\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 21, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2016/21\hfil Selfadjoint singular differential operators]
{Selfadjoint singular differential operators of first order
and their spectrum}
\author[Z. I. Ismailov, P. Ipek \hfil EJDE-2016/21\hfilneg]
{Zameddin I. Ismailov, Pembe Ipek}
\address{Zameddin I. Ismailov \newline
Karadeniz Technical University,
Department of Mathematics, 61080, Trabzon, Turkey}
\email{zameddin.ismailov@gmail.com}
\address{Pembe Ipek \newline
Karadeniz Technical University,
Department of Mathematics, 61080, Trabzon, Turkey}
\email{ipekpembe@gmail.com}
\dedicatory{Dedicated to Prof. E. S. Panakhov on his 60-th birthday}
\thanks{Submitted August 25, 2015. Published January 12, 2016.}
\subjclass[2010]{47A10, 47B38}
\keywords{Singular Selfadjoint differential operator; spectrum}
\begin{abstract}
Based on Calkin-Gorbachuk method, we describe all selfadjoint extensions
of the minimal operator generated by linear multipoint singular symmetric
differential-operator, as a direct sum of weighted Hilbert space of
vector-functions. Another approach to the investigation of this problem
has been done by Everitt, Zettl and Markus. Also we study the structure
of spectrum of these extensions.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks
\section{Introduction}
The first works devoted to the general theory of selfadjoint extensions
of symmetric operators having equal deficiency indexes in any
Hilbert space belong to von Neumann \cite{n2}. Generalization of this theory to
normal extensions has been done Coddington \cite{c1}.
Applications of this theory to two-point differential operators in Hilbert
spaces can be found in works of many mathematicians; see for example
\cite{d1,g1,g2,r1}.
It is known that for the existence of selfadjoint extension of any linear
closed densely defined symmetric $ T $ in a Hilbert space $ \mathcal{H} $
the necessary and sufficient condition is a equality of deficiency indexes
$ m(T)=n(T) $, where $ m(T)=\dim \ker( T^{*}+iE)$, $n(T)=\dim \ker(T^{*}-iE )$.
In the multipoint cases it may be faced with different views.
Let $ T_1 $ and $ T_2 $ be minimal operators generated by the linear
singular differential expression
$$
l(u)=itu'(t)
$$
in the weighted Hilbert $ L_{\alpha} ^2(-\infty,a )$ and
$ L_{\alpha} ^2(b,\infty )$, $ \alpha (t)=t$, $a,b \in \mathbb{R} $
of functions, respectively. In this case it is clear that
$$
\left( m(T_1),n(T_1)\right) =(0,1), \quad
\left( m(T_2),n(T_2)\right) =(1,0)
$$
Consequently, the operators $ T_1 $ and $ T_2 $ are maximal symmetric.
Hence they are not any selfadjoint extensions. However, direct sum
$T= T_1 \oplus T_2 $ of operators $ T_1 $ and $ T_2 $ in the direct
sum $ \mathcal{H}= L_{\alpha} ^2(-\infty,a )\oplus L_{\alpha} ^2(b,\infty ) $
have an equal defect numbers (1,1). Then by the general theory it has
at least one selfadjoint extension \cite{n2}.
Indeed it can be easily shown that the multipoint differential expression
$$
l(u)=\left( itu_1'(t),itu_2'(t)\right),\quad
u=(u_1,u_2),\quad u_1\in D(T_1^{*}), \quad u_2\in D(T_2^{*})
$$
with boundary condition $ u_2(b)=u_1(a) $
generates selfadjoint extension of $ T_1 \oplus T_2 $.
There are different methods for the description all selfadjoint
extensions of minimal symmetric operators. For example, in mathematical
literature the Eweritt-Zettl-Markus and Calkin-Gorbachuk methods are very
important of them.
The models of many physical and technical phenomena are expressed
as differential operators. Therefore operator theory plays an exceptionally
important role in modern mathematics, especially in the modelling of
processes of multi-particle quantum mechanics, quantum field theory,
the multipoint boundary value problems for differential equations
\cite{a1,z1}.
Although the first studies of the theory multipoint differential operators
were performed at the beginning of twentieth century, most of them which
are about the investigation of the theory and application to spectral problems,
have been found since 1950 \cite{e1,k1,l1,l2,z1,z2}.
In the article by Everitt and Zettl \cite{e1} in the scalar case,
all selfadjoint extensions of the minimal operator generated by
Lagrange-symmetric any order quasi-differential expression with equal
deficiency indexes in terms of boundary conditions are described by
Glazman-Krein-Naimark method for regular and singular cases in the
direct sum of corresponding Hilbert spaces of functions.
\section{Statement of the problem}
Let $ H $ be a separable Hilbert space and let $ a,b\in \mathbb{R} $.
In the Hilbert space
$ L^2_{1/\alpha}(H,(-\infty,a))\oplus L^2_{1/\beta}(H,(b,+\infty)) $
of a vector-functions
we consider the following linear multipoint differential-operator
expression for first order in a form
\begin{equation} \label{e1}
l(w)=(k(u),m(v)),
\end{equation}
where $ w=(u,v)$, $k(u)=i\alpha(t)u'(t)+Au(t)$,
$m(v)=i\beta(t)v'(t)+Bv(t)$,
$ \alpha\in C(-\infty,a)$, $\beta\in C(b,+\infty) $ and there exist positive
numbers $ \alpha_1,\alpha_2,\beta_1,\beta_2 $ such that
$\alpha_1\leq\alpha(t)\leq\alpha_2 $ for any $ tb $ and for simplicity assumed
that $ A $ and $ B $ are linear bounded selfadjoint operators in $ H $.
In similar way in \cite{g2} the minimal $ K_0(M_0) $ and maximal $ K(M) $ operators
associated with differential expression $ k(m) $ in
$ L^2_{1/\alpha}(H,(-\infty,a))(L^2_{1/\beta}(H,(b,+\infty))) $
can be constructed.
The operators $ L_0=K_0\oplus M_0 $ and $L=K\oplus M $
in the Hilbert space
$ \mathcal{H}=L^2_{1/\alpha}(H,(-\infty,a)) \oplus L^2_{1/\beta}(H,(b,+\infty)) $
are called minimal and maximal operators associated with differential
expression \eqref{e1}, respectively.
It is clear that operator $ L_0 $ is a symmetric and $ L_0^{*}=L $
in $ \mathcal{H}$. The minimal operator $ L_0 $ is not maximal.
Indeed, differential expression \eqref{e1} with boundary condition
$ u(a)=v(b) $ generates a selfadjoint extension of $ L_0$.
The main goal in this article is to describe all selfadjoint extensions
of minimal operator $ L_0 $ in $ \mathcal{H} $ in terms of boundary
values (Sec. 3). In section 4 the structure of spectrum of these extensions
will be investigated.
\section{Description of Selfadjoint Extensions}
In this section using by Calkin-Gorbachuk method will be investigated
the general representation of selfadjoint extensions of minimal operator $L_0 $.
Firstly, prove the following proposition.
\begin{lemma} \label{lem3.1}
The deficiency indexes of the operators $ K_0 $ and $ M_0 $ are of the form
$$
(m(K_0),n(K_0))=(\dim H,0), \quad
(m(M_0),n(M_0))=(0,\dim H).
$$
\end{lemma}
\begin{proof}
From the conditions under the weight functions $ \alpha ( \cdot)$ and
$ \beta ( \cdot)$ imply that
$$
\frac{1}{\alpha_2}\leq\frac{1}{\alpha(t)}
\leq \frac{1}{\alpha_1}, \; tb\,.
$$
From these relations we have
$$
\int_{-\infty}^{a}\frac{dt}{\alpha(t)}=\int_{b}^{\infty}\frac{dt}{\beta(t)}=+\infty
$$
On the other hand, it is clear that the general solutions of differential
equations
$$
i\alpha(t)u'_{\pm}\pm iu_{\pm}=0, \; tb
$$
in the $L^2_{1/\alpha}(H,(-\infty,a))$ and $ L^2_{1/\beta}(H,(b,+\infty)) $
are
\[
u_{\pm}(t)=\exp\Big( \mp \int_{-\infty}^{t}\frac{ds}{\alpha(s)}\Big)f, \quad
f\in H, \; tb
\]
respectively
From these representations we have
\begin{align*}
\| u_{+}\|^2_{L^2_{1/\alpha}(H,(-\infty,a))}
&= \int_{-\infty}^{a} \frac{1}{\alpha(t)}\| u_{+}(t)\|^2_{H}dt\\
&= \int_{-\infty}^{a} \frac{1}{\alpha(t)}
\exp\Big( -2\int_{-\infty}^{t}\frac{ds}{\alpha(s)}\Big)dt\| f \|^2_{H} \\
&= \int_{-\infty}^{a}\exp\Big( -2\int_{-\infty}^{t}\frac{ds}{\alpha(s)}\Big)
d\Big( \int_{-\infty}^{t}\frac{ds}{\alpha(s)} \Big) \| f \|^2_{H}\\
&= \frac{1}{2} \| f \|^2_{H} <\infty
\end{align*}
Consequently,
$$
\dim\ker(K+iE)=\dim H
$$
On the other hand it is clear that for any $ f\in H $ the solutions
$$
u_{-}(t)=\exp\Big( \int_{\ -\infty}^{t}\frac{ds}{\alpha(s)}\Big)f
\notin L^2_{1/\alpha}(H,(-\infty,a)).
$$
Hence
$\dim\ker (K-iE)=0$.
In a similar way we can show that
$$
m(M_0)=\dim\ker (M+iE)=0 \quad \text{and} \quad
n(M_0)=\dim\ker (M-iE)=\dim H
$$
This completes the proof of theorem.
\end{proof}
From last Lemma \ref{lem3.1} we have
$$
m(L_0)=m(K_0)+m(M_0)=\dim H \quad \text{and} \quad
n(L_0)=n(K_0)+n(M_0)=\dim H
$$
Consequently, $ (m(L_0),n(L_0))=(\dim H, \dim H)$.
Consequently, the minimal operator $ L_0 $ has a selfadjoint extensions;
see \cite{n2}.
\begin{definition}[\cite{g2}] \label{def3.2} \rm
Let $ \mathbf{H} $ be any Hilbert space and
$ S:D(S)\subset \mathbf{H}\to \mathbf{H} $ be a closed densely
defined symmetric operator in the Hilbert space $ \mathbf{H} $ having
equal finite or infinite deficiency indexes.
A triplet $( \mathcal{B},\gamma_1,\gamma_2)$,
where $ \mathcal{B} $ is a Hilbert space, $ \gamma_1 $ and $ \gamma_2 $
are linear mappings from $ D(S^{*}) $ into $ \mathcal{B} $, is called a
space of boundary values for the operator $ S $ if for any $ f,g\in D(S^{*}) $
$$
(S^{*}f,g)_{\mathbf{H}}-(f,S^{*}g)_{\mathbf{H}}
=(\gamma_1(f),\gamma_2(g))_{\mathcal{B}}-(\gamma_2(f),\gamma_1(g))_{\mathcal{B}}
$$
while for any $ F_1,F_2\in \mathcal{B} $, there exists an element
$ f\in D(S^{*}) $ such that $ \gamma_1(f)=F_1 $ and $ \gamma_2(f)=F_2 $.
\end{definition}
It is known that for any symmetric operator with equal deficiency indexes
have at least one space of boundary values \cite{g2}.
\begin{lemma} \label{lem3.3}
If $ u\in D(K) $ and $ v\in D(M)$, then $ \lim _{t\to-\infty}u(t)=0$,
$\lim _{t\to+\infty}v(t)=0 $ and $ u(a),v(b)\in H$.
\end{lemma}
\begin{proof}
Under the assumptions for $ \alpha(\cdot) $ and $ \beta(\cdot) $ we have
\begin{gather*}
\frac{1}{\alpha_2}\leq\frac{1}{\alpha(t)}\leq \frac{1}{\alpha_1}, \; tb\,.
\end{gather*}
Then for any pair of functions $ x(\cdot)\in L^2(H,(-\infty,a))$,
$y(\cdot)\in L^2(H,(b,+\infty))$, from the above relations,
\begin{gather*}
0\leq\int_{-\infty}^{a}\frac{1}{\alpha(t)}\| x(t)\|^2_{H}dt
\leq\frac{1}{\alpha_1}\int_{-\infty}^{a}\| x(t)\|^2_{H}dt, \\
0\leq\int_{b}^{\infty}\frac{1}{\beta(t)}\| y(t)\|^2_{H}dt
\leq\frac{1}{\beta_1}\int_{b}^{\infty}\| y(t)\|^2_{H}dt
\end{gather*}
Consequently,
$$
L^2(H,(-\infty,a))\subset L^2_{1/\alpha}(H,(-\infty,a)) \quad\text{and} \quad
L^2(H,(b,+\infty))\subset L^2_{1/\beta}(H,(b,+\infty))\,.
$$
Similarly, if $ x(\cdot)\in L^2_{1/\alpha}(H,(-\infty,a))$ and
$y(\cdot)\in L^2_{1/\beta}(H,(b,+\infty))$, then from the conditions
on the weight functions we have
\begin{gather*}
\frac{1}{\alpha_2}\int_{-\infty}^{a} \| x(t)\|^2_{H}dt
\leq\int_{-\infty}^{a}\frac{1}{\alpha(t)}\| x(t)\|^2_{H}dt,
\\
\frac{1}{\beta_2}\int_{b}^{\infty}\| y(t)\|^2_{H}dt
\leq\int_{b}^{\infty}\frac{1}{\beta(t)}\| y(t)\|^2_{H}dt
\end{gather*}
Hence,
$$
L^2_{1/\alpha}(H,(-\infty,a))\subset L^2(H,(-\infty,a)) \quad \text{and} \quad
L^2_{1/\beta}(H,(b,+\infty))\subset L^2(H,(b,+\infty)).
$$
Therefore it is obtained that
$$
L^2_{1/\alpha}(H,(-\infty,a))= L^2(H,(-\infty,a)) \quad \text{and} \quad
L^2_{1/\beta}(H,(b,+\infty))= L^2(H,(b,+\infty)).
$$
On the other hand, from the relations
$$
D(K)\subset L^2(H,(-\infty,a)) \quad \text{and} \quad
D(M)\subset L^2(H,(b,+\infty)),
$$
we have
$\lim _{t\to-\infty}u(t)=0$ and $\lim _{t\to+\infty}v(t)=0$ for any
$ u\in D(K) $ and $ v\in D(M)$.
Now we assume that $ u\in D(K)$. Then
$$
u\in L^2_{1/\alpha}(H,(-\infty,a)) \quad \text{and} \quad
\alpha u'\in L^2_{1/\alpha}(H,(-\infty,a))
$$
This implies
$$
\frac{1}{\sqrt{\alpha}}u\in L^2(H,(-\infty,a))\quad \text{and} \quad
\sqrt{\alpha} u'\in L^2(H,(-\infty,a))
$$
Then for any $ tb,
\end{gather*}
then it is clear that $ (u,v)\in D(L) $ and $ \gamma_1(w)=f$, $\gamma_2(w)=g$.
\end{proof}
Lastly, using the method in \cite{g2} it can be established the following results.
\begin{theorem} \label{thm3.5}
If $ \widetilde{L} $ is a selfadjoint extension of the minimal operator
$ L_0 $ in $ \mathcal{H} $, then it generates by the differential-operator
expression \eqref{e1} and boundary condition
$$
v(b)=Wu(a),
$$
where $ W:H\to H $ is a unitary operator. Moreover, the unitary operator
$ W $ in $ H $ is determined uniquely by the extension $ \widetilde{L} $,
i.e. $ \widetilde{L}=L_{W}$ and vice versa.
\end{theorem}
\section{Spectrum of the selfadjoint extensions}
In this section the structure of the spectrum of the selfadjoint extension
$ L_{W} $ in $ \mathcal{H} $ will be investigated.
Firstly, prove the following results.
\begin{theorem} \label{thm4.1}
The point spectrum of selfadjoint extension $ L_{W} $ is empty, i.e.
$$
\sigma_{p}(L_{W})=\emptyset.
$$
\end{theorem}
\begin{proof}
Consider the eigenvalue problem
$$
l(w)=\lambda w, \quad w=(u,v)\in \mathcal{H}, \; \lambda\in \mathbb{R}
$$
with boundary condition $ v(b)=Wu(a) $.
From this it is obtained that
\begin{gather*}
i\alpha(t)u'(t)+Au(t)=\lambda u(t), \; tb, \\
v(b)=Wu(a)
\end{gather*}
The general solutions of the above equations are
\begin{gather*}
u(\lambda;t)=\exp\Big(-i(A-\lambda)\int_{t}^{a}\frac{ds}{\alpha(s)} \Big)
f_{\lambda}, \quad f_{\lambda} \in H, \; tb, \\
v(\lambda;b)=Wu(\lambda;a)
\end{gather*}
It is clear that for $ f_{\lambda} \neq 0 $ and $ g_{\lambda} \neq 0 $
the solutions $ u(\lambda;\cdot)\notin L^2_{1/\alpha}(H,(-\infty,a))$ and
$ v(\lambda;\cdot)\notin L^2_{1/\beta}(H,(b,\infty))$.
Therefore, for every unitary operator $ W $ we have
$ \sigma_{p}(L_{W})=\emptyset$.
Since the residual spectrum for any selfadjoint operator in a Hilbert space
is empty, then we study the continuous spectrum of selfadjoint extensions $ L_{W} $
of the minimal operator $ L_0 $.
On the other hand from the general theory of linear selfadjoint operators
in Hilbert spaces for the resolvent set $ \rho(L_{W})$ of any selfadjoint
extension $ L_{W} $ is true
$$
\rho (L_{W})\supset \{ \lambda\in \mathbb{C}:
\operatorname{Im}\lambda \neq 0 \}.
$$
\end{proof}
For the continuous spectrum of selfadjoint extensions the following proposition
is true.
\begin{theorem} \label{thm4.2}
The continuous spectrum of the selfadjoint extension $ L_{W} $ is of the form
$$
\sigma_{c}(L_{W})=\mathbb{R}.
$$
\end{theorem}
\begin{proof}
For $ \lambda\in \mathbb{C}$ with $\lambda_{i}=\operatorname{Im}\lambda>0 $ and
$ f=(f_{a},f_{b})\in \mathcal{H} $ the norm of function
$ R_{\lambda} (L_{W})f(t) $ in $ \mathcal{H} $ satisfies
\begin{align*}
\| R_{\lambda} (L_{W}))f(t) \|_{\mathcal{H}}^2
& = \| \exp \Big( i(\lambda-A)\int_{t}^{a}\frac{ds}{\alpha(s)}\Big) f_{\lambda}
\\
& \quad + i\int_{t}^{a}\exp \Big( i(A-\lambda)\int_{s}^{t}\frac{d\tau}{\alpha(\tau)}
\Big) \frac{f_{a}(s)}{\alpha(s)}ds\|^2_{L^2_{1/\alpha}(H,(-\infty,a))} \\
& \quad + \| i \int_{t}^{\infty}\exp
\Big( i(B-\lambda)\int_{s}^{t}\frac{d\tau}{\beta(\tau)}\Big)
\frac{f_{b}(s)}{\beta(s)}ds\|^2_{L^2_{1/\beta}(H,(b,\infty))} \\
&\geq \| i \int_{t}^{\infty}\exp
\Big( i(B-\lambda)\int_{s}^{t}\frac{d\tau}{\beta(\tau)}\Big)
\frac{f_{b}(s)}{\beta(s)}ds\|^2_{L^2_{1/\beta}(H,(b,\infty))}
\end{align*}
The vector functions $ f^{*}(\lambda; t )$ in the form
\[
f^{*}(\lambda; t )=\Big( 0, \exp\Big(-i(\overline{\lambda}-B)
\int_{b}^{t}\frac{ds}{\beta(s)} \Big)f \Big),
\]
with $\lambda\in \mathbb{C}$, $ \lambda_{i}=\operatorname{Im}\lambda>0$,
$f\in H $ belong to $ \mathcal{H}$. Indeed,
\begin{align*}
\| f^{*}(\lambda; t )\|^2_{\mathcal{H}}
& = \int_{b}^{\infty} \frac{1}{\beta(t)}\| \exp
\Big(-i(\overline{\lambda}-B)\int_{b}^{t}\frac{ds}{\beta(s)} \Big)f \|^2_{H}dt \\
& = \int_{b}^{\infty} \frac{1}{\beta(t)}
\exp\Big(-2\lambda_{i}\int_{b}^{t}\frac{ds}{\beta(s)} \Big)dt\| f\|^2_{H} \\
& = \frac{1}{2\lambda_{i}}\| f\|^2_{H}<\infty
\end{align*}
For the such functions $ f^{*}(\lambda; \cdot )$ we have
\begin{align*}
&\| R_{\lambda} (L_{W})f^{*}(\lambda; \cdot ) \|^2_{\mathcal{H}}\\
& \geq \| i\int_{t}^{\infty}\frac{1}{\beta(s)}
\exp\Big( i(B-\lambda)\int_{s}^{t}\frac{d\tau}{\beta(\tau)}
-i(\overline{\lambda}-B)\int_{b}^{s}\frac{d\tau}{\beta(\tau)}\Big)
fds\|^2_{L^2_{1/\beta}(H,(b,\infty))} \\
& = \| \exp\Big( -i\lambda \int_{b}^{t}\frac{d\tau}{\beta(\tau)}
+iB\int_{b}^{t}\frac{d\tau}{\beta(\tau)}\Big) \\
&\quad\times \int_{t}^{\infty}\frac{1}{\beta(s)}\exp
\Big( -2\lambda_{i}\int_{b}^{s}\frac{d\tau}{\beta(\tau)}\Big)
fds\|^2_{L^2_{1/\beta}(H,(b,\infty))}\\
& = \| \exp\Big( \lambda_{i} \int_{b}^{t}\frac{d\tau}{\beta(\tau)}\Big)
\int_{t}^{\infty}\frac{1}{\beta(s)}
\exp \Big( -2\lambda_{i}\int_{b}^{s}\frac{d\tau}{\beta(\tau)}\Big)
ds\|^2_{L^2_{1/\beta}(H,(b,\infty))}\| f\|_{H}^2\\
& = \| \frac{1}{2\lambda_{i}}\exp\Big( -\lambda_{i} \int_{b}^{t}
\frac{d\tau}{\beta(\tau)}\Big)\|^2_{L^2_{1/\beta}(H,(b,\infty))}\| f\|_{H}^2\\
& = \frac{1}{4\lambda_{i}^2}\int_{b}^{\infty}\frac{1}{\beta(t)}
\exp\Big( -2\lambda_{i} \int_{b}^{t}\frac{d\tau}{\beta(\tau)}\Big)dt \| f\|_{H}^2 \\
& = \frac{1}{8\lambda_{i}^{3}}\| f\|_{H}^2\,.
\end{align*}
From this,
$$
\| R_{\lambda} (L_{W})f^{*}(\lambda; \cdot )\|_{\mathcal{H}}
\geq \frac{\| f\|_{H}^2}{2\sqrt{2}\lambda_{i}\sqrt{\lambda_{i}}}
=\frac{1}{2\lambda_{i}} \| f^{*}(\lambda; t )\|_{\mathcal{H}},
$$
i.e., for $ \lambda_{i}=\operatorname{Im}\lambda>0 $ and $ f\neq 0 $ is valid
$$
\frac{\| R_{\lambda} (L_{W})f^{*}(\lambda; \cdot )
\|_{\mathcal{H}} }{ \| f^{*}(\lambda; t )\|_{\mathcal{H}}}
\geq\frac{1}{2\lambda_{i}}
$$
On the other hand it is clear that
$$
\| R_{\lambda} (L_{W})\|\geq\frac{\| R_{\lambda} (L_{W})f^{*}(\lambda;
\cdot )\|_{\mathcal{H}} }{ \| f^{*}(\lambda; t )\|_{\mathcal{H}}}, \quad f\neq 0
$$
Consequently,
$$
\| R_{\lambda} (L_{W})\|\geq\frac{1}{2\lambda_{i}} \quad \text{for }
\lambda\in \mathbb{C},\ \lambda_{i}=\operatorname{Im}\lambda>0.
$$
\end{proof}
Using the above theorem, the spectrum of the singular differential operator
generated by differential expression
$$
l((u,v))=\Big( i\frac{t^{2\alpha}+1}{t^{2\alpha}} u'(t,x)
+xu(t,x),i\frac{t^{2\beta}+1}{t^{2\beta}} v'(t,x)+xv(t,x)\Big), \quad
\alpha,\beta>0
$$
with boundary condition
$$
u(1,x)=e^{i\varphi}u(-1,x), \ \varphi\in [0,2\pi)
$$
in the direct sum
$ L^2_{(t^{2\alpha}+1)/t^{2\alpha}}((-\infty,-1)
\times \mathbb{R} )\oplus L^2_{(t^{2\beta}+1)/t^{2\beta}}((1,\infty)
\times \mathbb{R} ) $ is purely continuous and coincides with $ \mathbb{R}$.
Note that another approach for the singular differential operators for
$n$th order in the scalar case has been given in \cite{s1}.
In special case of functions $ \alpha(\cdot) $ and $ \beta(\cdot) $
the analogous results have been obtained in \cite{b1}.
\subsection*{Acknowledgements}
The authors would like to thank to Prof. A. Ashyralyev (Fatih University, Istanbul)
for his interesting and encouragement discussions.
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