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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 136, pp. 1--17.\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-2017/136\hfil Spectral analysis]
{Spectral analysis of $q$-fractional Sturm-Liouville operators}

\author[B. P. Allahverdiev, H. Tuna \hfil EJDE-2017/136\hfilneg]
{Bilender P. Allahverdiev, H\"{u}seyin Tuna}

\address{Bilender P. Allahverdiev \newline
Department of Mathematics,
 Faculty of Arts and Sciences,
S\"{u}leyman Demirel University,
32260 Isparta, Turkey}
\email{bilenderpasaoglu@sdu.edu.tr}

\address{H\"{u}seyin Tuna \newline
Department of Mathematics,
Faculty of Arts and Sciences,
Mehmet Akif Ersoy University,
15030 Burdur, Turkey}
\email{hustuna@gmail.com}

\dedicatory{Communicated by Mokhtar Kirane}

\thanks{Submitted April 4, 20017. Published May 18, 2017.}
\subjclass[2010]{47A20, 47A40, 47A45, 34B05, 34B10, 34L10, 47E05}
\keywords{Dissipative $q$-fractional Sturm-Liouville operator; dilation;
\hfill\break\indent  eigenvector; scattering matrix; functional model; characteristic function}

\begin{abstract}
 In this article, we study $q$-fractional Sturm-Liouville operators. Using by
 the functional method, we pass to a new operator. Then, showing that this
 operator is a maximal operator and constructing a self-adjoint dilation
 of the maximal dissipative operator. We prove a theorem on the completeness
 of the system of eigenvectors and associated vectors of the dissipative
 $q$-fractional Sturm-Liouville operators.
\end{abstract}

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

\section{Introduction}

It is well known that many problems in mechanics, engineering and
mathematical physics lead to the concept of completeness of root functions
and basis properties of all or part of the eigenvectors and associated
vectors corresponding to some operators. In many engineering applications,
the Sturm-Liouville problems arise as boundary value problems. These
problems and many of the associated theories were presented in 1800's
(see \cite{rsj,bml,az}), and since then, the related fields such
as fractional Sturm-Liouville operators and $q$-fractional Sturm-Liouville
operators have attracted considerable interest in a variety of applied
sciences and mathematics (see \cite{rhe,aak,mkop,ksm} and the references therein).

Spectral theory is one of the major subjects of modern functional analysis
and its applications in mathematics. So there has recently been a noticeable
interest in spectral analysis of Sturm-Liouville boundary value problems
(see \cite{ku,p3,bpa,bpa6,man,ssb, ug1} and the references therein).

There are some methods to give the completeness of non-self-adjoint
(dissipative) operators, such as the method of contour integration of the
resolvent, Lidskij's method, functional model method, etc. In this paper, we
prove a theorem on the completeness of the system of eigenvectors and
associated vectors of dissipative operators by using functional model theory
that belongs to Sz.-Nagy-Foia\c{s}. It is related with the equivalence of
the Lax-Phillips scattering function technique and Sz.-Nagy-Foia\c{s}
characteristic function.\ By combining the results of Nagy-Foia\c{s}
\cite{18} and Lax-Phillips \cite{17}, the characteristic function is expressed
with a scattering matrix, and the dissipative operator in the spectral
representation of dilation becomes the model. By means of different spectral
representation of dilation, the given operator can be written very simply
and the functional models are obtained. The eigenvalues, eigenvectors and
the spectral projection of the model operator is expressed obviously by the
characteristic function. In the centre of this method, the problem on the
completeness of the system of eigenvectors is solved by writing the
characteristic function as a factorization. That is, the factorization of
the characteristic function gives us information about whether the system of
all eigenvectors and associated vectors is complete or not. This approach is
applied for dissipative Schr\"{o}dinger operators, Sturm-Liouville
operators, fractional Sturm-Liouville operators and difference
Sturm-Liouville operators (see \cite{bpa,bpa6,zm2,ssb, ug1}).

In this article, we apply it to the $q$-fractional Sturm-Liouville operator.
To do this, we form a new operator and show that this new operator is a
maximal dissipative operator. Then, we construct a functional model of the
dissipative operator by means of the incoming and outgoing spectral
representations, and define its characteristic function because this makes
it possible to determine the scattering matrix of dilation according to the
Lax and Phillips scheme. Finally, we prove a theorem on the completeness of
the system of eigenvectors and associated vectors of dissipative operators.

\section{Preliminaries}

In this section, we recall some basic definitions and properties of the
fractional calculus theory, which are useful in the following discussion.
These definitions and properties can be found in
\cite{ankitap,er,kac,ra,vla,vla1,zm2,ip,sgs} and the references therein.

Let $q\ $be a positive number with $0<$ $q<1$, $A\subset \mathbb{R}$,
\begin{gather*}
A_{t,q}:=\{ tq^{n}:n\in \mathbb{N}\} ,\text{ }A_{t,q}^{\ast
}:=A_{t,q}\cup \{ 0\} ,\quad t>0, \\
\mathcal{A}_{t,q}:=\{ \pm tq^{n}:n\in \mathbb{N}\} ,\quad t>0.
\end{gather*}

Let $y(\cdot) $ be\ a complex-valued function on $A$.
The $q$\emph{-difference operator} $D_{q}$ is defined by
\begin{equation*}
D_{q}y( x) =\frac{y( qx) -y( x) }{\mu( x) }\quad
\text{for all }x\in A\backslash \{ 0\} ,
\end{equation*}
where $\mu ( x) =( q-1) x$. The $q$-derivative at zero
is defined by
\begin{equation*}
D_{q}y( 0) =\lim_{n\to \infty }\frac{y(q^{n}x) -y( 0) }{q^{n}x}\quad
(x\in A),
\end{equation*}
if the limit exists and does not depend on $x$. A \emph{right-inverse} to
$D_{q}$, the \emph{Jackson} $q$-\emph{integration} is given by
\begin{equation*}
\int_0^{x}f( t) d_{q}t
=x( 1-q) \sum_{n=0}^{\infty}q^{n}f( q^{n}x) \quad (x\in A),
\end{equation*}
provided that the series converges, and
\begin{equation*}
\int_{a}^{b}f( t) d_{q}t=\int_0^{b}f( t)
d_{q}t-\int_0^{a}f( t) d_{q}t\quad (a,b\in A).
\end{equation*}

Let $L_{q}^2(0,a)$ be the space of all complex-valued functions defined
on $[0,a]$ such that
\begin{equation*}
\| f\| :=\Big( \int_0^{a}| f( x)| d_{q}x\Big) ^{1/2}<\infty .
\end{equation*}
The space $L_{q}^2(0,a)$ is a separable Hilbert space with the inner
product
\begin{equation}
( f,g) :=\int_0^{a}f( x) \overline{g( x) }d_{q}x,\quad f,g\in L_{q}^2(0,a),
 \label{inner}
\end{equation}
and the orthonormal basis
\begin{equation*}
\phi _n( x) =\begin{cases}
\frac{1}{\sqrt{x( 1-q) }}, & x=aq^{n}, \\
0, & \text{otherwise,}
\end{cases}
\end{equation*}
where $n=0,1,2,\dots$ (see \cite{ankitap}).

\begin{definition}\label{def1} \rm
A function $f$ which is defined on $A$, $0\in A$, is said to be $q$-regular
at zero if
\begin{equation*}
\lim_{n\to \infty }f( xq^{n}) =f( 0)
\end{equation*}
for every $x\in A$ (see \cite{ankitap}).
\end{definition}

Let $C( A) $ denote the space of all $q$-regular at zero
functions on $A$. This space is a normed space with the norm function
\begin{equation*}
\| f\| =\sup \{ | f( xq^{n})| : x\in A,\; n\in \mathbb{N}\} .
\end{equation*}
(see \cite{ankitap}).

\begin{definition}\label{def2} \rm
A $q$-regular at zero function $f$ which is defined on $A_{t,q}^{\ast }$ is
said to be $q$-absolutely continuous if
\begin{equation*}
\sum_{j=0}^{\infty }| f( uq^{j}) -f(uq^{j+1}) | \leq K,\quad
\forall u\in A_{t,q}^{\ast },
\end{equation*}
where $K$ is a constant depending on the function $f$ (see \cite{ankitap}).
\end{definition}

The space of $q$-absolutely continuous functions on $A_{t,q}^{\ast }$ is
denoted by $AC_{q}( A_{t,q}^{\ast }) $.

For $n\in \mathbb{N}$ and $\alpha ,a_1,\dots,a_n\in \mathbb{C}$,
 the $q$-\emph{shifted factorial}, the \emph{multiple} $q$-\emph{shifted
factorial} and the $q$-\emph{binomial coefficients} are defined by
\begin{gather*}
( a;q) _0=1,\quad
 ( a;q)_n=\prod_{k=0}^{n-1}( 1-aq^{k}) ,\quad
 ( a;q)_{\infty }=\prod_{k=0}^{\infty }( 1-aq^{k}) , \\
( a_1,a_2,\dots,a_k:q) =\prod_{j=1}^{k}(a_j;q) _n, \\
\begin{bmatrix} \alpha \\ 0 \end{bmatrix}_{q}=1,\quad
\begin{bmatrix} \alpha \\ n\end{bmatrix}_{q}
=\frac{(1-q^{\alpha }) ( 1-q^{\alpha -1}) \dots( 1-q^{\alpha
-n+1}) }{( q;q) _n},
\end{gather*}
respectively (see \cite{ankitap}). The \emph{generalized}
$q$-\emph{shifted factorial} is defined by
\begin{equation*}
( a;q) _{\nu }=\frac{( a;q) _{\infty }}{( aq^{\nu
};q) _{\infty }}\quad (\nu \in \mathbb{R)}
\end{equation*}
(see \cite{ankitap}). The $q$-\emph{Gamma function} is defined by
\begin{equation*}
\Gamma _{q}( z) =\frac{( q;q) _{\infty }}{(q^{z};q) _{\infty }}( 1-q) ^{1-z},\quad
 z\in \mathbb{C},\; | q| <1
\end{equation*}
(see \cite{ankitap}).

\begin{definition} \label{def3} \rm
Let $0<\alpha \leq 1$. The left-sided and right-sided Riemann-Liouville
$q$-fractional operator are given by the formulas
\begin{gather}
J_{q,a^{+}}^{\alpha }f( x)
=\frac{x^{\alpha -1}}{\Gamma_{q}( \alpha ) }
 \int_a^x ( \frac{qt}{x};q) _{\alpha -1}f( t) d_{q}t, \\
J_{q,b^{-}}^{\alpha }f( x)
=\frac{1}{\Gamma _{q}( \alpha ) } \int_{qx}^b t^{\alpha -1}( \frac{qx}{t}
;q) _{\alpha -1}f( t) d_{q}t,
\end{gather}
respectively \textbf{(}see \cite{zm2}\textbf{)}.
\end{definition}

\begin{definition} \label{def4} \rm
Let $\alpha >0$ and $\lceil \alpha \rceil =m$. The left-sided and
right-sided Riemann-Liouville fractional $q$-derivatives of order $\alpha $
are defined, respectively, as follows:
\begin{gather}
D_{q,a^{+}}^{\alpha }f( x)
=D_{q}^{m}J_{q,a^{+}}^{m-\alpha}f( x) ,\text{\ } \\
D_{q,b^{-}}^{\alpha }f( x) =\big( \frac{-1}{q}\big)
^{m}D_{q^{-1}}^{m}J_{q,b^{-}}^{m-\alpha }f( x) .
\end{gather}
Similar formulas give the left-sided and right-sided Caputo fractional
$q$-derivatives of order $\alpha $, respectively as follows:
\begin{gather*}
^{c}D_{q,a^{+}}^{\alpha }f( x)
=J_{q,a^{+}}^{m-\alpha}D_{q}^{m}f( x) , \\
^{c}D_{q,b^{-}}^{\alpha }f( x)
=( \frac{-1}{q})^{m}J_{q,b^{-}}^{m-\alpha }D_{q^{-1}}^{m}f( x)
\end{gather*}
(see \cite{zm2}).
\end{definition}

\begin{theorem} \label{thm1}
(i) The left-sided Riemann-Liouville $q$-fractional operator satisfies the
semi-group property
\begin{equation}
J_{q,a^{+}}^{\alpha }J_{q,a^{+}}^{\beta }=J_{q,a^{+}}^{\alpha +\beta
}f( x) ,\quad x\in A_{q,a},
\end{equation}
for any function defined on $A_{q,a}$  and for any values of $\alpha $ and
$\beta $.

(ii) The right-sided Riemann-Liouville $q$-fractional operator satisfies the
semi-group property
\begin{equation*}
J_{q,b^{-}}^{\alpha }J_{q,b^{-}}^{\beta }f( x)
=J_{q,b^{-}}^{\alpha +\beta }f( x) ,\quad x\in A_{q,b},
\end{equation*}
for any function defined on $A_{q,b}$ and for any values
of $\alpha $ and $\beta$ (see \cite{zm2}).
\end{theorem}

\begin{definition} \label{def5} \rm
An operator $T$ is called dissipative (accumulative) if
$\operatorname{Im}(Tx,x) \geq 0$, $(\operatorname{Im}( Tx,x) \leq 0)$ for all
$ x\in D( T)$ (see \cite{icg,mlg}).
\end{definition}


\begin{definition} \label{def6} \rm
The linear operator $T$ with domain $D( T) $ acting in the
Hilbert space $H$ is called simple if there is no invariant subspace
$N\subseteq D( T) $ $( N\neq \{ 0\} ) $ of
the operator $T$ on which the restriction $T$ to $N$ is self-adjoint
(see \cite{bpa2}).
\end{definition}

\begin{definition} \label{def7} \rm
A triple $( \mathcal{H},\Lambda _1,\Lambda _2) $ is called a
space of boundary values of a closed symmetric operator $T$ on a Hilbert
space $H$\ if $\Lambda _1,\Lambda _2$ are linear maps from
$D(T^{\ast }) $ to $\mathcal{H}$ with equal deficiency numbers such that:
\begin{itemize}
\item[(i)] Green's formula is valid,
\begin{equation*}
( T^{\ast }f,g) _H-( f,T^{\ast }g) _H=(
\Lambda _1f,\Lambda _2g) _{\mathcal{H}}-( \Lambda
_2f,\Lambda _1g) _{\mathcal{H}},\quad f,g\in D( A^{\ast }) .
\end{equation*}

\item[(ii)] For any $F_1,F_2\in \mathcal{H}$, there is a vector
$f\in D(T^{\ast }) $ such that $\Lambda _1f=F_1$, $\Lambda _2f=F_2$
\cite{mlg}.
\end{itemize}
\end{definition}

\begin{definition} \label{def8} \rm
Let $T:H\to H$ be an operator acting on a Hilbert space $H$, and
let $U:\mathcal{H}\to \mathcal{H}$ be acting on another Hilbert
space $\mathcal{H\supset }H$. The operator $U$ is called a dilation of $A$
if
\begin{equation*}
T^{n}h=P_HU^{n}h,\quad h\in H,n\geq 0,
\end{equation*}
where $P_H$ is the orthogonal projection of $\mathcal{H}$ onto $H$. The
space $\mathcal{H}$ is called a dilation space (see \cite{18}).
\end{definition}

\begin{definition} \label{def9} \rm
Let $T$ be a symmetric operator and $\lambda $ a non-real number. The
operator
\begin{equation*}
V=( T-\lambda I) ( T-\overline{\lambda }I) ^{-1}
\end{equation*}
is called the Cayley transform of the operator $T$ (see \cite{man}).
\end{definition}

\begin{definition} \label{def10} \rm
Let $H$ be a Hilbert space and $T:H\to H$ be a contraction
operator, i.e., $\|T\| <1$. The operator
$D_T=(I-T^{\ast }T) ^{1/2}$ is called the defect operator of $T$. The
characteristic function $\theta _T$ of the contraction $A$ is defined by
\begin{equation*}
\theta _T( \xi ) =D_{T^{\ast }}( I-\xi T^{\ast })^{-1}( \xi -T) D_T
\end{equation*}
(see \cite{18}).
\end{definition}

\begin{definition}[\cite{18}]  \label{def11} \rm
The analytic function $S( \lambda ) $ on the upper half-plane
$\mathbb{C}_{+}$ is called the \emph{inner function} on
$\mathbb{C}_{+}$ if $| S( \lambda ) | \leq 1$ for all
$\lambda \in \mathbb{C}_{+}$ and $| S( \lambda ) | =1$ for almost all
$\lambda \in (-\infty ,\infty )$.
\end{definition}

\begin{definition} \label{def12} \rm
A sequence of points $( a_n)$ inside the unit disk is said to
satisfy the Blaschke condition when
\begin{equation*}
\sum_n( 1-| a_n| ) <\infty .
\end{equation*}
Given a sequence obeying the Blaschke condition, the Blaschke product is
defined as
\begin{equation*}
B( z) =\prod_nB( a_n,z)
\end{equation*}
with factors
\begin{equation*}
B( a,z) =\frac{| a| }{a}\frac{a-z}{1-\overline{
a}z}
\end{equation*}
provided that $a\neq 0$. Here $\overline{a}$ is the complex conjugate of $a$
(see \cite{18}).
\end{definition}

\begin{definition} \label{def13} \rm
For a given countable set of points $a_i$, $i=1,2,\dots$. such that
$| a_i| <1$ and $\sum_i( 1-|a_i| ) <\infty $, and an ordered family
of orthogonal projections $\{ P_i\} $ in $E$ in the disk
$|a_i| <1$ we can construct the Blaschke-Potapov product
\begin{equation*}
\Pi ( \xi ) =\prod_{k=1}^{\infty }\{ \frac{a_k-\xi
}{1-\overline{a_k}\xi }\frac{\overline{a_k}}{| a_k|
}P_k+( I-P_k) \}
\end{equation*}
which is the multi-dimensional analogue of the Blaschke product
(see \cite{ku}).
\end{definition}

\begin{definition} \label{def14} \rm
The logarithmic capacity of a compact set $E$ in the complex plane is given
by
\begin{equation*}
\gamma ( E) =e^{-V( E) },
\end{equation*}
where
\begin{equation*}
V( E) =\inf_{\nu }\int_{E\times E}\ln \frac{1}{u-v}d\nu (
u) d\nu ( v)
\end{equation*}
and $\nu $ runs over each probability measure on $E$ (see \cite{ron}).
\end{definition}

\begin{definition}[\cite{ron}] \label{def15} \rm
Let $\widetilde{E}$ be an $n-$dimensional Hilbert space
$( n<\infty) $. In $\widetilde{E}$ we fix an orthonormal basis
$e_1,e_2,\dots,e_n$ and denote by $E_k$ $( k=1,2,\dots,n) $ the
linear span of the vectors $e_1,e_2,\dots,e_k$.
 If $L\subset E_k$, then the population of $x\in E_{k-1}$ with the property
\begin{equation*}
\operatorname{Cap}\{ \lambda :\lambda \in \mathbb{C},( x+\lambda e_k)
\subset L\} >0
\end{equation*}
will be shown by $\Gamma _{k-1}L$ ($\operatorname{Cap}G$ is the inner
logarithmic capacity of a set $G\subset \mathbb{C}$).
The $\Gamma$-capacity of a set $L\subset \widetilde{E}$ is a number
\begin{equation*}
\Gamma -\operatorname{Cap}L
:=\sup \operatorname{Cap}\{ \lambda :\lambda e_1\subset \Gamma _1\Gamma _2
\dots\Gamma _{n-1}L\} ,
\end{equation*}
where the supremum is taken with respect to all orthonormal basis in
$\widetilde{E}$.
\end{definition}

It is known that every set $L\subset \widetilde{E}$ of zero
$\Gamma$-capacity has zero $2n$-dimensional Lebesgue measure; however, the
converse is not true (see \cite{bpa2}).

Now denote by $[E]$ the set of all linear operators in $E$
($\dim E=m$).
To convert $[E]$ into an $m^2-$dimensional Hilbert space, we give the
inner product $\langle T,S\rangle =trS^{\ast }T$ for $T,S\in [ E]$
($trS^{\ast }T$ is the trace of the operators $S^{\ast }T$). Hence we may
give the $\Gamma $-capacity of a set in $E$ (see \cite{bpa2}).

\section{Dilation of $q$-fractional Sturm-Liouville operator}

In this section, we construct a space of boundary value for minimal
symmetric fractional $q$-Sturm-Liouville operator and describe all
extensions (dissipative, accumulative, self-adjoint and other) of such
operators.

By $q$-fractional differential expression
\begin{equation}
\tau _{q,\alpha }y( x) :=D_{q,a^{-}}^{\alpha }p( x)
^{c}D_{q,0^{+}}^{\alpha }y( x) +r( x) y( x)
,\quad \alpha \in ( 0,1) ,   \label{3.1}
\end{equation}
consider the fractional $q$-Sturm-Liouville equation
\begin{equation}
\tau _{q,\alpha }y( x) -\lambda y( x) =0,\quad x\in
A_{t,\alpha }^{\ast },  \label{322}
\end{equation}
\newline
where $p( x) \neq 0$ for all $x\in A_{t,\alpha }^{\ast }$ and
$p,r $ are real valued functions defined in $A_{t,\alpha }^{\ast }$.
For $q\to 1$, this problem was investigated by Ery{\i}lmaz and Tuna (see
\cite{ery}).

To pass from the differential expression $\tau _{q,\alpha }y$ to operators,
we introduce the Hilbert space
$H\subseteq L_{q}^2( A_{t,\alpha}^{\ast }) \cup C( A_{t,\alpha }^{\ast })$,
$\alpha \in ( 0,1) $ with the inner product \eqref{inner}.

Let $\mathcal{L}_0\ $denote the closure of the minimal operator generated
by \eqref{3.1} and $\mathcal{D}_0$ its domain. Besides, we denote by
$\mathcal{D}$ the set of all functions $f$ from $H$ such that $f\in
AC_{q}( A_{t,q}^{\ast }) $ and $\tau _{q,\alpha }y\in H$.$\
\mathcal{D}$ is the domain of the maximal operator $\mathcal{L}$.
Furthermore, $\mathcal{L}=\mathcal{L}_0^{\ast }\ $\cite{man}.

For two arbitrary functions $y,z\in \mathcal{D}$, we have Green's
identity \cite{zm2}:
\begin{equation}
\begin{aligned}
&\int_0^{a}\left[ y( x) \tau _{q,\alpha }z( x)
-z( x) \tau _{q,\alpha }y( x) \right] d_{q}x    \\
&= [y( x) ( J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha
}z) ( \frac{x}{q}) -z( x) (
J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha }y) ( \frac{x}{q}
) ]\mid _{x=0}^{a}.
\end{aligned} \label{3.5}
\end{equation}

Let us denote by $\Lambda _1,\Lambda _2$ the linear maps from
$\mathcal{D}$ to $E:=\mathbb{C}^2$ by the formulas
\begin{equation}
\Lambda _1f=\begin{pmatrix}
-y( 0) \\
y( a)
\end{pmatrix},\quad
 \Lambda _2f=\begin{pmatrix}
J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha }y( 0) \\
J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha }y( \frac{a}{q})
\end{pmatrix},\quad
 y\in \mathcal{D}.  \label{3.6}
\end{equation}

\begin{lemma} \label{lem2}
For arbitrary $y,z\in \mathcal{D}$, one has
\begin{equation*}
( \mathcal{L}y,z) _H-( y,\mathcal{L}z) _H=(
\Lambda _1y,\Lambda _2z) _{E}-( \Lambda _2y,\Lambda
_1z) _{E}.
\end{equation*}
\end{lemma}

\begin{proof}
From Green's identity and by
\begin{gather*}
[ y,z]_{x}
=y( x) J_{q,a^{-}}^{1-\alpha}p^{c}D_{q,0^{+}}^{\alpha }
\overline{z( \frac{x}{q}) }-\overline{z( x) }J_{q,a^{-}}^{1-\alpha }
p^{c}D_{q,0^{+}}^{\alpha }y(\frac{x}{q}) , \\
y,z \in \mathcal{D},\ x\in [ 0,a] ,
\end{gather*}
 we have
\begin{equation*}
( \mathcal{L}y,z) _H-( y,\mathcal{L}z)_H=[y,z]_{a}-[y,z]_0.
\end{equation*}
Then, we obtain
\begin{align*}
&( \Lambda _1y,\Lambda _2z) _{E}-( \Lambda
_2y,\Lambda _1z) _{E} \\
&= -y( 0) J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha }
\overline{z( 0) }-(-\overline{z( 0) }
)J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha }y( 0) \\
&\quad +y( a) J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha }
\overline{z( \frac{a}{q}) }-z( a)
J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha }y( \frac{a}{q})
\\
&=[y,z]_{a}-[y,z]_0.
\end{align*}
Hence we have
\begin{equation*}
( \mathcal{L}y,z) _H-( y,\mathcal{L}z) _H=(
\Lambda _1y,\Lambda _2z) _{E}-( \Lambda _2y,\Lambda_1z) _{E}.
\end{equation*}
\end{proof}

\begin{theorem} \label{thm3}
The triplet $( E,\Lambda _1,\Lambda _2) $ defined by \eqref{3.6}
 is a boundary space of the operator $\mathcal{L}_0$.
\end{theorem}

\begin{proof}
The first condition is obtained from Lemma \ref{lem2}. Now we will prove the second
condition. Let $u=\begin{pmatrix}
u_1 \\
u_2
\end{pmatrix}$,
$v=\begin{pmatrix}
v_1 \\
v_2
\end{pmatrix} \in E$. Then, the vector-valued function
\begin{equation*}
y( t) =\alpha _1( t) u_1( t) +\alpha
_2( t) v_1( t) +\beta _1( t)
u_2( t) +\beta _2( t) v_2( t) ,
\end{equation*}
where $\alpha _1(\cdot)$, $\alpha _2(\cdot)$,
$ \beta_1(\cdot)$, $\beta _2(\cdot) \in H$, satisfies the conditions
\begin{gather*}
\alpha _1( 0) = -1,\\
J_{q,a^{-}}^{1-\alpha
}p^{c}D_{q,0^{+}}^{\alpha }\alpha _1( 0) =\alpha _1(a)
= J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha }\alpha _1(
aq^{-1}) =0, \\
J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha }\alpha _2( 0)= 1,\\
\alpha _2( 0) =\alpha _2( a)
= J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha }\alpha _2(aq^{-1}) =0, \\
\beta _1( a) = 1,\\
 \beta _1( 0) =J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha }\beta _1( 0)
= J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha }\beta _1(
aq^{-1}) =0, \\
J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha }\beta _2(
aq^{-1})  = 1, \\
\beta _2( 0)  = \beta _2( a)
=J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha }\beta _2( 0)
=0.
\end{gather*}
Note that $y(\cdot) $ belongs to the set $\mathcal{D}$ and $\Lambda _1y=u$,
$\Lambda _2y=v$. Hence the proof is complete.
\end{proof}

\begin{corollary} \label{coro4}
For any contraction $K$ in $E$, the restriction of the operator
$\mathcal{L}$ to the set of functions $y\in \mathcal{D}$ satisfying
either the boundary conditions
\begin{equation}
( K-I) \Lambda _1y+i( K+I) \Lambda _2y=0
\label{3.7}
\end{equation}
or
\begin{equation}
( K-I) \Lambda _1y-i( K+I) \Lambda _2y=0
\label{3.8}
\end{equation}
is respectively, a maximal dissipative or a maximal accumulative extension
of the operator $\mathcal{L}_0$, where $\mathcal{L}_0$ is the
restriction of the operator $\mathcal{L}$\ to the domain $\mathcal{D}$.
Conversely, every maximal dissipative (accumulative) extension of the
operator $\mathcal{L}_0\ $is the restriction of $\mathcal{L}$ to the set
of functions $y\in \mathcal{D}$ satisfying \eqref{3.7} \eqref{3.8}, and the
extension uniquely determines the contraction $K$. Conditions \eqref{3.7}
\eqref{3.8}, in which $K$ is an isometry describe the maximal symmetric
extensions of $\mathcal{L}_0$ in $H$. If$\ K$ is unitary, these conditions
define self-adjoint extensions.
\end{corollary}

In particular, the boundary conditions
\begin{gather}
-y( 0) +\gamma _1J_{q,a^{-}}^{1-\alpha
}p^{c}D_{q,0^{+}}^{\alpha }y( 0) =0, \\
y( a) +\gamma _2J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha
}y( \frac{a}{q}) =0,
\end{gather}
with $\operatorname{Im}\gamma _1\geq 0$ or
$\gamma _1=\infty ,\ \operatorname{Im}\gamma_2\geq 0$ or
$\gamma _2=\infty $ ($\operatorname{Im}\gamma _1=0$ or
$\gamma_1=\infty ,\ \operatorname{Im}\gamma _2=0$ or
 $\gamma _2=\infty $) describe the maximal dissipative (self-adjoint)
extensions of $\mathcal{L}_0$ with separated boundary conditions.

Now we study the maximal dissipative operator $\mathcal{L}_{K}$, where
 $K$ is the strict contraction in $E$ generated by the expression
$\tau_{q,\alpha }y$ and the boundary condition \eqref{3.7}.
Since $K$ is a strict contraction, the operator $K+I$ must be invertible,
and the boundary condition \eqref{3.7} is equivalent to the condition
\begin{equation}
\Lambda _2y+\Omega \Lambda _1y=0,  \label{3.11}
\end{equation}
where $\Omega =-i( K+I) ^{-1}( K-I)$, $\operatorname{Im}\Omega>0$, and
 $-K$ is the Cayley transform of the dissipative operator $\Omega $.
We denote $\mathcal{L}_{\Omega }$ $( =\mathcal{L}_{K}) $ the
dissipative operator generated by the expression $\tau _{q,\alpha }y$ and
the boundary condition \eqref{3.11}.

Let
\begin{equation*}
\Omega =\begin{pmatrix}
\gamma _1 & 0 \\
0 & \gamma _2
\end{pmatrix}
\end{equation*}
where $\operatorname{Im}\gamma _1>0$, $\operatorname{Im}\gamma _2>0$ and
$\eta ^2=2\operatorname{Im}\Omega ,\ \eta >0$.
Then the boundary condition \eqref{3.11}
coincides with the separated boundary conditions \eqref{3.7} and \eqref{3.8}.

\section{Self-adjoint dilation, incoming and outgoing spectral representations}

In this section, we construct a self-adjoint dilation of the maximal
dissipative $q$-fractional Sturm-Liouville operator and its incoming and
outgoing spectral representations. Hence we determine the scattering matrix
of the dilation according to the Lax and Phillips scheme (\cite{17}, \cite
{18}). Later, we construct a functional model of this operator, using
incoming spectral representations. Finally, we determine the characteristic
function of this operator.

Now we consider the ``incoming'' and
``outgoing'' subspaces $L^2( (-\infty ,0) ;E) $ and
$L^2( ( 0,\infty );E) $. The orthogonal sum
$\mathcal{H}=L^2(( -\infty ,0);E)\oplus H\oplus L^2(( 0,\infty ) ;E)$
is called the \emph{main Hilbert space of the dilation}.

In the space $\mathcal{H}$, we define the operator $\Upsilon $ on the set
$D( \Upsilon ) $, where $D( \Upsilon ) $
consist of vectors $w=\langle \psi _{-},y,\mathcal{
\psi }_{+}\rangle $, generated by the expression
\begin{gather}
\Upsilon \langle \psi _{-},y,\psi _{+}\rangle
= \langle i\frac{d\psi _{-}}{d\xi },\tau _{q,\alpha }y,i\frac{d
\psi _{+}}{d\varsigma }\rangle ;  \label{311} \\
\Lambda _2y+\Omega \Lambda _1y = \eta \psi _{-}(0),\Lambda
_2y+\Omega _1^{\ast }\Lambda y=\eta \psi _{+}(0),\eta ^2:=2
\operatorname{Im}\Omega ,\eta >0,  \label{3.2}
\end{gather}
where $\psi _{-}\in W_2^{1}( ( -\infty ,0);E)$,
$\psi _{+}\in W_2^{1}( ( 0,\infty );E)$, $y\in H$ and $W_2^{1}$
is the Sobolev space.

\begin{theorem} \label{thm5}
The operator $\Upsilon $ is self-adjoint in $\mathcal{H}$.
\end{theorem}

\begin{proof}
We first prove that $\Upsilon$ is symmetric in $\mathcal{H}$.
Let $f,g\in D( \Upsilon ) $, $f=\langle \psi _{-},y,
\psi _{+}\rangle $ and $g=\langle \zeta _{-},z,\zeta
_{+}\rangle $. Then, we have
\begin{equation}
\begin{aligned}
&( \Upsilon f,g) _{\mathcal{H}}-( f,\Upsilon g) _{\mathcal{H}}\\
&=i( \psi _{-}( 0) ,\zeta _{-}(
0) ) _{E}-i( \psi _{+}( 0) ,\zeta
_{+}( 0) ) _{E}+[y,z]_{a}-[y,z]_0.
\end{aligned}  \label{3.3}
\end{equation}
By direct computation, we obtain
\begin{equation*}
i( \psi _{-}( 0) ,\zeta _{-}( 0)
) _{E}-i( \psi _{+}( 0) ,\zeta _{+}(
0) ) _{E}+[y,z]_{a}-[y,z]_0=0.
\end{equation*}
Thus, $\Upsilon \subseteq \Upsilon ^{\ast }$, i.e., $\Upsilon $ is a
symmetric operator.

It is easy to check that $\Upsilon $ and $\Upsilon ^{\ast }$ are generated
by the same expression \eqref{311}. Let us describe the domain of
$\Upsilon^{\ast }$. We shall compute the terms outside the integral sign,
which are obtained via integration by parts in the bilinear form
$( \Upsilon f,g) _{\mathcal{H}}$, $f\in D( \Upsilon )$,
$g\in D(\Upsilon ^{\ast }) $. Their sum is equal to zero, i.e.,
\begin{equation}
[ y,z]_{a}-[y,z]_0+i( \psi _{-}( 0) ,\zeta
_{-}( 0) ) _{E}-i( \psi _{+}( 0)
,\zeta _{+}( 0) ) _{E}=0.  \label{3.4}
\end{equation}
Further, solving the boundary conditions \eqref{3.2} for $\Lambda _1y$ and
$\Lambda _2y$, we find that
\begin{equation*}
\Lambda _1y=-i\eta ^{-1}( \psi _{-}( 0) -\psi _{+}( 0) ) ,\quad
\Lambda _2y=\eta \psi _{-}( 0) +i\mathcal{L}\eta ^{-1}( \psi
_{-}( 0) -\psi _{+}( 0) ) .
\end{equation*}
Therefore, using \eqref{3.6}, we find that \eqref{3.4} is equivalent to the
equality
\begin{align*}
&i( \psi _{+}( 0) ,\zeta _{+}( 0)
) _{E}-i( \psi _{-}( 0) ,\zeta _{-}(0) ) _{E} \\
&= [y,z]_{a}-[y,z]_0=( \Lambda _1y,\Lambda _2z) _{E}-(
\Lambda _2y,\Lambda _1z) _{E} \\
&= -i( \eta ^{-1}( \psi _{-}( 0) -\mathcal{
\psi }_{+}( 0) ) ,\Lambda _2z) _{E}-( \eta
\psi _{-}( 0) ,\Lambda _1z) _{E} \\
&\quad -i( \mathcal{L}\eta ^{-1}( \psi _{-}( 0) -
\psi _{+}( 0) ) ,\Lambda _1z) _{E}.
\end{align*}
Since the values $\psi _{\mp }( 0)$ can be arbitrary
vectors, a comparison of the coefficients of
$\psi _{i\mp }(0)$ $( i=1,2)$ on the left-hand side and the right-hand
side of the last equality proves that the vector
$g=\langle \zeta_{-},z,\zeta _{+}\rangle$ satisfies the boundary conditions
\eqref{3.2}, namely, $\Lambda _2z+\Omega \Lambda _1z=\eta \zeta _{-}( 0)$,
$\Lambda _2z+\Omega _1^{\ast }\Lambda _1z=\eta \zeta _{+}(0) $.
Therefore $D( \Upsilon ^{\ast }) \subseteq D(\Upsilon ) $, and hence
$\Upsilon =\Upsilon ^{\ast }$.
\end{proof}

Note that the self-adjoint operator $\Upsilon $ generates a unitary group
$U_t=\exp ( i\Upsilon t) $ $(t\in (-\infty ,\infty ))$ on
$\mathcal{H}$. Let us denote by $\mathcal{P}:\mathcal{H}\to H$ and
$\mathcal{P}_1:H\to \mathcal{H}$ the mappings acting according to
the formulae $\mathcal{P}:\langle \psi _{-},y,
\psi _{+}\rangle \to y$ and
$\mathcal{P}_1:y\to \langle 0,y,0\rangle $. Let
$Z_t:=\mathcal{P}U_t\mathcal{P}_1$, $t\geq 0$,
by using $U_t$. The family $\{ Z_t\} $ $( t\geq 0)$ of operators
is a strongly continuous semigroup of completely non-unitary
contraction on $H$. Let us denote by\ $B$\ the generator of this semigroup
$:By=\lim_{t\to +0}( \frac{Z_ty-y}{it}) $. The domain of
$B$ consists of all vectors for which the limit exists. The operator $B$ is
dissipative. The operator $\Upsilon $ is called the \emph{self-adjoint
dilation} of $B$. Then, we have the following result.

\begin{theorem} \label{thm6}
The operator $\Upsilon $ is a self-adjoint dilation of the
operator $\mathcal{L}_{\Omega }( =\mathcal{L}_{K}) $.
\end{theorem}

\begin{proof}
It is sufficient to prove the following equality (see \cite{ku}):
\begin{equation}
\mathcal{P}( \Upsilon -\lambda I) ^{-1}\mathcal{P}_1y=(
\mathcal{L}_{\Omega }-\lambda I) ^{-1}y,\quad y\in H,\;
\operatorname{Im} h<0.
\end{equation}
We set $( \Upsilon -\lambda I) ^{-1}\mathcal{P}_1y=g=\langle
\zeta _{-},z,\zeta _{+}\rangle $. Then
$( \Upsilon -\lambda I) g=\mathcal{P}_1y$, and hence
$\tau _{q,\alpha }z-\lambda z=y$,
$\zeta_{-}( \xi ) =\zeta _{-}( 0) e^{-i\lambda \xi }$ and
$\zeta _{+}( \xi ) =\zeta _{+}( 0) e^{-i\lambda \xi }$.
Since $g\in D( \Upsilon ) $, we have
$\zeta _{-}\in W_2^{1}((-\infty ,0);E)$. Thus it follows that
$\zeta _{-}( 0)=0$ and consequently, $z$ satisfies the boundary condition
$\Lambda_2z+\Omega \Lambda _1z=0$. Therefore
$z\in D( \mathcal{L}_{\Omega }) $, and since the point $\lambda $ with
$\operatorname{Im}\lambda <0$ cannot be an eigenvalue of the dissipative operator,
 it follows that $z=(\mathcal{L}_{\Omega }-\lambda I) ^{-1}y$. Thus
\begin{equation*}
( \Upsilon -\lambda I) ^{-1}\mathcal{P}_1y=\langle 0,(
\mathcal{L}_{\Omega }-\lambda I) ^{-1}y,\eta ^{-1}( \Lambda
_2y+\Omega ^{\ast }\Lambda _1y) e^{-i\lambda \xi }\rangle
\end{equation*}
for $y\in H$\ and $\operatorname{Im}\lambda <0$. By applying the
 mapping $\mathcal{P}$, we obtain
\begin{align*}
( \mathcal{L}_{\Omega }-\lambda I) ^{-1}
&= \mathcal{P}( \Upsilon -\lambda I) ^{-1}\mathcal{P}_1
 =-i\mathcal{P}\int_0^{\infty}U_te^{-i\lambda t}dt
 \mathcal{P}_1 \\
&= -i\int_0^{\infty}Z_te^{-i\lambda t}dt
=(B-\lambda I) ^{-1},\quad \operatorname{Im}\lambda <0,
\end{align*}
i.e., $\mathcal{L}_{\Omega }=B$.
\end{proof}

On the other hand, the unitary group $\{ U_t\} $ has an
important property which makes it possible to apply it to the Lax-Phillips
theory (see \cite{17}). It has orthogonal incoming and outgoing subspaces
$D_{-}=\langle L^2( -\infty ,0) ,0,0\rangle $ and
$D_{+}=\langle 0,0,L^2( 0,\infty ) \rangle $, and they have the following
properties.

\begin{lemma} \label{lem7}
$U_tD_{-}\subset D_{-}$, $t\leq 0$ and $U_tD_{+}\subset D_{+}$,
$t\geq0$.
\end{lemma}

\begin{proof}
We will just prove for $D_{+}$ since the proof for $D_{-}$ is similar. Set
$\mathcal{R}_{\lambda }=( \Upsilon -\lambda I) ^{-1}$. Then, for
all $\lambda $, with $\operatorname{Im}\lambda <0$, we have
\begin{equation*}
\mathcal{R}_{\lambda }f=\langle 0,0,-ie^{-i\lambda \xi }
\int_0^{\xi } e^{i\lambda s}\psi _{+}( s)
ds\rangle ,\quad  f=\langle 0,0,\psi _{+}\rangle \in D_{+}.
\end{equation*}
Hence we have $\mathcal{R}_{\lambda }f\in D_{+}$. If $g\perp D_{+}$, then we obtain
\begin{equation*}
0=( \mathcal{R}_{\lambda }f,g) _{\mathcal{H}}
=-i\int_0^{\infty } e^{-i\lambda t}( U_tf,g) _{\mathcal{H}}dt, \quad
\quad \operatorname{Im}\lambda <0.
\end{equation*}
Thus we have $( U_tf,g) _{\mathcal{H}}=0$ for all $t\geq 0$,
i.e., $U_tD_{+}\subset D_{+}$ for $t\geq 0$.
\end{proof}

\begin{lemma} \label{lem8}
$\cap_{t\leq 0} U_tD_{-}=\cap_{t\geq 0}U_tD_{+}=\{ 0\} $.
\end{lemma}

\begin{proof}
Let us define the mapping
$\mathcal{P}^{+}:\mathcal{H}\to L^2( ( 0,\infty ) ;E) $ and the mapping
$\mathcal{P}_1^{+}:L^2( ( 0,\infty ) ;E) \to D_{+}$ as
$\mathcal{P}^{+}:\langle \psi _{-},y,\psi
_{+}\rangle \to \psi _{+}$ and
$\mathcal{P}_1^{+}: \psi \to \langle 0,0,\psi \rangle $,
respectively. We consider that the semigroup of isometries
 $U_t^{+}:= \mathcal{P}^{+}U_t\mathcal{P}_1^{+}$ $( t\geq 0) $ is a
one-sided shift in $L^2( ( 0,\infty ) ;E) $. Indeed,
the generator of the semigroup of the one-sided shift $V_t$ in
$L^2( ( 0,\infty ) ;E) $ is the differential operator
$i\frac{d}{d\xi }$ with the boundary condition
$\psi (0) =0$. On the other hand, the generator $S$
of the semigroup of
isometries $U_t^{+}$ $( t\geq 0) $ is the operator
\begin{equation*}
S\psi =\mathcal{P}^{+}\Upsilon \mathcal{P}_1^{+}\psi
=\mathcal{P}^{+}\Upsilon \langle 0,0,\psi \rangle
=\mathcal{P} ^{+}\langle 0,0,i\frac{d\psi }{d\xi }\rangle
=i\frac{d\psi }{d\xi },
\end{equation*}
where $\psi \in W_2^{1}( ( 0,\infty ) ;E) $
and $\psi ( 0) =0$. Since a semigroup is uniquely
determined by its generator, it follows that $U_t^{+}=$ $V_t$, and hence
we obtain
\begin{equation*}
\cap_{t\geq 0} U_tD_{+}=\langle 0,0,\cap_{t\leq 0} V_tL^2( ( 0,\infty ) ;E)
\rangle =\{0\} .
\end{equation*}
\end{proof}

\begin{lemma} \label{lem9}
The operator $\mathcal{L}_{\Omega }$ is simple.
\end{lemma}

\begin{proof}
Let $H'\subset H$ be a nontrivial subspace in which
$\mathcal{L}_{\Omega }$ induces a self-adjoint operator
$\mathcal{L}_{\Omega }'$ with domain $D( \mathcal{L}_{\Omega }')
=H'\cap D( \mathcal{L}_{\Omega }) $.
If $f\in D( \mathcal{L}_{\Omega }') $, then
$f\in D( \mathcal{L}_{\Omega }^{\ast }) $ and
\begin{align*}
0 &= \frac{d}{dt}\Vert e^{i\mathcal{L}_{\Omega }t}f\Vert _H^2=\frac{d}{dt
}( e^{i\mathcal{L}_{\Omega }t}f,e^{i\mathcal{L}_{\Omega }t}f) _H\\
&= -2(\operatorname{Im}\Omega \Lambda _1e^{i\mathcal{L}_{\Omega }t}f,\Lambda
_1e^{i\mathcal{L}_{\Omega }t}f)_{E}.
\end{align*}
Consequently,  $\Lambda _1e^{i\mathcal{L}_{\Omega }t}f=0$.
For eigenvectors $y\in H'$ of the operator $\mathcal{L}_{\Omega }$ we
have $\Lambda _1y( \lambda ) =0$. Using this result with the
boundary condition $\Lambda _2y+\Omega \Lambda _1y=0$, we have
$\Lambda_2y=0$, i.e., $y=0$. Since all the solutions of
$\tau _{q,\alpha}y=\lambda y\ $belong to $H$, from this it can be concluded that the
resolvent operator $R_{\lambda }( \mathcal{L}_{\Omega}) $ is compact, and
the spectrum of $\mathcal{L}_{\Omega }$ is purely
discrete. Consequently, by the theorem on expansion in the
eigenvectors of the self-adjoint operator
 $\mathcal{L}_{\Omega }'$, we obtain $H'=\{ 0\} $. Hence the operator
$\mathcal{L}_{\Omega }$ is simple.
\end{proof}

Now we set
\begin{equation*}
H_{-}=\overline{\cup_{t\geq 0} U_tD_{-}},\quad  H_{+}=\overline{
\cup_{t\leq 0} U_tD_{+}}.
\end{equation*}

\begin{lemma} \label{lem10}
The equality $H_{-}+H_{+}=\mathcal{H}$ holds.
\end{lemma}

\begin{proof}
From Lemma \ref{lem9}, it is easy to show that the subspace
$\mathcal{H}'= \mathcal{H}\ominus ( H_{-}+H_{+}) $ is invariant relative to
the group $\{ U_t\} $, and has the form $\mathcal{H}'=\langle 0,H',0\rangle $
where $H'$ is a subspace of $H$.
Therefore, if the subspace $\mathcal{H}'$ (and hence also $H'$) were nontrivial,
then the unitary group $\{ U_t'\} $ restricted to this subspace would
 be a unitary part of the group
$\{ U_t\} $, and hence the restriction
$\mathcal{L}_{\Omega }'$ of $\mathcal{L}_{\Omega }$ to $H'$ would be a
self-adjoint operator in $H'$. Since the operator
$\mathcal{L}_{\Omega }$ is simple, it follows that $H'=\{ 0\} $.
\end{proof}

Suppose that $\chi ( \lambda ) $ and $\omega ( \lambda) $ are the solutions
of $\tau _{q,\alpha }y=\lambda y$, satisfying
the conditions
\begin{equation*}
\chi ( 0,\lambda ) =0,\quad
J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha }\chi ( 0,\lambda ) =-1,\quad
\omega (0,\lambda ) =1,\quad
J_{q,a^{-}}^{1-\alpha }p^{c}D_{q,0^{+}}^{\alpha}\omega ( 0,\lambda ) =0.
\end{equation*}
We denote by $m( \lambda )$ the matrix-valued function
satisfying the conditions
\begin{equation*}
m( \lambda ) \Lambda _1\chi =\Lambda _2\chi ,\quad
m(\lambda ) \Lambda _1\omega =\Lambda _2\omega .
\end{equation*}
$m( \lambda ) $ is a meromorphic function on the complex plane
$\mathbb{C}$ with a countable number of poles on the real axis.
Furthermore, it is possible to show that the function $m( \lambda ) $ possesses the
following properties: $\operatorname{Im}m( \lambda ) \leq 0$ for all
$\operatorname{Im}\lambda \neq 0$, and
$m^{\ast }( \lambda ) =m(\overline{\lambda }) $ for all
$\lambda \in\mathbb{C}$, except the real poles $m( \lambda ) $.

We denote by $\mu _j( x,\lambda )$ and
$\mathcal{\nu }_j( x,\lambda ) $ $(j=1,2) $ the solutions of the
system\ $\tau _{q,\alpha }y=\lambda y$, which satisfy the conditions
\begin{equation*}
\Lambda _1\mu _j=( m( \lambda ) +\Omega )^{-1}\eta e_j,\quad
\Lambda _1\mathcal{\nu }_j=( m( \lambda) +\Omega ^{\ast }) ^{-1}\eta e_j\quad
 ( j=1,2) ,
\end{equation*}
where $\{ e_1,e_2\} $ is an orthonormal basis for $E$.

We set
\begin{equation*}
U_{\lambda j}^{-}( x,\xi ,\rho ) =\langle e^{-i\lambda \xi
}e_j,\mu _j(x,\lambda ),\eta ^{-1}( m+\Omega ^{\ast }) (
m+\Omega ) ^{-1}\eta e^{-i\lambda \rho }e_j\rangle \quad (j=1,2) .
\end{equation*}
We note that the vectors $U_{\lambda j}^{-}( x,\xi ,\rho ) $
$( j=1,2)$ for real $\lambda $ do not belong to the space
$\mathcal{H}$. However, $U_{\lambda j}^{-}( x,\xi ,\rho ) $
 $( j=1,2)$ satisfies the equation $\Upsilon U=\lambda U$ and the
corresponding boundary conditions for the operator $\Upsilon $.

By means of the vector $U_{\lambda j}^{-}( x,\xi ,\rho ) $
$( j=1,2) $, we define the transformation
$\mathcal{F}_{-}:f\to \widetilde{f_{-}}( \lambda ) $ by
\begin{equation*}
( \mathcal{F}_{-}f) ( \lambda ) :=\widetilde{
f_{-}}( \lambda ) :=\frac{1}{\sqrt{2\pi }}\sum_{j=1}^2
( f,U_{\lambda j}^{-}) _{\mathcal{H}}e_j
\end{equation*}
on the vectors $f=\langle \psi _{-},y,\psi_{+}\rangle $ in which
$\psi _{-}$, $\psi _{+}$, $y$ are
smooth, compactly supported functions.

\begin{lemma} \label{lem11}
The transformation $\mathcal{F}_{-}$  maps isometrically
 $H_{-}$ onto  $L^2( ( -\infty ,\infty );E)$.
For all vectors $f,g\in H_{-}$ the Parseval equality
and the inversion formulae hold:
\begin{gather*}
( f,g) _{\mathcal{H}}=(\widetilde{f_{-}},
\widetilde{g_{-}})_{L^2}=\int_{-\infty }^{\infty }
\sum_{j=1}^2\widetilde{f_{j-}}( \lambda )
\overline{\widetilde{g_{j-}}( \lambda ) }d\lambda ,\\
f=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{\infty }
\sum_{j=1}^2\widetilde{f_{j-}}( \lambda ) U_{\lambda
j}^{-}d\lambda ,
\end{gather*}
where
$\widetilde{f_{-}}( \lambda ) =(\mathcal{F}_{-}f) ( \lambda ) $ and
$\widetilde{g_{-}}( \lambda ) =( \mathcal{F}_{-}g)( \lambda ) $.
\end{lemma}

\begin{proof}
By the Paley-Wiener theorem, we obtain
\begin{equation*}
\widetilde{f_{j-}}( \lambda )
=\frac{1}{\sqrt{2\pi }}( f,U_{\lambda j}^{-}) _{\mathcal{H}}
=\frac{1}{2\pi }\int_{-\infty }^0 ( \psi _{-}( \xi )
,e^{-i\lambda \xi }e_j) _{E}d\xi \in H_{-}^2( E) ,
\end{equation*}
where $f=\langle \psi _{-},0,0\rangle ,g=\langle \zeta _{+},0,0\rangle \in D_{-}$.
If we use the Parseval equality
for Fourier integrals, then we obtain
\begin{equation*}
( f,g) _{\mathcal{H}}
=\int_t{-\infty }^{\infty } ( \psi _{-}( \xi ) ,\zeta _{-}( \xi )
) _{E}d\xi
=\int_{-\infty }^{\infty } (\widetilde{f_{-}}( \lambda ) ,
\widetilde{g_{-}}(\lambda ) )_{E}d\lambda =( \mathcal{F}_{-}f,\mathcal{F}
_{-}g) _{L^2},
\end{equation*}
where $H_{\pm }^2(E)$ denote the Hardy classes in $L^2( (-\infty,\infty );E) $
 consisting of the functions analytically extendible to
the upper and lower half-planes, respectively. We now extend the Parseval
equality to the whole of $H_{-}$. We consider in $H_{-}$ the dense set
$H_{-}'$ consisting of smooth, compactly supported functions in
$D_{-}:f\in H_{-}'$ if $f=U_T$ $f_0$,
$f_0=\langle \psi_{-},0,0\rangle $,
$\psi _{-}\in C_0^{\infty }( (-\infty ,0) ;E) $, where $T=T_{f}$ is a
nonnegative number depending on $f$. If $f,g\in H_{-}'$,
 then for $T>T_{f}$ and $T>T_{g}$ we have $U_{-T}f$, $U_{-T}g\in D_{-}$.
Moreover, the first components of these vectors belong to
$C_0^{\infty}( ( -\infty ,0) ;E)$. Since the operators $U_t$
$( t\in (-\infty ,\infty )) $ are unitary, by the equality
\begin{equation*}
\mathcal{F}_{-}U_tf=\frac{1}{\sqrt{2\pi }}\sum_{j=1}^2(
U_tf,U_{\lambda j}^{-}) _{\mathcal{H}}e_j=e^{i\lambda t}\mathcal{F}
_{-}f,
\end{equation*}
we have
\begin{align*}
( f,g) _{\mathcal{H}}
&= ( U_{-T}f,U_{-T}g) _{\mathcal{H}}
 =( \mathcal{F}_{-}U_{-T}f,\mathcal{F}_{-}U_{-T}g) _{L^2} \\
&= (e^{-i\lambda T}\mathcal{F}_{-}f,e^{-i\lambda T}\mathcal{F}
_{-}g)_{L^2}
=(\overset{\sim }{f},\overset{\sim }{g})_{L^2}.
\end{align*}
By taking the closure, we obtain the Parseval equality for the space
$H_{-} $. The inversion formula is obtained from the Parseval equality if all
integrals in it are considered as limits in the mean of integrals over
finite intervals. Finally, we obtain the desired result
\begin{equation*}
\mathcal{F}_{-}H_{-}=\overline{\cup_{t\geq 0} \mathcal{F}
_{-}U_tD_{-}}
=\overline{\cup_{t\geq 0} e^{i\lambda t}H_{-}^2}
=L^2( (-\infty ,\infty );E) .
\end{equation*}
\end{proof}

Now we set
\begin{equation*}
U_{\lambda j}^{+}( x,\xi ,\rho ) =\langle S_{\Omega }(
\lambda ) e^{-i\lambda \xi }e_j,\mathcal{\nu }_j( x,\lambda
) ,e^{-i\lambda \rho }e_j\rangle \quad ( j=1,2) ,
\end{equation*}
where
\begin{equation}
S_{\Omega }( \lambda ) =\eta ^{-1}( m( \lambda )
+\Omega ) ( m( \lambda ) +\Omega ^{\ast })
^{-1}\eta .  \label{31}
\end{equation}
We note that the vectors $U_{\lambda j}^{+}( x,\xi ,\rho ) $ for
real $\lambda $ do not belong to the space $\mathcal{H}$. However,
$U_{\lambda j}^{+}( x,\xi ,\rho ) $ satisfies the equation
$\Upsilon U=\lambda U$ and the corresponding boundary conditions for the
operator $\Upsilon $. With the help of the vector
$U_{\lambda j}^{+}(x,\xi ,\rho ) $, we define the transformation
$\mathcal{F}_{+}:f\to \widetilde{f_{+}}( \lambda ) $ by
\begin{equation*}
( \mathcal{F}_{+}f) ( \lambda ) :=\widetilde{
f_{+}}( \lambda ) :=\sum_{j=1}^2\widetilde{
f_{j+}}( \lambda ) e_j:=\frac{1}{\sqrt{2\pi }}
\sum_{j=1}^2( f,U_{\lambda j}^{+}) _{\mathcal{H}}e_j
\end{equation*}
on the vectors $f=\langle \psi _{-},y,\psi_{+}\rangle $ in which
$\psi _{-},\ \psi _{+}$ and $y$ are smooth, compactly supported functions.

\begin{lemma} \label{lem12}
The transformation $\mathcal{F}_{+}$ isometrically maps
$H_{+}$ onto $L^2( (-\infty ,\infty )) $.
For all vectors $f,g\in H_{+}$ the Parseval equality
and the inversion formula hold:
\begin{gather*}
( f,g) _{\mathcal{H}}=(\widetilde{f_{+}},
\widetilde{g_{+}})_{L^2}=\int_{-\infty }^{\infty}
\sum_{j=1}^2\widetilde{f_{j+}}( \lambda )
\overline{\widetilde{g_{j+}}( \lambda ) }d\lambda ,\\
f=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{\infty}
\sum_{j=1}^2\widetilde{f_{j+}}( \lambda ) U_{\lambda j}^{+}d\lambda ,
\end{gather*}
where $\widetilde{f_{+}}( \lambda ) =(\mathcal{F}_{+}f) ( \lambda ) $ and
$\widetilde{g_{+}}( \lambda ) =( \mathcal{F}_{+}g)( \lambda ) $.
\end{lemma}

The proof of the above lemma is similar to that of Lemma \ref{lem11},
and it is omitted.
It is clear that the matrix-valued function
$S_{\Omega }( \lambda ) $\ is meromorphic in $\mathbb{C}$ and all the poles
are in the lower half-plane.
From \eqref{31}, we obtain $\| S_{\Omega }( \lambda) \| \leq 1$ for
$\operatorname{Im}\lambda >0$ and $S_{\Omega }(\lambda ) $ is
the unitary matrix for all $\lambda \in \mathbb{R}$.
Therefore, we have
\begin{equation}
U_{\lambda j}^{+}=\sum_{k=1}^2S_{jk}( \lambda ) U_{\lambda
k}^{-}\quad (j=1,2,\dots,n),  \label{32}
\end{equation}
where $S_{jk}$ $ ( j,k=1,2,\dots,n)$ are elements of the matrix
$S_{\Omega }( \lambda )$. From Lemmas \ref{lem11} and \ref{lem12}, we obtain
$H_{-}=H_{+}$. With Lemma \ref{lem10}, this shows that
 $H_{-}=H_{+}=\mathcal{H}$. Therefore we have proved the following lemma
for the incoming and outgoing subspaces (for $D_{-}$ and $D_{+}$).

\begin{lemma} \label{lem13}
$\overline{\cup_{t\geq 0} U_tD_{-}}=\overline{\cup_{t\leq 0} U_tD_{+}}
=\mathcal{H}$.
\end{lemma}

\begin{lemma} \label{lem14}
$D_{-}\perp D_{+}$.
\end{lemma}

The proof of the above lemma is straightforward, hence omitted.
Thus the transformation $\mathcal{F}_{-}$ isometrically maps $H_{-}$ onto
$L^2( (-\infty ,\infty );E) $ with the subspace $D_{-}$ mapped
onto $H_{-}^2(E)$ ,and the operators $U_t$ are transformed into the
operators of multiplication by $e^{i\lambda t}$.
This means that $\mathcal{F}_{-}$ is the incoming spectral representation
for the group $\{U_t\} $. Similarly, $\mathcal{F}_{+}$ is the outgoing spectral
representation for the group $\{ U_t\} $. It follows that the
passage from the $\mathcal{F}_{-}$ representation of an element
$f\in \mathcal{H}$ to its $\mathcal{F}_{+}\ $representation is accomplished as
$\widetilde{f_{+}}( \lambda ) =S_{\Omega }^{-1}(\lambda )
\widetilde{f_{-}}( \lambda ) $.
Consequently, according to \cite{17}, we have proved the following result.

\begin{theorem} \label{thm15}
The function $S_{\Omega }^{-1}( \lambda ) $ is
the scattering matrix of the group $\{ U_t\} $ (of the
self-adjoint operator $\Upsilon $).
\end{theorem}

Let $S( \lambda ) $ be an arbitrary non-constant inner function
on the upper half-plane. Let us define $K$ by the formula
$ K=H_{+}^2\ominus SH_{+}^2$. It is clear that
$K\neq \{0\} $ is a subspace of the Hilbert space $H_{+}^2$.
We consider the semigroup of operators $Z_t$ $( t\geq 0) $ acting in $K$
according to the formula
\begin{equation*}
Z_t\varphi =\mathcal{P}[ e^{i\lambda t}\varphi] ,\quad \varphi
=\varphi ( \lambda ) \in K,
\end{equation*}
where $\mathcal{P}$ is the orthogonal projection from $H_{+}^2$ onto $K$.
The generator of the semigroup $\{ Z_t\} $ is denoted by
\begin{equation*}
T\varphi =\lim_{t\to +0}( \frac{Z_t\varphi-\varphi }{it}) ,
\end{equation*}
where $T$ is a maximal dissipative operator acting in $K$, and with domain
$D(T)$ consisting of all the functions $\varphi \in K$ such that the limit
exists. The operator $T$ is called a \emph{model dissipative operator}.
This model dissipative operator is a special case of a more general model
dissipative operator constructed by Nagy and Foia\c{s} \cite{18}, which is
associated with the names of Lax-Phillips \cite{17}. Here the basic
assertion is that $S( \lambda ) $ is the \emph{characteristic
function} of the operator $T$.

Let $K=\langle 0,H,0\rangle $, so that $\mathcal{H=}D_{-}\oplus
K\oplus D_{+}$. From the explicit form of the unitary transformation
$\mathcal{F}_{-}$ under the mapping $\mathcal{F}_{-}$, we obtain
\begin{equation} \label{34}
\begin{gathered}
\mathcal{H} \to L^2( (-\infty ,\infty );E),\quad
 f\to \widetilde{f_{-}}( \lambda ) =(
\mathcal{F}_{-}f) ( \lambda ) ,    \\
D_{-} \to H_{-}^2( E) ,\quad  D_{+}\to S_{\Omega}H_{+}^2( E) ,    \\
K \to H_{+}^2( E) \ominus S_{\Omega}H_{+}^2( E) ,    \\
U_t \to (\mathcal{F}_{-}U_t\mathcal{F}_{-}^{-1}
\widetilde{f_{-}})( \lambda ) =e^{i\lambda t}\widetilde{f_{-}}( \lambda ) .
\end{gathered}
\end{equation}
The formulas in \eqref{34} show that the operator
$\mathcal{L}_{\Omega }( \mathcal{L}_{K})$ is unitarily equivalent to the model
dissipative operator with the characteristic function
$S_{\Omega }(\lambda)$. Since the characteristic functions of unitary equivalent
dissipative operators coincide (see \cite{18}), we have thus proved following theorem.

\begin{theorem} \label{thm16}
The function $S_{\Omega }( \lambda ) $ defined
by \eqref{31} coincides with the characteristic function of the maximal
dissipative operator $\mathcal{L}_{\Omega } ( \mathcal{L}_{K}) $.
\end{theorem}

\section{Completeness of root vectors}

In this section, we prove that all the root vectors of the maximal
dissipative $q$-fractional Sturm-Liouville operator are complete. We know
that the absence of the singular factor in the factorization of the
characteristic function guarantees the completeness of the system of root
vectors of maximal dissipative operators (\cite{18}). We will prove that the
characteristic function of the maximal dissipative $q$-fractional
Sturm-Liouville operator is a Blaschke-Potapov product.

\begin{lemma} \label{lem17}
The characteristic function $\widetilde{S}_{K}( \lambda ) $ of the
operator $\mathcal{L}_{K}$ has the form
\begin{align*}
\widetilde{S}_{K}( \lambda )
:=& S_{\Omega }( \lambda )\\
=& X_1( I-K_1K_1^{\ast }) ^{1/2}( \Theta ( \xi
) -K_1) ( I-K_1^{\ast }\Theta ( \xi )
) ^{-1}( I-K_1K_1^{\ast }) ^{1/2}X_2,
\end{align*}
where $K_1=-K$ is the Cayley transformation of the dissipative operator
 $\Omega $, and $\Theta ( \xi ) $ is the Cayley transformation of
the matrix-valued function $m( \lambda )$, where
\begin{gather*}
\xi =( \lambda -i) ( \lambda +i) ^{-1}, \\
X_1 :=( \operatorname{Im}\Omega ) ^{-1/2}( I-K_1)
^{-1}( I-K_1K_1^{\ast }) ^{1/2}, \\
X_2 :=( I-K_1^{\ast }K_1) ^{-1/2}( I-K_1^{\ast
}) ^{-1}( \operatorname{Im}\Omega ) ^{1/2}, \\
| \det X_1| | \det X_2| =1.
\end{gather*}
\end{lemma}

Recall that the inner matrix-valued function
$\widetilde{S}_{K}(\lambda ) $ is a Blaschke-Potapov product if and only if
$\det \widetilde{S}_{K}( \lambda ) $ is a Blaschke product
(see \cite{ginz,18}). By Lemma \ref{lem17}, the characteristic function
$\widetilde{S}_{K}( \lambda ) $is a Blaschke-Potapov product if and only if the
matrix-valued function
\begin{equation*}
X_{K}( \xi ) =( I-K_1K_1^{\ast }) ^{1/2}(
\Theta ( \xi ) -K_1) ( I-K_1^{\ast }\Theta (
\xi ) ) ^{-1}( I-K_1K_1^{\ast }) ^{1/2}
\end{equation*}
is a Blaschke-Potapov product in the unit disk.

We use the following result of \cite{ginz}.

\begin{lemma} \label{lem18}
Let $X( \xi )$ $( | \xi | <1) $ be
a analytic function with the values to be contractive operators in
$[E]$ $(\| X( \xi ) \| \leq 1)$. Then for $\Gamma$-quasi-every strictly
contractive operators (i.e., for all strictly
contractive $K\in [ E]$ possible with the exception of a set of
$\Gamma $ of zero capacity) the inner part of the contractive function
\begin{equation*}
X_{K}( \xi ) =( I-K_1K_1^{\ast }) ^{1/2}(X( \xi ) -K_1) ( I-K_1^{\ast }X( \xi )
) ^{-1}( I-K_1K_1^{\ast }) ^{1/2}
\end{equation*}
is a Blaschke-Potapov product.
\end{lemma}

By summing all the obtained results for the dissipative operator
$\mathcal{L}_{K}( \mathcal{L}_{\Omega }) $, we have proved the following result.

\begin{theorem} \label{thm19}
For$\Gamma$-quasi-every strictly contractive $K\in [ E]$ the
characteristic function $\widetilde{S}_{K}( \lambda ) $ of the
dissipative operator $\mathcal{L}_{K}$ is a Blaschke-Potapov product, and
the spectrum of $\mathcal{L}_{K}$ is purely discrete and belongs to the open
upper half-plane. For$\ \Gamma $-quasi-every strictly contractive
$K\in[ E]$ the operator $\mathcal{L}_{K}$ has a countable number of
isolated eigenvalues with finite multiplicity and limit points at infinity,
and the system of eigenvectors and associated vectors (or root vectors) of
this operator is complete in the space $H$.
\end{theorem}

\subsection*{Acknowledgements}
The authors want to thank the anonymous referee for the suggestions and
 remarks that helped us improve this article.

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\end{document}