\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 143, pp. 1--8.\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-2017/143\hfil Selfadjoint singular differential operators] {First-order selfadjoint singular differential operators in a Hilbert space of vector functions} \author[P. Ipek, B. Yilmaz, Z. I. Ismailov \hfil EJDE-2017/143\hfilneg] {Pembe Ipek, B\"ulent Yilmaz, Zameddin I. Ismailov} \address{Pembe Ipek \newline Karadeniz Technical University, Institute of Natural Sciences, 61080, Trabzon, Turkey} \email{ipekpembe@gmail.com} \address{B\"ulent Yilmaz \newline Marmara University, Department of Mathematics, Kad{\i}k\"oy, 34722, Istanbul, Turkey} \email{bulentyilmaz@marmara.edu.tr} \address{Zameddin I. Ismailov \newline Karadeniz Technical University, Department of Mathematics, 61080, Trabzon, Turkey} \email{zameddin.ismailov@gmail.com} \dedicatory{Communicated by Ludmila Pulkina } \thanks{Submitted March 17, 2018. Published June 17, 2017.} \subjclass[2010]{47A10, 47B25} \keywords{Multipoint singular differential expression; deficiency indeces; \hfill\break\indent symmetric and selfadjoint differential operator; spectrum} \begin{abstract} In this article, we give a representation of all selfadjoint extensions of the minimal operator generated by first-order linear symmetric multipoint singular differential expression, with operator coefficient in the direct sum of Hilbert spaces of vector-functions defined at the semi-infinite intervals. To this end we use the Calkin-Gorbachuk method. Finally, the geometry of spectrum set of such extensions is researched. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In the first years of the previous century, von Neumann \cite{rf11} and Stone \cite{rf10} investigated the theory of selfadjoint extensions of linear densely defined closed symmetric operators in a Hilbert spaces. Applications to scalar linear even order symmetric differential operators and description of all selfadjoint extensions in terms of boundary conditions due to Glazman in his seminal work \cite{rf5} and in the book of Naimark \cite{rf8}. In this sense the famous Glazman-Krein-Naimark (or Everitt-Krein-Glazman-Naimark) Theorem in the mathematical literature it is to be noted. In the mathematical literature there is another method co-called Calkin-Gorbachuk method. (see \cite{rf6,rf9}). Our motivation for this article originates from the interesting researches of Everitt, Markus, Zettl, Sun, O'Regan, Agarwal \cite{rf2,rf3,rf4,rf12} in scalar cases. Throughout this paper we consider Zettl and Suns's view about these topics \cite{rf12}. A selfadjoint ordinary differential operator in Hilbert space is generated by two things: \begin{itemize} \item[(1)] a symmetric (formally selfadjoint) differential expression; \item[(2)] a boundary condition which determined selfadjoint differential operators; \end{itemize} And also for a given selfadjoint differential operator, a basic question is: What is its spectrum? In this work in Section 3 the representation of all selfadjoint extensions of a multipoint symmetric quasi-differential operator, generated by first-order symmetric differential-operator expression (for the definition see \cite{rf4}) in the direct sum of Hilbert spaces of vector-functions defined at the semi-infinite intervals in terms of boundary conditions are described. In sec. 4 the structure of spectrum of these selfadjoint extensions is investigated. \section{Statement of the problem} In the direct sum $ \mathcal{H}=L^2(H,(-\infty, a_1)) \oplus L^2(H,(a_2,\infty))$, $H$ is a separable Hilbert space, and $ a_1, a_2 \in \mathbb{R} $ will be considered for the multipoint differential-operator expression in the form \begin{gather*} l(u)=(l_1(u_1),l_2(u_2)), \\ l_{k}(u_{k})=i\rho_{k}u_{k}'+\frac{1}{2}i\rho_{k}'u_{k}+A_{k}u_{k}, \quad k=1,2, \end{gather*} where \begin{itemize} \item[(1)] $ \rho_1: (-\infty,a_1)\to (0,\infty)$, $\rho_2: (a_2,\infty)\to (0,\infty)$; \item[(2)] $ \rho_1\in AC_{loc}(-\infty,a_1)$ and $ \rho_2\in AC_{loc}(a_2,\infty) $; \item[(3)] $ \int_{-\infty}^{a_1}\frac{ds}{\rho_1(s)}=\infty$, $\int_{a_2}^{\infty}\frac{ds}{\rho_2(s)}=\infty $; \item[(4)] $ A_{k}^{*}=A_{k}:D(A_{k})\subset H\to H$, $k=1,2$. \end{itemize} The minimal operators $ L_0^{1} $ and $ L_0^2 $ corresponding to differential-operator expressions $ l_1 $ and $ l_2 $ in $ L^2(H,(-\infty, a_1)) $ and $ L^2(H,(a_2, \infty)) $, respectively, can be defined by a standard processes, see\cite{rf7}. The operators $ L_1=(L_0^{1})^{*}$ and $L_2=(L_0^2)^{*} $ are maximal operators corresponding to $ l_1 $ and $ l_2 $ in $ L^2(H,(-\infty, a_1)) $ and $ L^2(H,(a_2, \infty)) $, respectively. In this case the operators $$ L_0=L_0^{1}\oplus L_0^2 \quad \text{and} \quad L=L^{1}\oplus L^2 $$ will be indicating the minimal and maximal operators corresponding to differential expression on $ \mathcal{H} $, respectively. It is clear that \begin{gather*} D(L^{1}) = \{ u_1\in L^2(H,(-\infty, a_1)): l_1(u_1) \in L^2(H,(-\infty, a_1)\}, \\ D(L^{1}_0) = \{ u_1\in D(L^{1}): (\sqrt{\rho_1}u_1)(a_1)=0 \} \end{gather*} and \begin{gather*} D(L^2) = \{ u_2\in L^2(H,(a_2,\infty)): l_2(u_2)\in L^2(H,(a_2,\infty)\}, \\ D(L^2_0) = \{ u_2\in D(L^2):(\sqrt{\rho_2}u_2)(a_2)=0\} . \end{gather*} \section{Description of Selfadjoint Extensions} In this section using the Calkin-Gorbachuk method will be investigated the general representation of selfadjoint extensions of minimal operator $ L_0 $. Firstly we prove the following result. \begin{lemma} \label{lem3.1} The deficiency indices of the operators $ L^{1}_0 $ and $ L^2_0 $ are of the form $$ (m(L^{1}_0),n(L^{1}_0))=(0,\dim H), \quad (m(L^2_0),n(L^2_0))=(\dim H,0). $$ \end{lemma} \begin{proof} Now for simplicity we assume that $ A_1=A_2=0$. It is clear that the general solutions of differential equations \begin{gather*} i\rho_1(t)u'_{1\pm}(t)+ \frac{1}{2}i\rho'_1(t)u_{1\pm}(t) \pm iu_{1\pm} (t)=0, \quad ta_2 \end{gather*} in $L^2(H,(-\infty,a_1))$ and $ L^2(H,(a_2,+\infty)) $ are in the form $$ u_{1\pm}(t)=\exp\Big( \pm \int_{t}^{c_1}\frac{2\pm \rho_1'(s)}{2\rho_1(s)} ds\Big)f_1, \quad f_1\in H, \; ta_2, \; c_2>a_2 $$ respectively. From these representations we have \begin{align*} &\| u_{1+}\|^2_{L^2(H,(-\infty,a_1))} \\ &= \int_{-\infty}^{a_1} \| u_{1+}(t)\|^2_{H}dt \\ &= \int_{-\infty}^{a_1} \exp\Big( \int_{t}^{c_1} \frac{2+ \rho_1'(s)}{\rho_1(s)}ds\Big)dt \| f_1\|_{H}^2 \\ &= \int_{-\infty}^{a_1} \frac{\rho_1 (c_1)}{\rho_1 (t)} \exp\Big( \int_{t}^{c_1} \frac{2}{\rho_1 (s)}ds \Big) dt \| f_1\|_{H}^2 \\ &= \frac{\rho_1 (c_1)}{2}\int_{-\infty}^{a_1} \exp\Big( \int_{t}^{c_1} \frac{2}{\rho_1 (s)}ds \Big) d\Big( -\int_{t}^{c_1} \frac{2}{\rho_1 (s)}ds \Big) \| f_1\|_{H}^2 \\ &= -\frac{\rho_1 (c_1)}{2} \Big[ \exp\Big( \int_{a_1}^{c_1} \frac{2}{\rho_1 (s)}ds \Big)- \exp\Big( \int_{-\infty}^{c_1} \frac{2}{\rho_1 (s)}ds \Big) \Big] \| f_1\|_{H}^2=\infty. \end{align*} Consequently, $$ \dim\ker(L_0^{1}+iE)=0 $$ On the other hand it is clear that \begin{align*} &\| u_{1-}\|^2_{L^2(H,(-\infty,a_1))}\\ &= \int_{-\infty}^{a_1} \| u_{1-}(t)\|^2_{H}dt \\ &= \int_{-\infty}^{a_1} \exp\Big( -\int_{t}^{c_1} \frac{2+ \rho_1'(s)}{\rho_1(s)}ds\Big)dt \| f_1\|_{H}^2 \\ &= \int_{-\infty}^{a_1} \frac{\rho_1 (c_1)}{\rho_1 (t)} \exp\Big( - \int_{t}^{c_1} \frac{2}{\rho_1 (s)}ds \Big) dt \| f_1\|_{H}^2 \\ &= \frac{\rho_1 (c_1)}{2}\int_{-\infty}^{a_1} \exp\Big( - \int_{t}^{c_1} \frac{2}{\rho_1 (s)}ds \Big) d\Big( - \int_{t}^{c_1} \frac{2}{\rho_1 (s)}ds \Big) \| f_1\|_{H}^2 \\ &= \frac{\rho_1 (c_1)}{2} \Big[ \exp\Big( - \int_{a_1}^{c_1} \frac{2}{\rho_1 (s)}ds \Big) - \exp\Big( - \int_{-\infty}^{c_1} \frac{2}{\rho_1 (s)}ds \Big) \Big] \| f_1\|_{H}^2 \\ &= \frac{\rho_1 (c_1)}{2} \exp\Big( - \int_{a_1}^{c_1} \frac{2}{\rho_1 (s)}ds \Big) \| f_1\|_{H}^2<\infty. \end{align*} Therefore, $$ u_{1-}(t)=\exp\Big( \int_{a_1}^{t}\frac{2 - \rho_1'(s)}{2\rho_1(s)}ds\Big)f_1 \in L^2(H,(-\infty,a_1)). $$ Hence $$ \dim\ker (L_0^{1}-iE)=\dim H $$ In a similar way it can be shown that $$ m(L^2_0)=\dim\ker (L_0^2+iE)=\dim H \quad \text{and} \quad n(L^2_0)=\dim\ker (L_0^2-iE)=0 $$ This completes the proof. \end{proof} Consequently, the minimal operator $ L_0 $ has selfadjoint extensions; see \cite{rf6}. To describe these extensions we need to obtain the space of boundary values. \begin{definition}[\cite{rf6}] \label{def3.2} \rm Let $ H $ be any Hilbert space and $ S:D(S)\subset H\to H $ be a closed densely defined symmetric operator in the Hilbert space $ \mathcal{H} $ having equal finite or infinite deficiency indices. 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^{*}) $ $$ (S^{*}f,g)_{H}-(f,S^{*}g)_{H} =(\gamma_1(f),\gamma_2(g))_{\mathcal{B}}-(\gamma_2(f),\gamma_1(g))_{\mathcal{B}} $$ 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} \begin{lemma} \label{lem3.3} Let \begin{gather*} \gamma_1:D(L)\to H, \quad \gamma_1(u)=\frac{1}{i\sqrt{2}}\big( (\sqrt{\rho_1}u_1) (a_1)+ (\sqrt{\rho_2}u_2)(a_2)\big), \\ \gamma_2:D(L)\to H, \quad \gamma_2(u)= \frac{1}{\sqrt{2}} \left( (\sqrt{\rho_1}u_1)(a_1) - (\sqrt{\rho_2}u_2)(a_2)\right), \end{gather*} where $u=(u_1,u_2) \in D(L)$. Then the triplet $ (H,\gamma_1,\gamma_2)$ is a space of boundary values of the minimal operator $ L_0 $ in $ \mathcal{H}$. \end{lemma} \begin{proof} For any $ u=(u_1,u_2)$, $v=(v_1,v_2) \in D(L) $ \begin{align*} & (Lu,v)_{\mathcal{H}} -(u,Lv)_{\mathcal{H}} \\ & = (L_1u_1,v_1)_{L^2(H,(-\infty, a_1))} + (L_2u_2,v_2)_{L^2(H,(a_2,\infty))} -(u_1,L_1v_1)_{L^2(H,(-\infty, a_1))} \\ &\quad -(u_2,L_2v_2)_{L^2(H,(a_2,\infty))} \\ & = \big[ (i\rho_1 u_1'+\frac{i}{2}\rho_1 'u_1+A_1u_1,v_1)_{L^2(H,(-\infty, a_1))}\\ &\quad -(u_1,i\rho_1 v_1'+\frac{i}{2}\rho_1 'v_1+A_1v_1)_{L^2(H,(-\infty,a_1))} \big] \\ & \quad + \big[ (i\rho_2 u_2'+\frac{i}{2}\rho_2 'u_2+A_2u_2,v_2) _{L^2(H,(a_2,\infty))} \\ &\quad -(u_2,i\rho_2 v_2'+\frac{i}{2}\rho_2 'v_2+A_2v_2)_{L^2(H,(a_2,\infty))} \big] \\ & = i \big[ (\rho_1 u_1',v_1)_{L^2(H,(-\infty,a_1))} +( u_1,\rho_1v_1')_{L^2(H,(-\infty,a_1))}\big] \\ &\quad + \frac{i}{2} \big[ (\rho_1'u_1,v_1)_{L^2(H,(-\infty,a_1))} + (u_1,\rho_1'v_1)_{L^2(H,(-\infty,a_1))}\big] \\ &\quad + i \big[ (\rho_2 u_2',v_2)_{L^2(H,(a_2,\infty))} +( u_2,\rho_2v_2')_{L^2(H,(a_2,\infty))}\big] \\ &\quad +\frac{i}{2} \big[ (\rho_2'u_2,v_2)_{L^2(H,(a_2,\infty))} + (u_2,\rho_2'v_2)_{L^2(H,(a_2,\infty))}\big] \\ & = i \big[ (\rho_1 u_1',v_1)_{L^2(H,(-\infty,a_1))} +( u_1,\rho_1v_1')_{L^2(H,(-\infty,a_1))}\big] + i (\rho_1'u_1,v_1) _{L^2(H,(-\infty,a_1))} \\ & \quad + i \big[ (\rho_2 u_2',v_2)_{L^2(H,(a_2,\infty))} +( u_2,\rho_1v_2')_{L^2(H,(a_2,\infty))}\big] +i (\rho_2'u_2,v_2)L_{(H,(a_2,\infty))} \\ & = i \big[ (\rho_1 u_1'+\rho_1u_1,v_1)_{L^2(H,(-\infty,a_1))} +( \rho_1u_1,v_1')_{L^2(H,(-\infty,a_1))}\big] \\ &\quad + i \big[ (\rho_2 u_2'+\rho_2u_2,v_2)_{L^2(H,(a_2,\infty))} +( \rho_2u_2,v_2')_{L^2(H,(a_2,\infty))}\big] \\ & = i [ ((\rho_1u_1),v_1)'] _{L^2(H,(-\infty,a_1))} +i \left[ ((\rho_2u_2),v_2)'\right] _{L^2(H,(a_2,\infty))} \\ & = i \big[ \left( (\sqrt{\rho_2}u_2)(a_2),(\sqrt{\rho_2}v_2)(a_2)\right) _{H} -\left( (\sqrt{\rho_1}u_1)(a_1),(\sqrt{\rho_1}v_1)(a_1)\right) _{H}\big] \\ & = (\gamma_1(u),\gamma_2(v))_{H}-(\gamma_2(u),\gamma_1(v))_{H}. \end{align*} Now for any element $ f_1, f_2 \in H $ let us find the function $ u=(u_1,u_2)\in D(L) $ such that \begin{gather*} \gamma_1(u)=\frac{1}{i\sqrt{2}}\left( (\sqrt{\rho_1}u_1)(a_1) + (\sqrt{\rho_2}u_2)(a_2)\right) =f_1 , \\ \gamma_2(u)=\frac{1}{\sqrt{2}}\left( (\sqrt{\rho_1}u_1)(a_1) - (\sqrt{\rho_2}u_2)(a_2)\right)=f_2 \end{gather*} From here the following two expressions are obtained $$ (\sqrt{\rho_1}u_1)(a_1)=(i f_1+f_2)/\sqrt{2}, \quad (\sqrt{\rho_2}u_2)(a_2)=(i f_1-f_2)/\sqrt{2}. $$ If we choose the functions $ u_1(\cdot), \ u_2(\cdot) $ in the following forms \begin{gather*} u_1(t)=\frac{1}{\sqrt{\rho_1(t)}}e^{t-a_1}(if_1+f_2)/\sqrt{2}, \quad ta_2, \end{gather*} then it is clear that $ u=(\sqrt{\rho_1}u_1,\sqrt{\rho_2}u_2)\in D(L) $ and $ \gamma_1(u)=f_1$, $\gamma_2(u)=f_2$. \end{proof} \begin{theorem} \label{thm3.4} If $ \widetilde{L} $ is a selfadjoint extension of the minimal operator $ L_0 $ in $ \mathcal{H} $ , then it is generated by the differential-operator expression $ l( \cdot ) $ and the following boundary condition $$ (\sqrt{\rho_2}u_2)(a_2)=W(\sqrt{\rho_1}u_1)(a_1), $$ 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 extensions $ L_{W} $ of the minimal operator $ L_0 $ in $ \mathcal{H} $ will be investigated. First let us prove the following results. \begin{theorem} \label{thm4.1} The point spectrum of the selfadjoint extension $ L_{W} $ is empty, i.e. $\sigma_{p}(L_{W})=\emptyset$. \end{theorem} \begin{proof} Consider the eigenvalue problem $$ l(u)=\lambda u, \quad u=(u_1,u_2)\in \mathcal{H}, \quad \lambda\in \mathbb{R} $$ with the boundary condition $$ (\sqrt{\rho_2}u_2)(a_2)=W(\sqrt{\rho_1}u_1)(a_1). $$ From here the following expressions are obtained \begin{gather*} i\rho_1(t)u_1'(t)+\frac{1}{2}i\rho_1'(t)u_1(t)+A_1u_1(t)=\lambda u_1(t), \quad ta_2,\\ (\sqrt{\rho_2}u_2)(a_2)=W(\sqrt{\rho_1}u_1)(a_1). \end{gather*} The general solutions of these equations are in the form \begin{gather*} u_1(t;\lambda)=\sqrt{\frac{\rho_1(c)}{\rho_1(t)}} \exp\Big(-i(A_1-\lambda)\int_{t}^{c}\frac{ds}{\rho_1(s)} \Big) f^{1}_{\lambda}, \quad f^{1}_{\lambda} \in H, \quad ta_2, \; c>a_2, \\ (\sqrt{\rho_2}u_2)(a_2)=W(\sqrt{\rho_1}u_1)(a_1). \end{gather*} It is clear that for the $ f^{1}_{\lambda} \neq 0 $ and $ f^2_{\lambda} \neq 0 $ the solutions are $ u_1(\cdot ;\lambda)\notin L^2(H,(-\infty,a_1))$ and $ u_2(\cdot ;\lambda)\notin L^2(H,(a_2,\infty))$. Therefore for every unitary operator $ W $ we have $ \sigma_{p}(L_{W})=\emptyset $. \end{proof} Since the residual spectrum for any selfadjoint operator in any Hilbert space is empty, then we have to investigate the continuous spectrum of selfadjoint extensions $ L_{W} $ of the minimal operator $ L_0 $ is investigated. 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 $$ \rho (L_{W})\supset \{ \lambda\in \mathbb{C}: \operatorname{Im} \lambda \neq 0 \}. $$ For the continuous spectrum of selfadjoint extensions we have the following statement. \begin{theorem} \label{thm4.2} The continuous spectrum of any selfadjoint extension $ L_{W} $ in of the form $$ \sigma_{c}(L_{W})=\mathbb{R}. $$ \end{theorem} \begin{proof} For $ \lambda\in \mathbb{C}$, $\lambda_{i}=\operatorname{Im}\lambda>0 $ and $ f=(f_1,f_2)\in \mathcal{H} $ the norm of function $ R_{\lambda} (L_{W})f(t) $ in $ \mathcal{H} $ we have \begin{align*} &\| R_{\lambda} (L_{W}))f(t) \|_{\mathcal{H}}^2 \\ & = \| \frac{1}{\rho_1(t)}\exp \Big( i(\lambda-A_1) \int_{t}^{a_1}\frac{ds}{\rho_1(s)}\Big) f^{1}_{\lambda} \\ &\quad + \frac{i}{\sqrt{\rho_1(t)}}\int_{t}^{a_1} \exp \Big( i(A_1-\lambda)\int_{s}^{t}\frac{d\tau}{\rho_1(\tau)}\Big) \frac{f_1(s)}{\sqrt{\rho_1(s)}}ds\|^2_{L^2(H,(-\infty,a_1))} \\ &\quad + \| \frac{i}{\sqrt{\rho_2(t)}} \int_{t}^{\infty} \exp \Big( i(A_2-\lambda)\int_{s}^{t}\frac{d\tau}{\rho_2(\tau)}\Big) \frac{f_2(s)}{\sqrt{\rho_2(s)}}ds\|^2_{L^2(H,(a_2,\infty))} \\ &\geq \| \frac{i}{\sqrt{\rho_2(t)}} \int_{t}^{\infty} \exp \Big( i(A_2-\lambda)\int_{s}^{t}\frac{d\tau}{\rho_2(\tau)}\Big) \frac{f_2(s)}{\sqrt{\rho_2(s)}}ds\|^2_{L^2(H,(a_2,\infty))}. \end{align*} The vector functions $ f^{*}(t; \lambda )$ in the form \[ f^{*}(t; \lambda )=\Big( 0, \frac{1}{\sqrt{\rho_2(t)}} \exp\Big(-i(\overline{\lambda}-A_2)\int_{a_2}^{t}\frac{ds}{\rho_2(s)} \Big)f \Big), \] with $\lambda\in \mathbb{C}$, $ \lambda_{i}=\operatorname{Im}\lambda>0$, $f\in H $ belong to $ \mathcal{H}$. Indeed, \begin{align*} \| f^{*}(t; \lambda )\|^2_{\mathcal{H}} & = \int_{a_2}^{\infty} \frac{1}{\rho_2(t)}\| \exp\Big(-i(\overline{\lambda}-A_2)\int_{a_2}^{t}\frac{ds}{\rho_2(s)} \Big) f \|^2_{H}dt \\ & = \int_{a_2}^{\infty} \frac{1}{\rho_2(t)} \exp\Big(-2\lambda_{i}\int_{a_2}^{t}\frac{ds}{\rho_2(s)} \Big)dt\| f\|^2_{H} \\ & = \frac{1}{2\lambda_{i}}\| f\|^2_{H}<\infty . \end{align*} For such functions $ f^{*}( \cdot; \lambda )$ we have \begin{align*} &\| R_{\lambda} (L_{W})f^{*}(\lambda; \cdot ) \|^2_{\mathcal{H}} \\ &\geq \| \frac{i}{\sqrt{\rho_2(t)}}\int_{t}^{\infty}\frac{1}{\rho_2(s)} \exp\Big( i(A_2-\lambda)\int_{s}^{t}\frac{d\tau}{\rho_2(\tau)} \\ &\quad -i(\overline{\lambda}-A_2)\int_{a_2}^{s}\frac{d\tau}{\rho_2(\tau)}\Big) fds\|^2_{L^2(H,(a_2,\infty))} \\ & = \| \frac{1}{\sqrt{\rho_2(t)}}\exp\Big( -i\lambda \int_{a_2}^{t} \frac{d\tau}{\rho_2(\tau)}+iA_2\int_{a_2}^{t}\frac{d\tau}{\rho_2(\tau)}\big) \\ & \quad \times \int_{t}^{\infty}\frac{1}{\rho_2(s)} \exp \Big( -2\lambda_{i}\int_{a_2}^{s}\frac{d\tau}{\rho_2(\tau)}\Big) fds\|^2_{L^2(H,(a_2,\infty))}\\ & = \| \frac{1}{\sqrt{\rho_2(t)}}\exp\Big( \lambda_{i} \int_{a_2}^{t} \frac{d\tau}{\rho_2(\tau)}\Big) \int_{t}^{\infty}\frac{1}{\rho_2(s)} \exp \Big( -2\lambda_{i}\int_{a_2}^{s}\frac{d\tau}{\rho_2(\tau)}\Big) ds\|^2_{L^2(H,(a_2,\infty))} \\ &\quad\times \| f\|_{H}^2 \\ & = \| \frac{1}{2\lambda_{i}}\exp \Big( -\lambda_{i} \int_{a_2}^{t}\frac{d\tau}{\rho_2(\tau)}\Big) \|^2_{L^2(H,(a_2,\infty))}\| f\|_{H}^2 \\ & = \frac{1}{4\lambda_{i}^2}\int_{a_2}^{\infty}\frac{1}{\rho_2(t)} \exp\Big( -2\lambda_{i} \int_{a_2}^{t}\frac{d\tau}{\rho_2(\tau)}\Big)dt \| f\|_{H}^2 \\ & = \frac{1}{8\lambda_{i}^{3}}\| f\|_{H}^2 . \end{align*} From this we have $$ \| R_{\lambda} (L_{W})f^{*}( \cdot; \lambda )\|_{\mathcal{H}} \geq \frac{\| f\|_{H}^2}{2\sqrt{2}\lambda_{i}\sqrt{\lambda_{i}}} =\frac{1}{2\lambda_{i}} \| f^{*}(t;\lambda )\|_{\mathcal{H}}. $$ Then for $ \lambda_{i}=\operatorname{Im}\lambda>0 $ and $ f\neq 0 $ the following inequality is valid $$ \frac{\| R_{\lambda} (L_{W})f^{*}( \cdot, \lambda )\|_{\mathcal{H}} } { \| f^{*}(\lambda; t )\|_{\mathcal{H}}}\geq\frac{1}{2\lambda_{i}}. $$ On the other hand it is clear that $$ \| R_{\lambda} (L_{W})\|\geq\frac{\| R_{\lambda} (L_{W})f^{*}(\cdot; \lambda )\|_{\mathcal{H}} } { \| f^{*}(\cdot; \lambda ) \|_{\mathcal{H}}}, \quad f\neq 0. $$ Consequently, for $ \lambda\in \mathbb{C}$ and $\lambda_{i}=\operatorname{Im}\lambda>0 $ we have $$ \| R_{\lambda} (L_{W})\|\geq\frac{1}{2\lambda_{i}}. $$ \end{proof} \begin{remark} \rm In the special case $ \rho_{k}=1$, $k=1,2 $, similar results have been obtained in \cite{rf1}. \end{remark} As an example all selfadjoint extensions $ L_{\varphi} $ of the minimal operator $ L_0 $, generated by the multipoint differential expression \begin{align*} l(u)& = (l_1(u_1),l_2(u_2)) \\ & = \Big( itu_1'(t,x)+\frac{1}{2}iu_1(t,x)-\frac{\partial^2 u_1} {\partial x^2}(t,x), i\sqrt{t}u_2'(t,x) \\ &\quad +\frac{1}{4\sqrt{t}}iu_2(t,x)-\frac{\partial^2 u_2}{\partial x^2}(t,x) \Big), \end{align*} with boundary conditions \begin{gather*} u_1(t,0) = u_1(t,1), \quad u_1'(t,0)=u_1'(t,1), \quad t<-1, \\ u_2(t,0) = u_2(t,1), \quad u_2'(t,0)=u_2'(t,1), \quad t>1 \end{gather*} in the direct sum $ L^2((-\infty,-1)\times (0,1))\oplus L^2((1,\infty)\times (0,1)) $ in terms of boundary conditions are described the boundary condition $$ (t^{1/4}u_2(t))(1,x)=e^{i\varphi}(\sqrt{t}u_1(t))(-1,x), \quad \varphi\in [0,2\pi), \; x\in (0,1). $$ Moreover, the spectrum of such extension is $$ \sigma (L_{\varphi})=\sigma_{c} (L_{\varphi})= \mathbb{R}. $$ \begin{thebibliography}{00} \bibitem{rf1} Bairamov, E; \"{O}zt\"{u}rk Mert, R; Ismailov, Z.; \emph{Selfadjoint extensions of a singular differential operator}. J. Math. Chem. 2012; 50: 1100-1110. \bibitem{rf2} El-Gebeily, M. A.; O'Regan, D.; Agarwal, R.; \emph{Characterization of self-adjoint ordinary differential operators}. Mathematical and Computer Modelling 2011; 54: 659-672. \bibitem{rf3} Everitt, W. N.; Markus, L.; \emph{The Glazman-Krein-Naimark Theorem for ordinary differential operators}. Operator Theory, Advances and Applications 1997; 98: 118-130. \bibitem{rf4} Everitt, W. N.; Poulkou. A.; \emph{Some observations and remarks on differential operators generated by first order boundary value problems}. Journal of Computational and Applied Mathematics 2003; 153: 201-211. \bibitem{rf5} Glazman, I. M.; \emph{On the theory of singular differential operators}. Uspehi Math. Nauk 1962; 40, 102-135, (1950). English translation in Amer. Math. Soc. Translations (1), 4: 1962, 331-372. \bibitem{rf6} Gorbachuk, V. I.; Gorbachuk, M. I.; \emph{Boundary value problems for operator differential equations}. Kluwer, Dordrecht: 1991. \bibitem{rf7} H\"ormander, L.; \emph{On the theory of general partial differential operators}. Acta Mathematica 1955; 94: 161-248. \bibitem{rf8} Naimark, M. A.; \emph{Linear differential operators, II}. NewYork: Ungar, 1968. \bibitem{rf9} Rofe-Beketov, F. S.; Kholkin, A. M.; \emph{Spectral analysis of differential operators}. World Scientific Monograph Series in Mathematics 2005; 7. \bibitem{rf10} Stone, M. H.; \emph{Linear transformations in Hilbert space and their applications in analysis}. Amer. Math. Soc. Collag. 1932; 15: 49-31. \bibitem{rf11} von Neumann, J.; \emph{Allgemeine eigenwerttheories hermitescher funktionaloperatoren}. Math. Ann. 1929-1930; 102: 49-31. \bibitem{rf12} Zettl, A.; Sun, J.; \emph{Survey Article: Self-Adjoint ordinary differential operators and their spectrum}. Rosky Mountain Journal of Mathematics 2015; 45,1: 763-886. \end{thebibliography} \end{document}