be given.
Denote by $Z_m([a,b])$ the set of the $m\times m$ complex matrix-valued functions $A$
whose entries $(a_{k,s})$ satisfy
\begin{equation}\label{Quasi cond}
\begin{gathered}
a_{k,s} \equiv 0, \quad s > k + 1;\\
 a_{k,s} \in L_1 ([a,b], \mathbb{C} ), 
\quad a_{k, k + 1} \neq 0 \text{a.e. on } [a,b],
\quad k = 1,2,\dots,m;  \; s = 1,2,\dots,k + 1;
\end{gathered}
\end{equation}
such matrices will be referred to as Shin-Zettl matrices of order
 $m$ on $[a,b]$.
Any Shin-Zettl matrix $A$ defines recursively the associated quasi-derivatives
 of orders $k \leq m$ of a function  $y \in \operatorname{Dom}(A)$
in the following way:
\begin{gather*}
D^{[0]}y := y,\\
D^{[k]}y := a^{-1}_{k,k+1}(t)
\Big((D^{[k - 1]}y)' - \sum_{s = 1}^{k} {a_{k,s}(t)D^{[s - 1]}}y\Big) ,
\quad k = 1,2,\dots,m - 1,\\
D^{[m]}y  := (D^{[m - 1]}y)' - \sum_{s = 1}^{m} {a_{m,s}(t)D^{[s - 1]}y},
\end{gather*}
and the associated domain $\operatorname{Dom}(A)$ is defined
by
\[
\operatorname{Dom}(A) := \{y : D^{[k]}y \in AC([a,b], \mathbb{C}),\,
 k=\overline{0,m-1}\}.
\]
The above yields $D^{[m]}y \in L_1([a,b], \mathbb{C})$.
The quasi-differential expression $l(y)$ of order $m$ associated with $A$
is defined by
\begin{equation}\label{Q-d_expr}
l(y) := i^mD^{[m]}y.
\end{equation}

Let $c \in [a,b]$ and $\alpha_k \in \mathbb{C}$, $k=\overline{0, m-1}$.
We say that a function $y$ solves the Cauchy problem 
\begin{equation}\label{cauchy pr 1}
l(y) - \lambda y = f \in L_2([a,b], \mathbb{C}), \quad
 (D^{[k]}y)(c) = \alpha_k, \quad k=\overline{0,m-1},
\end{equation}
if $y$ is the first coordinate of the vector function $w$ solving
the Cauchy problem for the associated the first order matrix equation
\begin{equation}\label{cauchy pr 2}
w'(t)=A_\lambda(t)w(t) + \varphi(t),
\quad
w(c) = (\alpha_0, \alpha_1, \dots \alpha_{m - 1})
\end{equation}
where we denote 
\begin{equation}\label{A matrix}
A_\lambda(t):=A(t) - 
\begin{pmatrix}
0&0&\dots &0 \\
0&0&\dots&0 \\
\vdots  &\vdots &\ddots &\vdots  \\
0&0&\dots &0 \\
i^{-m}\lambda&0&\dots&0
\end{pmatrix} \in L_1([a,b], \mathbb{C}^{m\times m}),
\end{equation}
and $\varphi(t) := \big(0, 0, \dots, 0, i^{-m}f(t)\big)^T 
\in L_1([a,b], \mathbb{C}^m)$.
The following statement is proved in \cite{Zettl-75}.

\begin{lemma}\label{lm_cauchy_pr_unique}
Under the assumptions \eqref{Quasi cond},
the problem \eqref{cauchy pr 1} has a unique solution defined on  $[a,b]$.
\end{lemma}

The quasi-differential expression $l(y)$ gives rise to the associated
\emph{maximal} quasi-differential operator
$L_{\rm max}:y  \mapsto l(y)$, defined on
\[
\operatorname{Dom}(L_{\rm max}) = \{y \in \operatorname{Dom}(A)  :
 D^{[m]} y \in L_2([a,b], \mathbb{C})\},
\]
in the Hilbert space $L_2([a,b], \mathbb{C})$.
The associated \emph{minimal} quasi-differential operator is defined as 
the restriction of $L_{\rm max}$ onto the set
\[
\operatorname{Dom}(L_{\rm min}) :=
\{y \in  \operatorname{Dom}(L_{\rm max}) : D^{[k]}y(a) = D^{[k]}y(b) = 0,\,
k = \overline{0,m - 1}\}.
\]
If the functions $a_{k,s}$ are sufficiently smooth, then all the brackets 
in the definition of the quasi-derivatives can be expanded, and we arrive 
at the usual ordinary differential expressions,
and the associated quasi-differential operators become differential ones.

Let us recall the definition of the formally adjoint quasi-differential 
expression $l^+(y)$.
The formally adjoint (also called the Lagrange adjoint) matrix $A^+ $
 for $A \in Z_m([a,b])$ is defined by
\[
A^+ := - \Lambda_m^{-1}\overline{A^T}\Lambda_m,
\]
where $\overline{A^T}$ is the conjugate transposed matrix of $A$, and
\[
\Lambda_m :=
\begin{pmatrix}
  0 & 0  & \dots & 0 & -1 \\
  0 & 0  & \dots & 1 & 0 \\
  \vdots  & \vdots  & \vdots  & \vdots  & \vdots  \\
  0 & (-1)^{m - 1}  & \dots & 0 & 0 \\
  (-1)^m & 0 & \dots & 0 & 0
\end{pmatrix}.
\]
One can easily see that $\Lambda_m^{-1} = (-1)^{m - 1}\Lambda_m$.

We we can define the Shin-Zettl quasi-derivatives associated with $A^+$
which will be denoted by
\[
D^{\{0\}}y, D^{\{1\}}y, \dots, D^{\{m\}}y;
\]
they act on the domain
\[
\operatorname{Dom}(A^+) := \{y : D^{\{k\}}y \in AC([a,b], \mathbb{C}),\,
 k=\overline{0,m-1}\}.
\]
The formally adjoint quasi-differential expression is now defined as
\[
l^+(y) := i^mD^{\{m\}}y\,.
\]
We denote the associated maximal and minimal operators by
$L^+_{\rm max}$ and $L^+_{\rm min}$ respectively
The following theorem is proved in \cite{Zettl-75}.

\begin{theorem}\label{thm_L_adjoint}
The operators $L_{\rm min}$, $L^+_{\rm min}$, $L_{\rm max}$,
 $L^+_{\rm max}$ are closed and densely defined in 
$L_2([a,b], \mathbb{C})$, and satisfy
\[
L_{\rm min}^* = L^+_{\rm max},\quad
L_{\rm max}^* = L^+_{\rm min}.
\]
If  $l(y) = l^+(y)$, then the operator $L_{\rm min} = L^+_{\rm min}$ 
is symmetric with the deficiency indices $({m,m} )$, and
\[
L_{\rm min}^* = L_{\rm max},\quad 
L_{\rm max}^* = L_{\rm min}.
\]
\end{theorem}

We also require the following two lemmas whose proof can be found 
 in \cite{EverMark-book}.

\begin{lemma}\label{lm_Lagrange}
For any $y, z \in \operatorname{Dom}(L_{\rm max})$ there holds
\[
\int_a^b \Big(D^{[m]}y\cdot\overline z  -
y\cdot\overline{D^{[m]}z} \Big)dt =
\sum_{k = 1}^{m}
(-1)^{k - 1}{D^{[m - k]}y\cdot\overline {D^{[k -1]}z}}\big|_{t = a}^{t = b}
\]
\end{lemma}

\begin{lemma}\label{lm_surj}
For any $(\alpha _0 ,\alpha _1, \dots, \alpha _{m - 1}),(\beta _0,\beta _1, 
\dots, \beta _{m - 1})\in\mathbb{C}^m$
there exists a function ${y \in \operatorname{Dom}(L_{\rm max})}$ such that
\[
D^{[k]}y(a) = \alpha _k , \quad D^{[k]}y(b) = \beta _k, \quad
 k = 0,1, \dots, m - 1.
\]
\end{lemma}


\section{Regularizations by quasi-derivatives}

Let us consider some classes of formal differential expressions with singular
coefficients admitting a regularisation with the help of the Shin-Zettl 
quasi-derivatives.
Consider first the formal Sturm-Liouville expression
\[
l(y) = -(p(t)y')'(t) + q(t)y(t), \quad t \in [a, b].
\]
The classical definition of the quasi-derivatives
\[
D^{[0]}y := y, \quad D^{[1]}y = py', \quad D^{[2]}y = (D^{[1]}y)' - qD^{[0]}y
\]
allows us to interpret the above expression $l$ as a regular quasi-differential 
expression if the function $p$ is finite almost everywhere and, in addition,
\begin{equation}\label{St-L_cond_class}
\frac{1}{p}, \, q \in L_1( [a,b], \mathbb{C}).
\end{equation}
Some physically interesting coefficients $q$ (i.e. having
non-integrable singularities or being a measure) are not covered by 
the preceding conditions, and this can be corrected using another 
set of quasi-derivatives as proposed in \cite{GMSt-L, SavShkal-99}.
Set
\begin{equation}\label{St-L reg}
\begin{gathered}
D^{[0]} y = y, \quad D^{[1]} y = py' - Qy, \\
D^{[2]} y = (D^{[1]} y)' + {\frac{Q}{p}}D^{[1]} y + {\frac{Q^2}{p}}y,
\end{gathered}
\end{equation}
where function $Q$ is chosen so that $Q' = q$ and the derivative is 
understood in the sense of distributions.
Then the expression
\[
l[y] = - D^{[2]} y
\]
is a Shin-Zettl quasi-differential one if the following conditions are satisfied:
\begin{equation}\label{St-L_cond_GM}
\frac{1}{p},\, \frac{Q}{p},\, \frac{Q^2}{p} \in L_1( [a,b], \mathbb{C}).
\end{equation}
In this case the expression $l$ generates the associated
quasi-differential operators $L_{\rm min}$ and $L_{\rm max}$.
One can easily see that if $p$ and $q$ satisfy conditions \eqref{St-L_cond_class},
then these operators  coincide with the classic Sturm-Liouville operators, but
the conditions \eqref{St-L_cond_GM} are considerably weaker than
\eqref{St-L_cond_class}, and the class of admissible coefficients is much larger
if one uses the quasi-differential machinery. 
This can be illustrated with an example.

\begin{example}\label{ex_t^a} \rm
Consider the differential expression  \eqref{St-L expr} with  
$p(t) = t^{\alpha}$ and $q(t) = ct^\beta$, and assume $c\neq 0$.
The conditions \eqref{St-L_cond_class} are reduced to the set of the 
inequalities $\alpha < 1$ and $\beta > -1$,
while the conditions \eqref{St-L_cond_GM} hold for
\[
 \alpha < 1 \text{ and } \beta > \max\Big\{\alpha - 2, \frac{\alpha - 3}{2}\Big\}.
\]
So we see that the use of the quasi-derivatives allows one to consider
the Sturm-Liouville expressions with any power singularity of the potential $q$
if it is compensated by an appropriate function $p$.
\end{example}

\begin{remark} \label{rmk1} \rm
The formulas  \eqref{St-L reg} for the quasi-derivatives contain a certain 
arbitrariness due to the non-uniqueness of the function $Q$ which is only 
determined up to a constant. However, one can show that if 
$\widetilde{Q} := Q + c$, for some constant $c \in \mathbb{C}$, then
$L_{\rm max}(Q) =  L_{\rm max}(\widetilde{Q})$ and
$L_{\rm min}(Q) =  L_{\rm min}(\widetilde{Q})$;
i.e., the maximal and minimal operators do not depend
on the choice of $c$. 
\end{remark}

One can easily see that the expression
\[ 
l^+(y) = -(\overline{p}y')' + \overline{q}y 
\]
defines the quasi-differential expression which is formally adjoint to one
generated by \eqref{St-L expr}.
It brings up the associated maximal and minimal operators 
$L^+_{\rm max}$ and $L^+_{\rm min}$.
Theorem \ref{thm_L_adjoint} shows that if $p$ and $q$ in \eqref{St-L expr} 
are real-valued, then the operator
$L_{\rm min} = L^+_{\rm min}$ is symmetric.

It is well-known that for the particular case $p \equiv 1$ and
$q \in  L_2([a,b], \mathbb{C})$ one has
\[
\operatorname{Dom}(L_{\rm max}) = W^2_2([a,b], \mathbb{C}) 
\subset C^1([a,b], \mathbb{C}).
\]
The following example shows that in some cases all functions
in $\operatorname{Dom}(L_{\rm max})\setminus\{0\}$
are non-smooth.

\begin{example}\label{ex_Non-smooth Dom} \rm
Consider the differential expression \eqref{St-L expr} with
\[
p(t) \equiv 1, \quad q(t) 
= \sum_{\mu \in \mathbb{Q}\cap(a,b)}{\alpha_\mu \delta(t - \mu)},
\]
where $\mathbb{Q}$ is the set of real rational numbers and
\[
\alpha_\mu \neq 0 \text{ for all } \mu\in\mathbb{Q}\cap(a,b), 
\text{ and } \sum_{\mu\in\mathbb{Q}\cap(a,b)}|\alpha_\mu| < \infty.
\]
Then one can take
\[
Q(t) = \sum_{\mu\in\mathbb{Q}\cap(a,b)}{\alpha_\mu H(t - \mu)},
\]
with $H(t)$ being Heaviside function, and $Q$ is a function of a bounded 
variation having discontinuities at every rational point of $(a,b)$. 
Therefore, for every subinterval $[\alpha, \beta] \subset (a,b)$ and any
$y \in \operatorname{Dom}(L_{\rm max})\,\cap\, C^1([\alpha,\beta], \mathbb{C})$
we have
\[
y'(\mu_+) - y'({\mu_-}) = \alpha_\mu y(\mu),\quad
 \mu \in\mathbb{Q}\cap[\alpha,\beta]. %\right.\right\}.
\]
Then $\alpha_\mu y(\mu) = 0$ for all $\mu \in\mathbb{Q}\cap[\alpha,\beta]$, 
which gives $y(\mu) = 0$, and the density of  
$\{\mu\}\cap[\alpha,\beta]$ in $[\alpha,\beta]$ implies
$y(t) = 0$ for all $t \in [\alpha,\beta]$.
\end{example}

Now consider the expression
\[
l(y) = i^m y^{(m)}(t) + q(t)y(t),  \quad m \geq 3,
\]
assuming that
\begin{equation}\label{m_cond_GM}
\begin{gathered}
 q = Q^{(k)}, \quad 1 \leq k \leq [\frac{m}{2}], \\
Q \in
  \begin{cases}
   L_2([a,b], \mathbb{C}), & m = 2n, \, k = n; \\
   L_1([a,b], \mathbb{C}) &\text{ otherwise, }
  \end{cases}
 \end{gathered}
\end{equation}
where the derivatives of $Q$ are understood in the sense of distributions.
Introduce the quasi-derivatives as follows:
\begin{equation}\label{m reg}
\begin{gathered}
D^{[r]}y = y^{(r)}, \quad 0 \leq r \leq m - k - 1;\\
D^{[m - k + s]}y = (D^{[m - k + s - 1]}y)' 
 + i^{-m}(-1)^{s} {k\choose s} \,Q D^{[s]}y, \quad 0 \leq s \leq k - 1;\\
D^{[m]}y = \begin{cases}
  (D^{[m - 1]}y)' + i^{-m}(-1)^{k} { k \choose k} Q D^{[k]}y, 
   & 1 \leq k < m/2, \\
  (D^{[m - 1]}y)' + Q D^{[\frac{m}{2}]}y + (-1)^{\frac{m}{2} + 1} Q^2 y, 
   & m = 2n = 2k;
           \end{cases}
\end{gathered}
\end{equation}
where $k \choose j$ are the binomial coefficients.
It is easy to verify that for sufficiently smooth functions $Q$ the 
equality $l(y) = i^m D^{[m]} y$ holds.
Also one can easily see that, under assumptions \eqref{m_cond_GM},
all the coefficients of the quasi-derivatives \eqref{m reg} are integrable 
functions.
The Shin-Zettl matrix corresponding to \eqref{m reg} has the form
\begin{equation}\label{m matr}
A(t):= \begin{pmatrix}
0 &1 &0 &\dots &0 &\dots &0 &0 \\
0 &0 &1 &\dots &0 &\dots &0 &0 \\
\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots\\
- i^{-m} {k\choose 0} Q &0 &0 &\dots &0 &\dots &0 &0 \\
0& i^{-m} {k \choose 1} Q &0 &\dots &0 &\dots &0 &0 \\
\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots\\
0 &0 &0 &\dots &0 &\dots &0 &1 \\
(-1)^{\tfrac{m}{2}} Q^2 \delta_{2k, m} &0 &0 &\dots &i^{-m}(-1)^{k + 1} { k\choose k} Q &\dots &0 &0 \\
\end{pmatrix},
\end{equation}
where $\delta_{ij}$ is the Kronecker symbol.
Similarly to the previous case the initial formal differential 
expression \eqref{m expr} can be defined in the quasi-differential form
\[
l[y] := i^{m} D^{[m]} y,
\]
and it generates the corresponding quasi-differential operators
$L_{\rm min}$ and $L_{\rm max}$.

\begin{remark} \label{rmk2} \rm
Again, the formulas for the quasi-derivatives depend on the choice of 
the antiderivative $Q$ of order $k$ of
the distribution $q$ which is not only defined up to a
a polynomial of order $\leq k - 1$.
However, one can show that the maximal and minimal operators do not depend
on the choice of this polynomial.
\end{remark}

For $k = 1$ the above regularization was proposed in \cite{SavShkal-99},
and for even $m$ they were announced in \cite{MirzShkal-11}.
The general case is presented here for the first time.
Note that if the distribution $q$ is real-valued, then the operator 
$L_{\rm min}$ is symmetric.


\section{Extensions of symmetric quasi-differential operators}

Throughout the rest of the paper we assume the Shin-Zettl matrix is formally 
self-adjoint, i.e.
$A = A^{+}$. The associated quasi-differential expression $l(y)$ is then
formally self-adjoint, $l(y) = l^+(y)$, and the minimal quasi-differential 
operator  $L_{\rm min}$
is symmetric with equal deficiency indices by Theorem \ref{thm_L_adjoint}.
So one may pose a problem of describing (by means of boundary triplets) 
various classes of extensions of  $L_{\rm min}$ in  $L_2([a,b], \mathbb{C})$.

For the reader's convenience we give a very short summary of the theory 
of boundary triplets based on the results of Rofe-Beketov \cite{RofeB-69-eng} 
and Kochubei \cite{Koch-75},
see also the monograph \cite{Gorb-book-eng} and references therein.

Let $T$ be a closed densely defined symmetric operator in a Hilbert 
space $\mathcal{H}$ with equal (finite or infinite) deficiency indices.

\begin{definition}[\cite{Gorb-book-eng}]\label{PGZdef} \rm
The triplet $( H, \Gamma _1 ,\Gamma _2 )$, where $H$ is an auxiliary Hilbert 
space and $\Gamma_1$, $\Gamma_2$ are the linear maps from 
$\operatorname{Dom}(T^*)$ to $H$, is called  a \emph{boundary triplet} 
for $T$, if the following two conditions are satisfied:
\begin{enumerate}
\item for any $ f,g \in \operatorname{Dom} ( {L^*} )$ there holds
\[
( {T^ *  f,g} )_\mathcal{H} - ( {f,T^ *  g}
)_\mathcal{H} = ( {\Gamma_1 f,\Gamma_2 g} )_H  -
( {\Gamma_2 f,\Gamma_1 g} )_H,
\]

\item for any $ g_1, g_2 \in H$ there is a vector 
$ f\in \operatorname{Dom} ( {T^*} )$ such that
$ \Gamma_1 f = g_1$ and $ \Gamma_2 f = g_2$.
\end{enumerate}
\end{definition}

The above definition implies that $ f \in \operatorname{Dom} ( {T} )$
if and only if $\Gamma_1f = \Gamma_2f = 0$.
A boundary triplet $( {H,\Gamma _1 ,\Gamma _2 } )$ with 
$\operatorname{dim} H = n$
exists for any symmetric operator $T$ with equal non-zero deficiency 
indices  $(n, n)$\, $(n \leq \infty)$, but it is not unique.


Boundary triplets may be used to describe all maximal dissipative,
 maximal accumulative and self-adjoint
extensions of the symmetric operator in the following way.
Recall that a densely defined linear operator $T$ on a complex Hilbert 
space $\mathcal{H}$ is called
\emph{dissipative} (resp. \emph{accumulative}) if
\[
\Im ( Tf, f )_\mathcal{H} \geq 0 \quad (\text{resp.} \leq 0), 
\quad \text{for all } f \in \operatorname{Dom} (T)
\]
and it is called \emph{maximal dissipative} 
(resp. \emph{maximal accumulative}) if, in addition,
$T$ has no non-trivial dissipative/accumulative extensions in $\mathcal{H}$.
Every symmetric operator is both dissipative and accumulative, 
and every self-adjoint operator is a maximal dissipative and maximal 
accumulative one.
Thus, if one has a symmetric operator $T$, then one can state the problem 
of describing its maximal dissipative and maximal accumulative extensions.
According to Phillips' Theorem \cite{Phil} (see also \cite[p. 154]{Gorb-book-eng})
every maximal dissipative or accumulative extension of a symmetric operator is a restriction of its
adjoint operator.  Let $(H, \Gamma _1,\Gamma _2 )$ be a boundary triplet for $T$.
The following theorem is proved in \cite{Gorb-book-eng}.

\begin{theorem}\label{thm_Gorb}
If $K$ is a contraction on $H$, then the restriction of $T^*$ to the 
set of the vectors
$f \in \operatorname{Dom} (T^*)$ satisfying the condition
\begin{equation}\label{absrt_ext_diss}
( {K - I} )\Gamma_{1} f + i( {K + I} )\Gamma_{2} f = 0
\end{equation}
or
 \begin{equation}\label{absrt_ext_akk}
( {K - I} )\Gamma_{1} f - i( {K + I} )\Gamma_{2} f = 0
\end{equation}
is a maximal dissipative, respectively, maximal accumulative extension of $T$.
Conversely, any maximal dissipative (maximal accumulative) extension of $L$ is 
the restriction of $T^*$ to the set of vectors $f \in \operatorname{Dom} (T^*)$,
satisfying \eqref{absrt_ext_diss} or \eqref{absrt_ext_akk}, respectively,
and the contraction $K$ is uniquely
defined by the extension.
The maximal symmetric extensions of  $T$ are described by the conditions
\eqref{absrt_ext_diss} and \eqref{absrt_ext_akk}, where $K$ is an isometric 
operator.
These conditions define a self-adjoint extension if $K$ is unitary.
\end{theorem}


\begin{remark}\label{remark_K} \rm
Let $K_1$ and $K_2$ be the unitary operators on $H$ and let the boundary 
conditions
 $$ 
( {K_1 - I} )\Gamma_{1}y + i( {K_1 + I} )\Gamma_{2} y = 0
$$
and
 $$ 
( {K_2 - I} )\Gamma_{1}y - i( {K_2 + I} )\Gamma_{2} y = 0
$$
define self-adjoint extensions. 
These are two different bijective parameterizations, which reflects the fact
that each self-adjoint operators is maximal dissipative and a maximal 
accumulative one at the same time.
The extensions, given by these boundary conditions coincide if $K_1 = K_2^{-1}$.
Indeed, the boundary conditions can be written in another form:
\begin{gather*}
K_1(\Gamma _{1}y + i \Gamma _{2}y ) = \Gamma _{1}y - i\Gamma _{2}y, \quad
	  \Gamma _{1}y - i\Gamma _{2}y \in \operatorname{Dom}(K) = H,\\
K_2(\Gamma _{1}y - i \Gamma _{2}y ) = \Gamma _{1}y + i\Gamma _{2}y, \quad
	  \Gamma _{1}y + i\Gamma _{2}y \in \operatorname{Dom}(K) = H,
\end{gather*}
and the equivalence of the boundary conditions reads as $K_1K_2 =  K_2K_1 = I$.
\end{remark}

Let us go back to the quasi-differential operators.
The following result is crucial for the rest of the paper as it allows 
to apply the boundary triplet machinery
to the symmetric minimal quasi-differential operator $L_{\rm min}$.


\begin{lemma}  \label{PGZth}
  Define linear maps $\Gamma_{[1]}$, $\Gamma_{[2]}$  from 
$\operatorname{Dom}(L_{\rm max})$ to $\mathbb{C}^{m}$ as follows:
for $m = 2n$ and $n \geq 2$ we set
\begin{equation}\label{Gamma 2n}
\Gamma_{[1]}y := i^{2n}   \begin{pmatrix}
   - D^{[2n - 1]}y(a),\\
   \dots,\\
   (-1)^nD^{[n]}y(a),\\
   D^{[2n - 1]}y(b),\\
   \dots,\\
   (-1)^{n - 1}D^{[n]}y(b)
   \end{pmatrix},
 \quad\Gamma_{[2]}y :=
   \begin{pmatrix}
   D^{[0]}y(a),\\
   \dots,\\
   D^{[n - 1]}y(a),\\
   D^{[0]}y(b),\\
   \dots,\\
   D^{[n - 1]}y(b)
   \end{pmatrix}
 \end{equation}
and for $m = 2n + 1$ and $n \in \mathbb{N}$, we set
\begin{equation}\label{Gamma 2n+1}
\Gamma_{[1]}y := i^{2n + 1}
   \begin{pmatrix}
    - D^{[2n]}y(a),\\
    \dots,\\
    (-1)^{n} D^{[n + 1]}y(a),\\
    D^{[2n]}y(b),\\
    \dots.,\\
    (-1)^{n - 1}D^{[n + 1]}y(b),\\
    \alpha D^{[n]}y(b) + \beta D^{[n]}y(a)
    \end{pmatrix},
  \quad
 \Gamma_{[2]}y :=
   \begin{pmatrix}
   D^{[0]}y(a),\\
   \dots,\\
   D^{[n -1]}y(a),\\
   D^{[0]}y(b),\\
   \dots,\\
   D^{[n - 1]}y(b),\\
   \gamma D^{[n]}y(b) + \delta D^{[n]}y(a)
   \end{pmatrix},
\end{equation}
where
\[
\alpha = 1, \quad
\beta = 1, \quad
\gamma = \frac{(-1)^n}{2} + i,\quad
 \delta = \frac{(-1)^{n + 1}}{2} + i.
 \]
Then $(\mathbb{C}^{m}, \Gamma_{[1]}, \Gamma_{[2]})$ is a boundary triplet 
for $L_{\rm min}$.
\end{lemma}

\begin{remark} \rm
The values of the coefficients $\alpha$, $\beta$, $\gamma$, $\delta$
for the odd case may be replaced by an arbitrary set of numbers satisfying 
the conditions
\begin{equation}\label{PGZ coef}
\begin{gathered}
    \alpha\overline{\gamma} + \overline{\alpha}\gamma = (-1)^{n},\quad
    \beta\overline{\delta} + \overline{\beta}\delta = (-1)^{n + 1}, \quad
    \alpha\overline{\delta} + \overline{\beta}\gamma = 0, \\
    \beta\overline{\gamma}  + \overline{\alpha}\delta = 0, \quad
    \alpha\delta - \beta\gamma \neq 0.
\end{gathered}
\end{equation}
\end{remark}

\begin{proof}
We need to check that the triplet $(\mathbb{C}^{m}, \Gamma_{[1]}, \Gamma_{[2]})$
satisfies the conditions $1)$ and $2)$
in Definition \ref{PGZdef} for  $T=L_{\rm min}$
and $\mathcal{H} = L_2([a,b], \mathbb{C})$.
Due to Theorem \ref{thm_L_adjoint}, $L^*_{\rm min} = L_{\rm max}$.

Let us start with the case of even order.
Due to Lemma \ref{lm_Lagrange}, for $m =2n$:
\[
( {L_{\rm max}y,z} ) - ( y,L_{\rm max}z)  =
i^{2n}\sum_{k = 1}^{2n}(-1)^{k - 1}{D^{[2n - k]}y\cdot\overline {D^{[k -1]}z}}
\big|_{t = a}^{t = b}.
\]
Denote
\[
\Gamma_{[1]} =: (\Gamma_{1a}, \Gamma_{1b}),\quad
\Gamma_{[2]} =: (\Gamma_{2a}, \Gamma _{2b}),
\]
where
\begin{gather*}
\Gamma_{1a}y = i^{2n}( - D^{[2n - 1]}y(a), \dots , (-1)^{n} D^{[n]}y(a)),\\
\Gamma_{1b}y = i^{2n}(D^{[2n - 1]}y(b), \dots , (-1)^{n - 1} D^{[n]}y(b)), \\
\Gamma_{2a}y = (D^{[0]}y(a), \dots, D^{[n - 1]}y(a)),\\
\Gamma_{2b}y = (D^{[0]}y(b), \dots, D^{[n - 1]}y(b)).
\end{gather*}
One calculates
\begin{gather*}
( {\Gamma _{1a} y,\Gamma _{2a} z} )  = i^{2n}
    \sum_{k = 1}^{n}(-1)^{k}{D^{[2n - k]}y(a)\cdot\overline {D^{[k -1]}z(a)}},\\
({\Gamma _{2a} y, \Gamma _{1a} z} ) = i^{2n}
    \sum_{k = n + 1}^{2n}(-1)^{k - 1}{D^{[2n - k]}y(a)\cdot\overline {D^{[k -1]}z(a)}},\\
( {\Gamma _{1b} y,\Gamma _{2b} z} )  = i^{2n}
    \sum_{k = 1}^{n}(-1)^{k - 1}{D^{[2n - k]}y(b)\cdot\overline {D^{[k -1]}z(b)}},\\
({\Gamma _{2b} y, \Gamma _{1b} z} ) = i^{2n}
    \sum_{k = n + 1}^{2n}(-1)^{k}{D^{[2n - k]}y(b)\cdot\overline {D^{[k -1]}z(b)}},
\end{gather*}
which results in
\begin{align*}
( {\Gamma _{[1]} y,\Gamma _{[2]} z} ) &= i^{2n}\sum_{k = 1}^{n}
(-1)^{k - 1}{D^{[2n - k]}y\cdot\overline {D^{[k -1]}z}}\big|_{t = a}^{t = b},\\
({\Gamma _{[2]} y,\Gamma _{[1]} z} ) &= i^{2n}\sum_{k = n + 1}^{2n}
(-1)^{k}{D^{[2n - k]}y\cdot\overline {D^{[k -1]}z}}\big|_{t = a}^{t = b}.,
\end{align*}
and this means that the condition (1) of the Definition \ref{PGZdef} is fulfilled,
and the surjectivity condition (2) is true due to Lemma \ref{lm_surj}.

The case of odd order is treated similarly.
Due to Lemma \ref{lm_Lagrange}, for $m = 2n + 1$ we have
\[
( {L_{\rm max}y,z} ) - ( y,L_{\rm max}z)  =
i^{2n + 1}\sum_{k = 1}^{2n + 1}(-1)^{k - 1}{D^{[2n - k]}y
\cdot\overline {D^{[k -1]}z}}\big|_{t = a}^{t = b}.
\]
Denote
\[
\Gamma_{[1]} =: (\Gamma_{1a}, \Gamma_{1b}, \Gamma_{1ab}),\quad
\Gamma_{[2]} =: (\Gamma_{2a}, \Gamma_{2b}, \Gamma_{2ab}),
\]
where
\begin{gather*}
\Gamma_{1a}y = i^{2n + 1}( - D^{[2n]}y(a), \dots , (-1)^{n} D^{[n + 1]}y(a)),\\
\Gamma_{1b}y = i^{2n + 1}(D^{[2n]}y(b), \dots , (-1)^{n + 1} D^{[n + 1]}y(b)), \\
\Gamma _{1ab}y = i^{2n + 1}(\alpha D^{[n]}y(b) + \beta D^{[n]}y(a)),\\
\Gamma _{2a}y = (D^{[0]}y(a), \dots, D^{[n - 1]}y(a)),\\
\Gamma _{2b}y = (D^{[0]}y(b), \dots, D^{[n - 1]}y(b)),\\
\Gamma _{2ab}y = \gamma D^{[n]}y(b) + \delta D^{[n]}y(a).
\end{gather*}
One calculates
\begin{gather*}
( {\Gamma _{1a} y,\Gamma _{2a} z} )  = i^{2n + 1}
    \sum_{k = 1}^{n}(-1)^{k - 1}{D^{[2n - k]}y(a)\cdot\overline {D^{[k -1]}z(a)}},
\\
({\Gamma _{2a} y, \Gamma _{1a} z} ) = i^{2n + 1}
    \sum_{k = n + 2}^{2n + 1}(-1)^{k}{D^{[2n - k]}y(a)
\cdot\overline {D^{[k -1]}z(a)}},
\\
( {\Gamma _{1b} y,\Gamma _{2b} z} )  = i^{2n + 1}
    \sum_{k = 1}^{n}(-1)^{k - 1}{D^{[2n - k]}y(b)\cdot\overline {D^{[k -1]}z(b)}},
\\
({\Gamma _{2b} y, \Gamma _{1b} z} ) = i^{2n + 1}
    \sum_{k = n + 2}^{2n + 1}(-1)^{k}{D^{[2n - k]}y(b)\cdot
\overline {D^{[k -1]}z(b)}},\\
\begin{aligned}
&( {\Gamma_{1ab} y,\Gamma_{2ab} z} ) - ( {\Gamma _{2ab} y,\Gamma _{1ab} z} ) \\
&= i^{2n + 1}(-1)^{n} (D^{[n]}y(b)\cdot\overline {D^{[n]}z(b)} - D^{[n]}
y(a)\cdot\overline {D^{[n]}z(a)}),
\end{aligned}
\end{gather*}
which shows that the condition (1) of Definition \ref{PGZdef} is satisfied.
Now take arbitrary vectors $f_1=(f_{1,k})_{k = 0}^{2n}$,
$f_2=(f_{2,k})_{k = 0}^{2n} \in \mathbb{C}^{2n + 1}$.
The last condition in \eqref{PGZ coef} means that the system
\[
\begin{gathered}
\alpha \beta_n + \beta \alpha_n = f_{1, n}\\
\gamma \beta_n + \delta \alpha_n = f_{2, n}
\end{gathered}
\]
has a unique solution $(\alpha_n,\beta_n)$.
Denoting
\begin{gather*}
\alpha_k := f_{1, k}, \quad\beta_k := f_{2, k} \quad \text{for }k < n,\\
\alpha_k  := (-1)^{2n + 1 - k}f_{1, k},\quad 
\beta_k := (-1)^{2n - k}f_{2, k}\quad  \text{for } n + 1 < k < 2n
\end{gather*}
we obtain two vectors
$(\alpha _0 ,\alpha _1, \dots, \alpha _{m - 1})$,
$(\beta _0 ,\beta _1, \dots, \beta_{m - 1})\in\mathbb{C}^m
\equiv \mathbb{C}^{2n+1}$.
By Lemma \ref{lm_surj},
there exists a function ${y \in \operatorname{Dom}(L_{\rm max})}$ such that
\[
D^{[k]}y(a) = \alpha _k , \quad D^{[k]}y(b) = \beta _k, \quad
 k = 0,1, \dots, m - 1,
\]
and due to above special choice of $\alpha$ and $\beta$ one has
$\Gamma_{[1]}y=f_1$ and $\Gamma_{[2]}y=f_2$, so the surjectivity condition
of Definition \ref{PGZdef} holds.
\end{proof}

For the sake of convenience, we introduce the following notation. 
Denote by $L_K$ the restriction of
$L_{\rm max}$ onto the set of the functions
$y(t) \in \operatorname{Dom}(L_{\rm max})$ satisfying the homogeneous 
boundary condition in the canonical form
\begin{equation} \label{diss_ext}
( K - I )\Gamma_{[1]} y + i( K + I )\Gamma_{[2]} y = 0.
\end{equation}
Similarly, denote by $L^K$ the restriction of $L_{\rm max}$ onto the set 
of the functions
$y(t) \in \operatorname{Dom}(L_{\rm max})$ satisfying the boundary condition
\begin{equation} \label{akk_ext}
 ( K - I )\Gamma_{[1]} y - i( K + I )\Gamma_{[2]} y = 0.
\end{equation}
Here  $K$ is an arbitrary bounded operator on the Hilbert space 
$\mathbb{C}^{m}$, and the maps $\Gamma_{[1]}$ è $\Gamma_{[2]}$ 
are defined by the formulas \eqref{Gamma 2n} or \eqref{Gamma 2n+1}
depending on $m$.
Theorem \ref{thm_Gorb} and Lemma \ref{PGZth} lead to the following description
of extensions of  $L_{\rm min}$.

\begin{theorem}\label{thm_ext}
Every $L_K$ with $K$ being a contracting operator in $\mathbb{C}^{m}$,
 is a maximal dissipative extension of   $L_{\rm min}$.
Similarly every $L^K$ with $K$ being a contracting operator in $\mathbb{C}^{m}$,
is a maximal accumulative extension of the operator $L_{\rm min}$.
Conversely, for any maximal dissipative (respectively, maximal accumulative) 
extension $\widetilde{L}$
of the operator $L_{\rm min}$ there exists a contracting operator $K$ such that
$\widetilde{L} = L_K$\,\, (respectively, $\widetilde{L} = L^K$).
The extensions $L_K$ and $L^K$ are self-adjoint if and only if $K$ 
is a unitary operator on $\mathbb{C}^{m}$.
These correspondences between operators $\{K\}$ and the extensions 
$\{\widetilde{L}\}$ are all bijective.
\end{theorem}


\begin{remark} \rm
Self-adjoint extensions of a symmetric minimal quasi-differential operator 
were described by means of the Glasman-Krein-Naimark theory in the 
work \cite{Zettl-75} and several subsequent papers.
However, the description by means of boundary triplets has important advantages, 
namely, it gives a bijective parametrization of extensions by unitary operators,
and one can describe the maximal dissipative and the maximal accumulative 
extensions in a similar way.
\end{remark}

\section{Real extensions}

Recall that a linear operator $L$ acting in $L_2([a,b], \mathbb{C})$ 
is called \textit{real} if:
\begin{enumerate}
\item  For every function $f$ from  $\operatorname{Dom}(L)$ 
the complex conjugate function $\overline{f}$
also lies in $\operatorname{Dom}(L)$.

\item The operator $L$ maps complex conjugate functions into complex
 conjugate functions, that is
$L(\overline{f}) = \overline{L(f)}$.
\end{enumerate}

If the minimal quasi-differential operator is real, one arrives at the 
natural question on how to describe its real extensions. 
The following theorem holds.

\begin{theorem} \label{thm4}
Let $m$ be even, and let the entries of the Shin-Zettl matrix $A = A^+$ 
be real-valued, then the maximal and minimal quasi-differential operators
 $L_{\rm max}$ and $L_{\rm min}$ generated by $A$ are real. 
All real maximal dissipative and maximal accumulative extensions of the
 real symmetric quasi-differential operator
$L_{\rm min}$ of the even order are self-adjoint.
The self-adjoint extensions $L_K$ or $L^K$ are real if and only 
if the unitary matrix $K$ is symmetric.
\end{theorem}

\begin{proof}
As the coefficients of the quasi-derivatives are real-valued functions, 
one has
\[
D^{[i]}\overline{y} = \overline{D^{[i]}y}, \quad i = \overline{1, 2n},
\]
which implies $l(\overline{y}) = \overline{l(y)}$.
Thus for any $y \in \operatorname{Dom}(L_{\rm max})$ we have
\[
D^{[i]}\overline{y} \in AC([a,b], \mathbb{C}), \quad
i = \overline{1, 2n - 1}, \quad 
l(\overline{y})  \in L_2([a,b], \mathbb{C}), \quad
L_{\rm max}(\overline{y}) = \overline{L_{\rm max}(y)}.
\]
This shows that the operator $L_{\rm max}$ is real.
Similarly, for $y \in \operatorname{Dom}(L_{\rm min})$ we have
\[
D^{[i]}\overline{y}(a) =  \overline{D^{[i]}y}(a) = 0, \quad 
D^{[i]}\overline{y}(b) =  \overline{D^{[i]}y}(b) = 0,
\quad i = \overline{1, 2n - 1},
\]
which proves that $L_{\rm min}$ is a real as well.

Due to the coefficients of the quasi-derivatives being real-valued,
 the equalities \eqref{Gamma 2n} imply
\[
\Gamma_{[1]}\overline{y} = \overline{\vphantom{\Gamma^1}\Gamma_{[1]}y}, \quad
\Gamma_{[2]}\overline{y} = \overline{\vphantom{\Gamma^1}\Gamma_{[2]}y}.
\]
As the maximal operator is real, any of its restrictions satisfies the condition 
(2) of the above definition of a real operator, so we are reduced to 
check the condition (1).

Let $L_K$ be an arbitrary real maximal dissipative extension given by 
the boundary conditions \eqref{diss_ext},
then for any $y \in \operatorname{Dom}(L_K)$ the complex conjugate 
$\overline{y} $ satisfies \eqref{diss_ext} too;
that is,
\[
( K - I )\Gamma_{[1]} \overline{y}  + i( K + I )\Gamma_{[2]} \overline{y}  = 0.
\]
By taking the complex conjugates we obtain
\[
(\overline{K\vphantom{K^1}} - I)\Gamma_{[1]} y  -
	i(\overline{K\vphantom{K^1}} + I)\Gamma_{[2]}y  = 0,
\]
and $L_K \subset L^{\overline{K}}$ due to Theorem \ref{thm_ext}.
Thus, the dissipative extension $L_K$ is also accumulative,
which means that it is symmetric. But $L_K$ is a maximal
dissipative extension of $L_{\rm min}$.
As the deficiency indices of $L_{\rm min}$ are finite,
the operator $L_K = L^{\overline{K}}$ must be self-adjoint.
Furthermore, due to Remark \ref{remark_K} the equality 
$L_K = L^{\overline{K}}$ is equivalent
to $K^{-1} = \overline{K}$. As $K$ is unitary, we have 
$K^{-1} = \overline{K^T}$, which gives
$K= K^T$.
In a similar way one can show that a maximal accumulative extension 
$L^K$ is real if and only if it is
self-adjoint and $K$ = $K^T$. 
\end{proof}

\section{Separated boundary conditions}

Now we would like to discuss the extensions defined by the so-called 
separated boundary conditions.
Denote by $\mathbf{f_a}$ the germ of a continuous function $f$ at the 
point $a$. We recall that the boundary conditions that define an operator 
$L \subset L_{\rm max}$ are called \emph{separated} if
for any $y \in \operatorname{Dom}(L)$ and any
$g, h \in \operatorname{Dom}(L_{\rm max})$ with
\[
\mathbf{g_a} = \mathbf{y_a},\quad  \mathbf{g_b} = 0,\quad
\mathbf{h_a} = 0,\quad  \mathbf{h_b} = \mathbf{y_b}.
\]
we have $g, h \in \operatorname{Dom}(L)$.

The following statement gives a description of the operators $L_K$ and
 $L^K$ with separated boundary conditions in the case of an even order $m = 2n$ .

\begin{theorem}\label{thm_divided adj}
The boundary conditions \eqref{diss_ext} and \eqref{akk_ext} defining 
$L_K$ and $L^K$ respectively are separated if and only if
the matrix $K$ has the block form
\begin{equation} \label{separable cond}
K = \begin{pmatrix}
  K_a & 0 \\
  0 & K_b 
\end{pmatrix},
\end{equation}
where $K_a$ and $K_b$ are $n\times n$ matrices.
\end{theorem}

\begin{proof}
We consider the operators $L_K$ only, the case of $L^K$ can be considered 
in a similar way.
We start with the following observation.
Let $y, g, h \in \operatorname{Dom}(L_{\rm max})$.
Then one can prove by induction that
 $\mathbf{y_a} = \mathbf{g_a}$ if and only if
$\mathbf{D^{[k]}y_a} = \mathbf{D^{[k]}g_a}$, $k = 0, 1, \dots, m$
and, similarly, $\mathbf{y_b} = \mathbf{h_b}$ if and only if
$\mathbf{D^{[k]}y_b} = \mathbf{D^{[k]}h_b}$, $k = 0, 1, \dots, m$.
Therefore, the equality $\mathbf{y_a} = \mathbf{g_a}$ implies 
$\Gamma_{1a}y = \Gamma_{1a}g$ and $\Gamma_{2a}y = \Gamma_{2a}g$,
and the equality $\mathbf{y_b} = \mathbf{h_b}$ implies 
$\Gamma_{1b}y = \Gamma_{1b}h$ and $\Gamma_{2b}y = \Gamma_{2b}h$.

If $K$ has the form \eqref{separable cond}, then the boundary
condition \eqref{diss_ext} can be rewritten as a system:
\begin{gather*}
(K_a - I)\Gamma_{1a}y + i(K_a + I)\Gamma_{2a}y  = 0,\\
 -(K_b - I)\Gamma_{1b}y + i(K_b + I)\Gamma_{2b}y  = 0,
\end{gather*}
and these boundary conditions are obviously separated.

Inversely, let the boundary conditions \eqref{diss_ext} be separated.
Let us represent $K \in \mathbb{C}^{2n \times 2n}$ in the block form
$$
K = \begin{pmatrix}
  K_{11} & K_{12} \\
  K_{21} & K_{22} 
\end{pmatrix}.
$$
with $n\times n$ blocks $K_{jk}$. We need to show that $K_{12} = K_{21} = 0$.
The boundary conditions \eqref{diss_ext} take the form
\begin{gather*}
(K_{11} - I)\Gamma_{1a}y + K_{12}\Gamma_{1b}y + i( K_{11} + I)\Gamma_{2a}y 
+ iK_{12}\Gamma_{2b}y  = 0,\\
K_{21}\Gamma_{1a}y + (K_{22} - I)\Gamma_{1b}y + iK_{21}\Gamma_{2a}y 
+ i(K_{22} + I)\Gamma_{2b}y  = 0.
\end{gather*}
By definition, any function $g$ with
$\mathbf{g_a} = \mathbf{y_a}$ and $\mathbf{g_b} = 0$ must also satisfy 
this system, which gives
\begin{gather*}
K_{11} [\Gamma_{1a}y + i\Gamma_{2a}y]= \Gamma_{1a}y - i\Gamma_{2a}y,\\
K_{21}[\Gamma_{1a}y + i\Gamma_{2a}y] = 0.
\end{gather*}
 Therefore, $\Gamma_{1a}y + i\Gamma_{2a}y \in \operatorname{Ker}(K_{21})$ for any
$y \in \operatorname{Dom}(L_K)$.

Now rewrite  \eqref{diss_ext} in a parametric form.
 For any $F = (F_1, F_2) \in \mathbb{C}^{2n}$
 consider the vectors $-i (K + I)F$ and $(K - I)F$.
Due to Lemma \ref{lm_surj} there is a function 
$y_F \in \operatorname{Dom}(L_{\rm max})$ such that
\begin{equation}\label{ext_parametric}
\begin{gathered}
-i (K + I)F   = \Gamma _{[1]} y_F,\\
(K - I)F  = \Gamma _{[2]} y_F.
\end{gathered}
\end{equation}
A simple calculation shows that $y_F$ satisfies the boundary 
conditions \eqref{diss_ext} and, therefore,
${y_F \in \operatorname{Dom}(L_K)}$.
We can rewrite \eqref{ext_parametric} as a system,
\begin{gather*}
- i(K_{11} + I)F_1 - iK_{12}F_2  = \Gamma_{1a}y_F,\\
-iK_{21}F_1 - i(K_{22} + I)F_2  = \Gamma_{1b}y_F,\\
(K_{11} - I)F_1 + K_{12}F_2  = \Gamma_{2a}y_F,\\
K_{21}F_1 + (K_{22} - I)F_2  = \Gamma_{2b}y_F.
\end{gather*}
The first and the third equations show that
$\Gamma_{1a}y + i\Gamma_{2a}y = -2iF_1$ for any $F_1 \in \mathbb{C}^{n}$.
Therefore, $\operatorname{Ker}(K_{21}) = \mathbb{C}^{n}$ which means 
$K_{21} = 0$. The equality  $K_{12} = 0$ is proved in the same way.
\end{proof}

\section{Generalized resolvents}

Let us recall that a \emph{generalized resolvent} of a closed symmetric 
operator $L$ in a Hilbert space $\mathcal{H}$ is an operator-valued 
function $\lambda\mapsto R_\lambda$ defined on $\mathbb{C} \setminus \mathbb{R}$
which can be represented as
\[
R_\lambda f = P^+ ( L^+ - \lambda I^+)^{- 1}f, \quad f \in \mathcal{H},
\]
where $L^+$ is a self-adjoint extension  $L$ which acts a certain Hilbert 
space $\mathcal{H}^+\supset\mathcal{H}$,
$I^+$ is the identity operator on $\mathcal{H}^+$,
and $P^+$ is the orthogonal projection operator from $\mathcal{H}^+$ 
onto $\mathcal{H}$.
It is known \cite{Ahiezer-eng} that an operator-valued function
$R_\lambda$ is a generalized resolvent of a symmetric operator $L$ 
if and only if it can be represented as
\[
( R_\lambda f, g )_\mathcal{H} 
= \int_{-\infty}^{+\infty}\frac{d(F_\mu f, g)}{\mu - \lambda},
  \quad f, g \in \mathcal{H},
\]
where $F_\mu$ is a generalized spectral function of the operator $L$; i.e.,
$\mu\mapsto F_\mu$ is an operator-valued function
 $F_\mu$ defined on $\mathbb{R}$ and taking
values in the space of continuous linear operators in $\mathcal{H}$
with the following properties:
\begin{enumerate}
\item  For $\mu_2 > \mu_1$, the difference $F_{\mu_2} - F_{\mu_1}$
 is a bounded non-negative operator.
\item $F_{\mu +} = F_\mu$ for any real $\mu$.
\item For any $x \in \mathcal{H}$ there holds
\[ 
\lim_{\mu \rightarrow - \infty}^{}||F_\mu x ||_\mathcal{H} = 0,
 \quad \lim_{\mu \rightarrow + \infty}^{} ||{F_\mu x - x} ||_\mathcal{H} = 0.
\]
\end{enumerate}

The following theorem provides a description of all generalized resolvents of
 the operator $L_{\rm min}$.

\begin{theorem}\label{thm_gener resolvent}
(1) Every generalized resolvent $R_\lambda$ of the operator $L_{\rm min}$
in the half-plane $\operatorname{Im}\lambda < 0$
acts by the rule $R_\lambda h = y$, where $y$ is the solution of the 
boundary-value problem
\begin{gather*}
l(y) = \lambda y + h,\\
 ( {K(\lambda) - I} )\Gamma _{[1]} f + i( {K(\lambda) + I} )\Gamma _{[2]} f = 0.
\end{gather*}
Here $h(x) \in L_2([a,b], \mathbb{C})$ and
$K(\lambda)$ is an $m\times m$ matrix-valued function which is holomorph
in the lower half-plane and satisfy $||K(\lambda)|| \leq 1$.

(2) In the half-plane $\operatorname{Im}\lambda > 0$, every generalized 
resolvent of $L_{\rm min}$ acts by $R_\lambda h = y$,
where $y$ is the solution of the boundary-value problem
\begin{gather*}
l(y) = \lambda y + h,\\
 ( {K(\lambda) - I} )\Gamma _{[1]} f - i( {K(\lambda) + I} )\Gamma _{[2]} f = 0.
\end{gather*}
Here $h(x) \in L_2([a,b], \mathbb{C})$ and
$K(\lambda)$ and
$K(\lambda)$ is an $m\times m$ matrix-valued function which is holomorph
in the lower half-plane and satisfy $\|K(\lambda)\| \leq 1$.

The parametrization of the generalized resolvents by the matrix-valued 
functions $K$ is bijective.
\end{theorem}

\begin{proof}
The Theorem is just an application of Lemma \ref{PGZth} and 
\cite[Theorem 1, Remark 1]{Bruk-76}
which prove a description of generalized resolvents in terms of 
boundary triplets.
Namely, one requires to take as an auxiliary Hilbert space $\mathbb{C}^m$ 
and as the operator $\gamma y := \{\Gamma_{[1]}y, \Gamma_{[2]}y\}$.
\end{proof}

\subsection*{Acknowledgments}
A. Goriunov was partially supported by the grant no. 01/01-12 from the
 National Academy of Sciences of Ukraine 
(under the joint Ukrainian-Russian project of NAS of Ukraine 
and Russian Foundation for Basic Research). 
V. Mikhailets was partially supported by the grant no. 03/01-12 from the
National Academy of Sciences of Ukraine 
(under the joint Ukrainian-Russian project of NAS of Ukraine and 
Syberian Branch of Russian Academy of Sciences). 


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\end{document}