\documentclass[twoside]{article} \usepackage{amssymb,amsmath} \pagestyle{myheadings} \markboth{\hfil Completeness of elementary solutions \hfil EJDE--2002/03} {EJDE--2002/03\hfil Yakov Yakubov \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 03, pp. 1--21. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Completeness of elementary solutions of second order elliptic equations in a semi-infinite tube domain % \thanks{ {\em Mathematics Subject Classifications:} 47E05, 35J25, 35P10. \hfil\break\indent {\em Key words:} Abstract differential equations, second order elliptic equations, \hfil\break\indent semi-infinite tube domains, isomorphism, completeness. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Submitted October 29, 2001. Published January 2, 2002.} } \date{} % \author{Yakov Yakubov} \maketitle \begin{abstract} Boundary-value problems for second order abstract differential equations on a semi-axis are considered in this article. We find isomorphisms for the corresponding operators and prove completeness of elementary solutions corresponding to subsets of eigenvalues. As an application of the abstract results, we study second order elliptic equations in semi-infinite tube domains. Our results can be applied to pure differential, integro-differential, functional-differential and equations with a shift. \end{abstract} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12 \section*{Introduction} The question of completeness for systems of eigenvectors corresponding to the whole spectrum arises when solving non-stationary equations. However, when solving stationary equations the question changes to the completeness of systems corresponding to subsets of the spectrum. For general equations this question can be very difficult. For thermal conduction and elasticity systems \cite{TY1,TY2,TY3}, it is an open question. In this article, we consider only equations without mixed derivatives. There are many articles and monographs devoted to the solvability of regular elliptic boundary-value problems in non-smooth bounded and unbounded domains \cite{KO,NP,KMR,KM}. In this article, we obtain algebraic conditions for the solvability of boundary-value problems for second order elliptic equations on semi-infinite cylindrical domains. We also obtain conditions for the completeness of elementary solutions corresponding to subsets of eigenvalues. The presence of an abstract operator in our equation allows us to consider integro-differential equations, functional-differential equations, equations with a shift, in addition to pure differential equations. To the best of our knowledge, our results are new. Similar questions were considered by Yakubov and Yakubov \cite{YY} for fourth order elliptic equations and by Shkalikov \cite{S} for second order elliptic equations in semi-infinite tube domains. In contrast to Shkalikov \cite{S} who assumes that two supplementary Kondratiev problems do not have eigenvalues on the line $\mathop{\rm Re}\lambda=1$, we find sufficient conditions for the completeness of root functions and elementary solutions corresponding to eigenvalues with $\mathop{\rm Re}\lambda_i<0$. We start by giving the notation and definitions to be used in this paper. Let $E$ be a Banach space and $n$ a non-negative integer. Let $W_p^n((0,1);E)$ denote the Banach space of functions with values from $E$ which have generalized derivatives up to order $n$ on $(0,1)$. In this space, we consider the norm $$ \|u\|_{W_p^n((0,1);E)}:=\sum_{k=0}^n\big(\int_0^1\|u^{(k)}(x)\|^p \,dx\big)^{1/p}\,. $$ Let the standard Sobolev space be $W^n_p(0,1):=W^n_p((0,1);\mathbb{ C})$. Let $E_0$ and $E_1$ be two Banach spaces continuously embedded into the Banach space $E$: $E_0\subset E$, $E_1\subset E$. Such spaces are called an {\it interpolation couple} $\{E_0,E_1\}$. We also consider the Banach space \begin{gather*} E_0+E_1:=\big\{u \mid u\in E: u=u_0+u_1\mbox{ with } u_j\in E_j,\; j=0,1 \big\},\\ \|u\|_{E_0+E_1}:=\inf \{\|u_0\|_{E_0}+\|u_1\|_{E_1}: u=u_0+u_1,\, u_j\in E_j\}. \end{gather*} Due to Triebel \cite[1.3.1]{T}, the functional $$ K(t,u):=\inf\{ \|u_0\|_{E_0}+t\|u_1\|_{E_1}: u=u_0+u_1,\, u_j\in E_j\} $$ is continuous on $(0,\infty)$ in $t$, and $$\min\{1,t\}\|u\|_{E_0+E_1}\le K(t,u)\le\max\{1,t\}\|u\|_{E_0+E_1}. $$ An {\it interpolation space} for $\{E_0,E_1\}$ by the $K$-method is defined as follows: \begin{gather*} \|u\|_{(E_0,E_1)_{\theta,p}} :=\big(\int_0^\infty t^{-1-\theta p}K^p(t,u) \,dt \big)^{1/p} ,\quad 0<\theta<1,\ 1\le p<\infty,\\ (E_0,E_1)_{\theta,p} :=\{ u \mid u\in E_0+E_1, \|u\|_{(E_0,E_1)_{\theta,p}}<\infty \}, \\ \|u\|_{(E_0,E_1)_{\theta,\infty}} :=\sup_{t\in(0,\infty)}t^{-\theta}K(t,u), \quad 0<\theta<1,\\ (E_0,E_1)_{\theta,\infty} :=\{u \mid u\in E_0+E_1, \|u\|_{(E_0,E_1)_{\theta,\infty}}<\infty\} . \end{gather*} When $\ell$ is an non-negative integer, $W^\ell_p(G)$ is a standard Sobolev space. Let $$ B^s_{p,q}(G):=(W^{s_0}_p(G),W^{s_1}_p(G))_{\theta,q}, $$ where $s_0,s_1$ are non-negative integers, $0<\theta<1$, $1
0$ \footnote{Singular numbers $s_j$ of the compact operator $A$ from a Hilbert space $H$ into a Hilbert space $H_1$ are eigenvalues $\lambda_j$ of the compact selfadjoint non-negative operator $(A^*A)^{\frac 12}$ in $H$.}; \item There exist \footnote {For $p>4$ the existence of one such ray is enough.} rays $\ell_k$ with angles between neighboring rays less than $\frac {p \pi}2$ and a number $\eta$ such that numbers $\lambda$ from $\ell_k$ and with sufficiently large moduli are regular points for the operator pencil $L(\lambda)$ and $$ \|L(\lambda)^{-1}\|_{B(H,H_1)}\le C|\lambda|^\eta,\quad \lambda\in\ell_k, \ |\lambda|\to\infty. $$ \end{enumerate} Then the spectrum of pencil (\ref{1.1}) is discrete and a system of root vectors of pencil (\ref{1.1}), corresponding to the eigenvalues $\lambda_i$ with $\mathop{\rm Re}\lambda_i\le 0$, is complete in the spaces $H_1$ and $H(B)$. \end{theorem} \paragraph{Proof} When applying a theorem from \cite[p.65]{YY} or \cite[Theorem 3.6, p.71]{Y}) to operator pencil (\ref{1.1}), we have two-fold completeness of a system of root vectors of (\ref{1.1}) in $H_1\oplus H$ and $H(B)\oplus H_1$. Let $v^0, v^1, v^2, \dots, v^s $ be a chain of root vectors of the operator pencil (\ref{1.1}) corresponding to the eigenvalue $\lambda_0$, i.e., \begin{gather} (\lambda_0^2 I+B)v^0=0, \label{1.2}\\ (\lambda_0^2 I+B)v^1+2\lambda_0v^0=0, \label{1.3}\\ (\lambda_0^2 I+B)v^k+2\lambda_0v^{k-1}+v^{k-2}=0,\quad k=2,\dots,s. \label{1.4} \end{gather} Then $-v^0, v^1, -v^2, \dots, (-1)^{s-1}v^s $ is a chain of root vectors of the operator pencil $L(\lambda)$ corresponding to $-\lambda_0$, i.e., $$[(-\lambda_0)^2 I+B](-v^0)=0, $$ which follows from (\ref{1.2}), $$[(-\lambda_0)^2 I+B]v^1+2(-\lambda_0)(-v^0)=0, $$ which follows from (\ref{1.3}), and \begin{gather*} \big[(-\lambda_0)^2 I+B\big](-v^k)+2(-\lambda_0)v^{k-1}+2(-v^{k-2})=0,\quad \text{if } k \text{ is even},\\ \big[(-\lambda_0)^2 I+B\big]v^k+2(-\lambda_0)(-v^{k-1})+2v^{k-2}=0,\quad \text{if } k \text{ is odd}, \end{gather*} which follow from (\ref{1.4}). Let $v(t)$ be an elementary solution of the equation $u''(t)+Bu(t)=0$, $t>0$. Then $$ v(0)=\begin{cases} v^j & \text{if } v(t) \text{ corresponds to } \lambda_0,\\ (-1)^{j+1}v^j & \text{if } v(t) \text{ corresponds to } -\lambda_0. \end{cases} $$ By virtue of the above-mentioned two-fold completeness, $$ \Big\|\begin{pmatrix} F_1\\ F_2 \end{pmatrix} -\sum_{k=1}^N C_{kN} \begin{pmatrix} v_k(0)\\ v_k'(0) \end{pmatrix} \Big\|_{H_1\oplus H} < \varepsilon \quad \text{for eigenvalues } \lambda_k $$ and $$ \Big\|\begin{pmatrix} F_1\\ F_2 \end{pmatrix} -\sum_{k=1}^N C_{kN} \begin{pmatrix} v_k(0)\\ v_k'(0) \end{pmatrix} \Big\|_{H(B)\oplus H_1} < \varepsilon \quad \text{for eigenvalues } \lambda_k, $$ then $$ \|F_1-\sum_{k=1}^N \tilde C_{kN} v_k^j\|_{H_1}<\varepsilon \quad \text{for }\lambda_k \text{ with } \mathop{\rm Re}\lambda_k\le 0$$ and $$ \|F_1-\sum_{k=1}^N \tilde C_{kN} v_k^j\|_{H(B)}<\varepsilon \quad \text{for } \lambda_k \text{ with } \mathop{\rm Re}\lambda_k\le 0 .$$ \hfill $\Box$ \subsection{Isomorphism of problems on the semi-axis} In a Hilbert space $H$, consider a boundary-value problem in $[0,\infty)$ for the second order elliptic equation \begin{gather} L(D)u := u''(x) + Bu(x)= f(x), \quad x >0,\label{1.5}\\ L u := \alpha u(0)+\beta u'(0)=\varphi,\label{1.6} \end{gather} where $\alpha$ and $\beta$ are complex numbers. Denote $L(\lambda):=\lambda^2 I+B$. \begin{theorem} \label{thm2} Let the following conditions be satisfied: \begin{enumerate} \item $B$ is a densely defined and closed operator in a Hilbert space $H$; \item $(1+|\lambda|^{2})\| L(\lambda)^{-1}\|_{B(H)}\leq C$, $\mathop{\rm Re}\lambda=0$; \item $|\alpha|+|\beta|\neq 0$; $\mathop{\rm Re}\alpha\beta^{-1}\leq 0$ if $\beta\neq 0$. \end{enumerate} Then the operator $ \mathbb{L} : u\to \mathbb{L}u := (L(D)u,L u)$ from $W^2_p((0,\infty);H(B),H)$ onto $L_{p}((0,\infty);H) \dot +(H(B),H)_{\frac m{2}+\frac 1{2p},p}$, where $m=0$ if $\beta=0$ and $m=1$ if $\beta\neq 0$, and $p>1$ is an isomorphism.\footnote{Isomorphism means that the operator and its inverse are bounded.} \end{theorem} \paragraph{Proof} By Theorem 1.8.2 in \cite{T}, the operator $\mathbb{L}$ is continuous from the space $W^2_p((0,\infty);H(B),H)$ into $L_{p}((0,\infty);H) \dot +(H(B),H)_{\frac m{2}+\frac 1{2p},p}$. Let us prove that for any $f\in L_p((0,\infty);H)$ and any $\varphi \in (H(B),H)_{\frac m{2}+\frac 1{2p},p}$ problem (\ref{1.5})--(\ref{1.6}) has a unique solution that belongs to $W^2_p((0,\infty);H(B),H)$. Let us show that a solution of problem (\ref{1.5})--(\ref{1.6}) is represented in the form $u(x)=u_1(x)+u_2(x)$, where $u_1(x)$ is the restriction on $[0,\infty)$ of a solution ${\tilde u}_1(x)$ of the equation \begin{equation} {\tilde u}_1''(x)+B{\tilde u}_1(x)= {\tilde f}(x), \quad x\in \mathbb{R},\label{1.7} \end{equation} where ${\tilde f}(x):=f(x)$ if $x\in[0,\infty)$ and ${\tilde f}(x):= 0$ if $x \in (-\infty,0)$, and $u_2(x)$ is a solution of the problem \begin{equation} \begin{gathered} u''_2(x) + Bu_2(x)= 0, \quad x >0,\\ \alpha u_2(0)+\beta u'_2(0)=-L u_1+\varphi . \end{gathered}\label{1.8} \end{equation} Apply Theorem 1 of \cite[p.250]{YY} to equation (\ref{1.7}). Let $H_1:=(H(B),H)_{\frac 1{2},2}$, $H_2:=H(B)$, $A_1:=0$, $A_2:=B$. Then, by virtue of \cite[formula (1), p.39]{YY}, we have $$ \| L(\lambda)^{-1}f \|_{H_1}=\| L(\lambda)^{-1}f \|_{(H(B),H)_{\frac 1{2},2}} \leq C \| L(\lambda)^{-1}f \|^{1/2}_{H(B)}\| L(\lambda)^{-1}f \|^{1/2}_H. $$ From condition (2) it follows that \begin{equation} |\lambda|^{2}\| L(\lambda)^{-1}f\|_{H}+\| L(\lambda)^{-1}f\|_{H(B)} \leq C\| f\|_{H}, \quad f \in H,\ \mathop{\rm Re}\lambda=0.\label{1.9} \end{equation} Using the last inequality and the Young inequality \cite[p.53]{YY}, we have \begin{align*} |\lambda|\| L(\lambda)^{-1}f \|_{H_1} &\leq C \| L(\lambda)^{-1}f \|^{1/2}_{H(B)} \big(|\lambda|^2\| L(\lambda)^{-1}f \|_H \big)^{1/2}\\ &\leq C\big(\| L(\lambda)^{-1}f \|_{H(B)}+|\lambda|^2\| L(\lambda)^{-1}f \|_H \big)\leq C\| f \|. \end{align*} Therefore, conditions (1)--(3) of \cite[Theorem 1, p.250]{YY} are satisfied and, hence, (\ref{1.7}) has a solution ${\tilde u}_1 \in W_p^2(\mathbb{R};H(B),H_1,H)$. Then $u_1 \in W^2_p((0,\infty);H(B),H)$. Let us now prove that for any $\varphi \in (H(B),H)_{\frac m{2}+\frac 1{2p},p}$ problem (\ref{1.8}) has a unique solution $u_2(x)$ that belongs to $W^2_p((0,\infty)$; $H(B),H)$. By the above inequality (\ref{1.9}), we have for $f\in H$, $\mathop{\rm Re}\lambda=0$ $$ \|(\lambda^2I+B)^{-1}f\|_H\le C|\lambda|^{-2}\|f\|_H,\ \|(\lambda^2I+B)^{-1}f\|_H\le C\|f\|_H . $$ This implies that $\|(\lambda^2I+B)^{-1}f\|_H\le C(1+|\lambda|^{2})^{-1}\|f\|_H$, $\mathop{\rm Re}\lambda=0$, i.e., $$ \|R(\lambda,-B)\|\leq C(1+|\lambda|)^{-1},\quad \mathop{\rm arg}\lambda=\pi\,. $$ Hence, as shown in Balakrishnan \cite{B}, there exists an operator $\text{e}^{-x (-B)^{1/2}}$ and for some $\omega >0$, $$ \| \text{e}^{-x (-B)^{1/2}}\|\leq C \text{e}^{-\omega x}, \quad x\geq 0. $$ Repeating the proof of \cite[Lemma 1, p.263]{YY}, one can show that an arbitrary solution of the equation in (\ref{1.8}) that belongs to $W^2_p((0,\infty);H(B), H)$ has the form \begin{equation} u_2(x)=\text{e}^{-x (-B)^{1/2}}g,\label{1.10} \end{equation} where $g\in (H(B),H)_{\frac 1{2p},p}$ (and conversely). To this end one should use Theorem 3.2.11 in Krein \cite{K}. Function (\ref{1.10}) satisfies the boundary condition in (\ref{1.8}) if \begin{equation} \alpha g-\beta (-B)^{1/2}g=\Phi,\label{1.11} \end{equation} where $\Phi=-Lu_1+\varphi$. Since $u_1\in W^2_p((0,\infty);H(B),H)$, by Theorem 1.8.2 in \cite{T}, $L u_1 \in (H(B),H)_{\frac m{2}+\frac 1{2p},p}$. Then $\Phi\in (H(B),H)_{\frac m2+\frac 1{2p},p}$. For $\beta = 0$, a solution of problem (\ref{1.8}) has the form $$ u_2(x)=\alpha^{-1} \text{e}^{-x (-B)^{1/2}} \Phi. $$ Since $\Phi\in (H(B),H)_{\frac 1{2p},p}$, $u_2 \in W^2_p((0,\infty);H(B),H)$. Let $\beta\neq 0$. From conditions (2) and (3), by T. Kato's theorem \cite[p.31]{YY}, it follows that (\ref{1.11}) has a unique solution $g=(\alpha I-\beta(-B)^{\frac 12})^{-1}\Phi$. Then solutions of (\ref{1.8}) have the form $$ u_2(x)=\text{e}^{-x (-B)^{1/2}}(\alpha I-\beta (-B)^{1/2})^{-1} \Phi. $$ By Theorem 1.15.2 in \cite{T}, the operator $(-B)^{1/2}$ from $(H(B),H)_{\frac 1{2p},p}$ onto $(H(B)$, $H)_{\frac {p+1}{2p},p}$ is an isomorphism. Then $(\alpha I-\beta (-B)^{1/2})^{-1} \Phi \in (H(B),H)_{\frac 1{2p},p}$, i.e., $u_2 \in W^2_p((0,\infty);H(B),H)$. The uniqueness of a solution of problem (\ref{1.5})--(\ref{1.6}) follows from the uniqueness of a solution of problem (\ref{1.8}). Indeed, if problem (\ref{1.5})--(\ref{1.6}) has two solutions $u(x)$, $\tilde u(x)$, then functions $u_2(x):= u(x)-u_1(x)$ and $\tilde u_2(x):= \tilde u(x)-u_1(x)$, where $u_1(x)$ is the restriction on $[0,\infty)$ of the solution $\tilde u_1(x)$ of (\ref{1.7}), are two different solutions of problem (\ref{1.8}), which is a contradiction. \hfill $\Box$ \subsection{Completeness of elementary solutions of a problem on the semi-axis} In those cases when it is difficult to prove the applicability of the Fourier method, it is desirable at least to establish that a solution of an initial boundary-value problem may be approximated by linear combinations of elementary solutions. In a Hilbert space $H$, consider a boundary-value problem in $[0,\infty)$ for the second order elliptic equation \begin{gather} u''(x)+B u(x)=0,\quad x>0,\label{1.12} \\ \alpha u(0)+\beta u'(0)=\varphi.\label{1.13} \end{gather} Let us find conditions that allow building an approximation of a solution of (\ref{1.12})--(\ref{1.13}) by linear combinations of elementary solutions of (\ref{1.12}). As it was mentioned in the introduction, the function \begin{equation} u_i(x) := \text{e}^{\lambda_i x} \big(\frac {x^{k_i}}{k_i!} u_{i0}+ \frac {x^{k_i-1}}{(k_i-1)!} u_{i1}+\cdots+u_{i k_i} \big) \label{1.14} \end{equation} is a solution of (\ref{1.12}) if and only if $u_{i0},u_{i1}, \dots,u_{ik_i}$ is a chain of root vectors of the characteristic operator pencil (\ref{1.1}) corresponding to the eigenvalue $\lambda_i$ and (\ref{1.14}) is called an elementary solution of (\ref{1.12}). Let $u_{10},u_{11},\dots,u_{1,r-1}$ be one of the maximal chains of root vectors of (\ref{1.1}) corresponding to the eigenvalue $\mu$. Then $\lambda_1=\lambda_2=\dots =\lambda_r=\mu$ and $k_1=0, k_2=1,\dots,k_r=r-1$. Note, it may happen that $\lambda_{r+1}=\mu$. \begin{lemma} \label{lm3} Let $|\alpha|+|\beta|\neq 0$ and $\mathop{\rm Re}\alpha\beta^{-1}\leq 0$ if $\beta\neq 0$. Then, if a system of root vectors $\{u_{ip}\}$ of (\ref{1.1}) corresponding to eigenvalues $\lambda_i$ with $\mathop{\rm Re}\lambda_i<0$ is complete (a basis) in a Hilbert space $H$ then a system of vectors $\{(\alpha+\beta\lambda_i)u_{ip}+\beta u_{i,p-1}\}$, where $u_{i,-1}=0$, is also complete (a basis) in $H$. \end{lemma} \paragraph{Proof} Let $u_{10},u_{11},\dots,u_{1,r-1}$ be one of the maximal chains of root vectors of (\ref{1.1}) corresponding to the eigenvalue $\mu$ with $\mathop{\rm Re}\mu<0$. Show that we can uniquely define coefficients $M_i$ with respect to coefficients $C_i$ from the equation \begin{align*} C_1u_{10}+C_2u_{11}+\cdots+C_ru_{1,r-1} =&M_1(\alpha+\beta\mu)u_{10}+M_2((\alpha+\beta\mu)u_{11}+\beta u_{10})\\ &+\cdots+M_r((\alpha+\beta \mu)u_{1,r-1}+\beta u_{1,r-2}). \end{align*} Rewrite the last equation in the form \begin{align*} C_1u_{10}+C_2u_{11}+\cdots+C_ru_{1,r-1} =&u_{10}(M_1(\alpha+\beta\mu)+\beta M_2) +u_{11}(M_2(\alpha+\beta\mu)\\ &+\beta M_3) +\cdots+u_{1,r-1} M_r(\alpha+\beta\mu). \end{align*} Therefore, \begin{gather*} M_1(\alpha+\beta\mu)+\beta M_2=C_1,\\ M_2(\alpha+\beta\mu)+\beta M_3=C_2,\\ \vdots\\ M_{r-1}(\alpha+\beta\mu)+\beta M_r=C_{r-1},\\ M_{r}(\alpha+\beta\mu)=C_r. \end{gather*} If $\beta=0$ then $\alpha\neq 0$ and $M_i=\frac 1\alpha C_i,\ i=1,\dots,r$. If $\beta\neq 0$ then $\alpha+\beta\mu\neq 0$ (since $\mathop{\rm Re}\alpha\beta^{-1}\leq 0$ and $\mathop{\rm Re}\mu<0$). Therefore, starting from the last equation of the previous system we find that $$ M_r=\frac{C_r}{\alpha+\beta\mu},\quad M_{r-1}=\frac{C_{r-1}-\beta M_r} {\alpha+\beta\mu},\;\dots\; ,\ M_1=\frac{C_1-\beta M_2}{\alpha+\beta \mu}.$$ \hfill $\Box$ \begin{theorem} \label{thm4} Let the following conditions be satisfied: \begin{enumerate} \item $B$ is a densely defined and closed operator in a Hilbert space $H$; \item $s_j(J;H(B),H)\le Cj^{-q}$, $j=1,\dots,\infty$, for some $q>0$; \item For $L(\lambda):=\lambda^2I+B$, $(1+|\lambda|^{2})\| L(\lambda)^{-1}\|_{B(H)}\leq C$, $\mathop{\rm Re}\lambda=0$; \item $|\alpha|+|\beta|\neq 0;$ $\mathop{\rm Re}\alpha\beta^{-1}\leq 0$ if $\beta\neq 0$; \item For $q\le 4$ there exist rays $\ell_k$ with angles between neighboring rays less than $\frac {q \pi}{4}$ and $\eta$ such that $$\|L(\lambda)^{-1}\|_{B(H,H(B))}\le C |\lambda|^\eta, \qquad \lambda\in\ell_k,\ |\lambda|\to\infty; $$ \item $\varphi\in(H(B),H)_{\frac m2+\frac 1{2p},p}$ for some $p>1$, where $m=0$ if $\beta=0$ and $m=1$ if $\beta\neq 0$. \end{enumerate} % Then problem (\ref{1.12})--(\ref{1.13}) has a unique solution $u\in W^2_p((0, \infty);H(B),H)$ and there exist numbers $C_{in}$ such that \begin{equation} \lim_{n\to\infty}\sum_{k=0,2}\int^\infty_0 \|u^{(k)}(x)- \sum^n_{i=1}C_{in}u^{(k)}_i(x)\|^p_{H_{2-k}} \ dx=0,\label{1.15} \end{equation} where $H_0=H, H_2=H(B)$, $u_i(x)$ are elementary solutions (\ref{1.14}) of equation (\ref{1.12}) corresponding to the eigenvalue $\lambda_i$ with $\mathop{\rm Re}\lambda_i<0 $. \end{theorem} \paragraph{Proof} Consider in $H$ an operator $S$ such that $D(S)=H(B)$, $S=S^*\ge c^2 I$ (see, for example, Lions and Magenes \cite[1.2.1]{LM}). By Lemma 1 in \cite[p.15]{YY} and condition (2), we have $$ s_j(J;H(B),H)=s_j(J;H(S),H)=s_j(JS^{-1};H,H)=\lambda_j(S^{-1})\le Cj^{-q}. $$ Let $H_1:=(H(B),H)_{\frac 12 ,2}$. Then $H_1=(H(S),H)_{\frac 1{2},2}=H(S^{1/2})$ and by Lemma~1 in \cite[p.15]{YY}, \begin{gather*} s_j(J;H(B),H_1)=s_j(J;H(S),H(S^{1/2}))\\ =s_j(S^{1/2}JS^{-1};H,H) =\lambda_j(S^{-\frac 1{2}})\le Cj^{- q/2},\\ s_j(J;H_1,H)=s_j(J;H(S^{1/2}),H)=s_j(JS^{-\frac 1{2}};H,H) =\lambda_j(S^{-\frac 1{2}})\le Cj^{- q/2}. \end{gather*} Hence, by Theorem 1, a system of root vectors of pencil (\ref{1.1}) corresponding to the eigenvalues $\lambda_i$, $\{u_i(0)\}=\{u_{ik_i}\}$, with $\mathop{\rm Re}\lambda_i<0 $ is complete in the spaces $H_1$ and $H(B)$. On the other hand, $\overline{H_1}|_H=H$. Then the same system of root vectors is complete in the space $H$ and, therefore, in $(H(B),H)_{\theta,p}$, $0<\theta<1$ (see \cite[Theorems 1.3.3 and 1.6.2]{T}). Therefore, by virtue of Lemma 3, a system $\{\alpha u_i(0)+\beta u'_i(0)\}$ is also complete in $(H(B),H)_{\theta,p}$, $0<\theta<1$. Hence, there exist numbers $C_{in} $ such that $$ \lim_{n\to\infty} \|\varphi-\sum^n_{i=1}C_{in}(\alpha u_i(0)+\beta u'_i(0))\|_{(H(B),H)_{\frac m2 +\frac 1{2p},p}} =0.$$ On the other hand, from Theorem 2 we have \begin{multline} \|u-\sum^n_{i=1}C_{in}u_i \|_{W^2_p((0,\infty);H(B),H)}\\ \le C \|\varphi-\sum^n_{i=1}C_{in}(\alpha u_i(0)+\beta u'_i(0))\|_{(H(B),H)_{\frac m2 +\frac 1{2p},p}}. \end{multline} \hfill $\Box$ \section{Boundary-value problems for second order elliptic equations} In this section we apply abstract results of section 1 to boundary-value problems for second order elliptic equations in semi-infinite tube domains. The corresponding isomorphism and completeness theorems are proved. Completeness theorems apply to eigenvalues $\lambda_i$ with $\mathop{\rm Re}\lambda_i<0$. \subsection{An isomorphism } In the semi-infinite strip $\Omega:=[0,\infty)\times [0,1]$, consider a principally boundary-value problem for an elliptic equation of the second order, \begin{gather} L u := D^{2}_{x}u(x,y) + b(y)D^{2}_{y}u(x,y)+M u(x,\cdot)\big|_y = f(x,y), \quad (x,y) \in \Omega,\label{2.1} \\ P u := \gamma u(0,y) + \delta D_x u(0,y) =\varphi(y),\qquad y \in [0,1],\label{2.2} \\ \begin{gathered} L_{1}u := \alpha_{1}D_{y}u(x,0) + \alpha_{0}u(x,0) = 0,\quad x \in [0,\infty),\\ L_{2}u := \beta_{1}D_{y}u(x,1) + \beta_{0}u(x,1)= 0,\quad x \in [0,\infty), \end{gathered} \label{2.3} \end{gather} where $\alpha_{\nu}$, $\beta_{\nu}$, $\gamma$, $\delta$ are complex numbers, $D_{x}:= {\partial\over\partial x}$, $D_{y}:= {\partial \over\partial y}$. The corresponding spectral problem is \begin{gather} \lambda^2 u(y)+b(y) u''(y) +Mu \big|_y=0,\quad y\in [0,1],\label{2.4} \\ \begin{gathered} {\tilde L}_{1}u := \alpha_{1} u'(0) + \alpha_{0}u(0)= 0,\\ {\tilde L}_{2}u := \beta_{1} u'(1) + \beta_{0}u(1) = 0. \end{gathered} \label{2.5} \end{gather} Let $m_\nu:=\mathop{\rm ord}L_\nu$, $W^{\ell,s}_{p,q}(\Omega):=W^\ell_p((0,\infty);W^s_q(0,1),L_q(0,1))$,\\ $L_{p,q}(\Omega):=W^{0,0}_{p,q}(\Omega)$. \begin{theorem} \label{thm5} Let the following conditions be satisfied: \begin{enumerate} \item $b\in C[0,1],\ b(y)>0$; \item $|\alpha_1|+|\alpha_0|\neq 0$ and $|\beta_1|+|\beta_0|\neq 0$; \item The operator $M$ from $W^2_2(0,1)$ into $L_2(0,1)$ is compact.\\ This is equivalent to $\forall\varepsilon>0$, $\|Mu\|_{L_2(0,1)}\le\varepsilon\|u\|_{W^2_2(0,1)} +C(\varepsilon)\|u\|_{L_2(0,1)}$, $u\in W^2_2(0,1)$ (see Lemma 4 and Remark 5 \cite[p.45]{YY}); \item The spectral problem (\ref{2.4})--(\ref{2.5}) does not have eigenvalues on the straight line $\mathop{\rm Re}\lambda=0;$ \item $|\gamma|+|\delta|\neq 0;$ $\ \mathop{\rm Re} \gamma\delta^{-1}\le 0$ when $\delta\neq 0$. \end{enumerate} Then the operator ${\mathbb{L}}:u\to {\mathbb{L}}u:=(L u,P u)$ from $W^{2,2}_{p,2}(\Omega;L_\nu u=0,\nu=1,2)$ onto $L_{p,2}(\Omega)\dot+ B_{2,p}^{2-m-\frac 1p}((0,1);{\tilde L}_\nu u=0, m_\nu<\frac 32-m-\frac 1p)$, if $p>1$ and $p\ne 2$, or $p=2$ and $m_\nu\neq 1-m$, is an isomorphism, where $m=0$ if $\delta=0$; $m=1$ if $\delta\neq 0$. \end{theorem} \paragraph{Remark} In the case $p=2$ and $m_\nu=1-m$, $(W^2_2((0,1);{\tilde L}_\nu u=0,\nu=1,2), L_2(0,1))_{\frac m2+\frac 14 ,2}=B^{\frac 32-m}_{2,2}((0,1)$; ${\tilde L}_{\nu}u=0$, $m_{\nu}<1-m$; ${\tilde L}_{\nu}u\in \widetilde{B}^{\frac 12}_{2,2}(0,1)$, $m_{\nu}=1-m)$ (see Triebel \cite[4.3.3]{T}) should be written instead of \hfil\break $B^{\frac 32-m}_{2,2}((0,1);{\tilde L}_{\nu}u=0,m_\nu<1-m)$. $\widetilde{B}^{s}_{p,q}(G) := \{u \mid u \in B^{s}_{p,q} (\mathbb{R}^{r}), \mathop{\rm supp}(u)\subset \overline G \}$. From the introduction, $B_{2,2}^s=W^s_2$. Moreover, by virtue of Theorem 6 of Grisvard and Seeley \cite[p.45]{YY}, $(W^2_2((0,1);{\tilde L}_\nu u=0,\nu=1,2), L_2(0,1))_{\frac m2+\frac 14 ,2}\supset (W^2_2((0,1); {\tilde L}_\nu u=0,\nu=1,2), L_2(0,1))_{\frac 15 ,2}=W^{\frac 85}_2((0,1);{\tilde L}_\nu u=0,\nu=1,2)$. Then, for a unique solvability (and not an isomorphism) it is enough to take $\varphi \in W^{\frac 85}_2((0,1);{\tilde L}_\nu u=0, \nu=1,2)$. \paragraph{Proof} Let us denote $H := L_{2}(0,1)$. Consider an operator $B$ defined by \begin{equation}\begin{gathered} D(B):=W^{2}_{2}((0,1);{\tilde L}_{\nu}u=0,\nu=1,2),\\ Bu:= b(y)u''(y)+M u\big|_y.\end{gathered}\label{2.6} \end{equation} Then problem (\ref{2.1})--(\ref{2.3}) can be rewritten in the form \begin{equation}\begin{gathered} u''(x) + Bu(x)=f(x),\\ \gamma u(0)+\delta u'(0)=\varphi, \end{gathered} \label{2.7} \end{equation} where $u(x):= u(x,\cdot ), f(x):= f(x,\cdot )$ are functions with values in the Hilbert space $H := L_{2}(0,1)$ and $\varphi:= \varphi(\cdot )$ is an element of H. Let us apply Theorem 2 to problem (\ref{2.7}). From Theorem 1 \cite[p.111]{YY} (or Theorem 1.7 \cite[p.100]{Y}) it follows that the operator $(\lambda^2 I+B)^{-1}$ is bounded in $L_2(0,1)$ (see below for the proof). A bounded operator is closed. The inverse operator to a closed operator is also closed. Therefore, $\lambda^2 I+B$ is a closed operator. This implies that the operator $B$ is closed, i.e., condition (1) of Theorem 2 is fulfilled. Consider the problem \begin{equation} \begin{gathered} \lambda^{2}u(y) + b(y)u''(y)+Mu \big|_y= f(y), \quad y \in(0,1), \\ \alpha_{1} u'(0) + \alpha_{0}u(0)= 0,\\ \beta_{1} u'(1) + \beta_{0}u(1) = 0. \end{gathered} \label{2.8} \end{equation} By condition (1), the equation $1 + b(y)\omega^{2}= 0$ has roots $\omega _{1}(y)=i\frac 1{\sqrt{b(y)}}$ and $\omega _{2}(y)=-i\frac 1{\sqrt{b(y)}}$. Then \begin{gather*} \underline\omega:=\inf_{y\in [0,1]} \min\{\mathop{\rm arg}\omega _{1}(y), \mathop{\rm arg}\omega _{2}(y)+\pi \}=\frac\pi 2,\\ \overline\omega:=\sup_{y\in [0,1] }\max \{\mathop{\rm arg}\omega _{1}(y), \mathop{\rm arg}\omega _{2}(y)+\pi \}=\frac\pi 2. \end{gather*} When choosing $\omega _{1}(y)=-i\frac 1{\sqrt{b(y)}}$, $\omega _{2}(y)=i\frac 1 {\sqrt{b(y)}}$ we get that $\underline\omega=\overline\omega=-\frac\pi 2$. Therefore, from Theorem 1 \cite[p.111]{YY} it follows that condition (2) of Theorem 2 is satisfied. Indeed, for a solution of problem (\ref{2.8}), from formula (5) \cite[p.112]{YY} for $\ell=2, \ q=2$, $\gamma=0$ and $\underline\omega=\overline\omega=\frac\pi 2$ we have \begin{gather*} |\lambda|^2 \| u\|_{L_{2}(0,1)}+\|u\|_{W_{2}^2(0,1)} \le C\| f\|_{L_{2}(0,1)}, \\ f\in L_{2}(0,1), \quad \varepsilon<\arg \lambda<\pi-\varepsilon, \quad |\lambda|\to \infty. \end{gather*} and for $\underline\omega=\overline\omega=-\frac\pi 2$, \begin{gather*} |\lambda|^2 \| u\|_{L_{2}(0,1)}+\|u\|_{W_{2}^2(0,1)} \le C\| f\|_{L_{2}(0,1)}, \\ f\in L_{2}(0,1), \quad \pi+\varepsilon<\arg \lambda<2\pi-\varepsilon, \quad |\lambda|\to \infty. \end{gather*} These two inequalities and condition (4) give us condition (2) of Theorem 2. By a theorem of Grisvard and Seeley (see, e.g., \cite[Theorem 6, p.45]{YY}), we have \begin{align*} (H(B),H)_{\theta,p} &= (W^{2}_{2}((0,1);{\tilde L}_{\nu}u=0,\nu=1,2), L_{2}(0,1))_{\theta,p} \\ &=B^{2(1-\theta) }_{2,p}((0,1);{\tilde L}_{\nu}u=0,m_{\nu}<2(1-\theta) -\frac 12), \end{align*} if there does not exist a number $m_\nu$ such that $m_\nu = 2(1-\theta)-\frac 12$. Consequently, $$(H(B),H)_{\frac m2+\frac 1{2p},p}= B^{2-m-\frac 1p}_{2,p}((0,1);{\tilde L}_{\nu}u= 0,m_{\nu}<\frac 32-m-\frac 1p).$$ If there exists $m_\nu = 2(1-\theta)-\frac 12$ then see the corresponding remark to Theorem 5. So, for problem (\ref{2.7}) all conditions of Theorem 2 are fulfilled, from which the statement of Theorem 5 follows. \hfill$\Box$ \smallskip In the semi-infinite domain $\Omega:=[0,\infty)\times G$, where $G\subset\mathbb{R}^r$, $r\ge 2$, is a bounded domain with an $(r-1)$-dimensional smooth boundary $\partial G$, consider a principally boundary-value problem for an elliptic equation of the second order \begin{gather} L u := D^{2}_{x}u(x,y) +\sum^{r}_{j,k=1}b_{jk}(y)D_{j}D_{k}u(x,y) +M u(x,\cdot)\big|_y=f(x,y),\label{2.9} \\ P u := \gamma u(0,y) + \delta D_x u(0,y) =\varphi(y),\quad y \in G, \label{2.10} \\ L_{1}u :=\sum_{|\alpha | \le m_{1}} b_{1 \alpha } (y')D^{\alpha }_{y}u(x,y' )= 0, \ (x,y' ) \in [0,\infty)\times \partial G ,\label{2.11} \end{gather} where $\gamma, \delta$ are complex numbers, $m_1\le 1$, $y:=(y_1,\dots,y_r)$, $\ D_{x}:={\partial\over\partial x},\ D^{\alpha }_{y} := D^{\alpha _{1}}_{1}\cdots D^{\alpha _{r}}_{r},\ D_{j}= {\partial\over\partial y_{j}}$. Let $W^{\ell,s}_{p,q} (\Omega):=W^\ell_p((0,\infty);W^s_q(G),L_q(G))$, $L_{p,q}(\Omega):=W^{0,0}_{p,q}(\Omega)$. The corresponding spectral problem is \begin{gather} \lambda^2 u(y)+\sum^{r}_{j,k=1}b_{jk}(y)D_{j}D_{k}u(y)+ M u \big|_y=0,\ \ y\in G,\label{2.12}\\ {\tilde L}_{1}u :=\sum_{| \alpha | \leq m_{1}}b_{1 \alpha } (y')D^{\alpha }_{y}u(y') = 0,\quad y' \in \partial G .\label{2.13} \end{gather} Let us denote $H := L_{2}(G)$ and consider the operator $B$ which is defined by \begin{equation}\begin{gathered} D(B) := W^{2}_{2}(G;{\tilde L}_{1}u=0),\\ Bu := \sum^{r}_{j,k=1}b_{jk}(y)D_{j}D_{k}u(y) + M u\big|_y.\end{gathered} \label{2.14} \end{equation} \begin{theorem} \label{thm6} Let the following conditions be satisfied: \begin{enumerate} \item $ b_{jk} \in C(\overline G),\ b_{1\alpha } \in C^{2-m_{1}}(\overline G),\ \partial G \in C^{2};$ \item If $y \in \overline G,\ \sigma:=(\sigma_1,\dots,\sigma_r) \in \mathbb{R}^r,\ | \sigma | + | \lambda | \ne 0$ then $$ \lambda^2- \sum^{r}_{j,k=1}b_{jk}(y)\sigma _{j}\sigma _{k}\neq 0,\quad \mathop{\rm Re}\lambda= 0; $$ \item $\sum_{|\alpha|=m_1}b_{1\alpha}(y')\sigma^\alpha\neq 0$ for any vector $\sigma$ normal to $\partial G$ at the point $y'\in \partial G ;$ \item Let $y' $ be any point on $\partial G $, the vector $\sigma ' $ tangent and the vector $\sigma $ normal to $\partial G $ at the point $y' \in \partial G $. Consider the following ordinary differential problem \begin{gather} \big[\lambda^2- \sum^{r}_{j,k=1}b_{jk}(y' ) \big(\sigma'_j+\sigma_j \frac d{dt}\big)\big(\sigma'_k +\sigma _{k}\frac d{dt}\big)\big]u(t) = 0,\; t >0,\; \mathop{\rm Re}\lambda=0,\label{2.15}\\ \sum_{| \alpha | =m_{1}}b_{1 \alpha }(y') \big(\sigma ' +\sigma \frac d{dt} \big)^{\alpha }u(t) \big|_{t=0}= h_{1};\label{2.16} \end{gather} problem (\ref{2.15})--(\ref{2.16}) should have only one solution that with all its derivatives tend to zero as $t\to \infty $ for any number $h_{1} \in \mathbb{C};$ \item $|\gamma|+|\delta|\neq 0;\ $ $\mathop{\rm Re} \gamma\delta^{-1}\le 0$ when $\delta\neq 0;$ \item The spectral problem (\ref{2.12})--(\ref{2.13}) does not have eigenvalues on the line $\mathop{\rm Re}\lambda=0;$ \item The operator $M$ from $W^{2}_2(G)$ into $L_2(G)$ is compact. \end{enumerate} Then the operator ${\mathbb{L}}:u\to {\mathbb{L}}u:=(L u,P u)$ from $W^{2,2}_{p,2}(\Omega;L_1 u=0)$ onto $L_{p,2}(\Omega)\dot+ $ $B_{2,p}^{2-m-\frac 1p}(G;{\tilde L}_1 u=0,m_1< \frac 32-m-\frac 1p)$, if $p>1$ and $p\ne 2$, or $p=2$ and $m_1 \neq 1-m$, \footnote{See the corresponding remark of Theorem 5.} is an isomorphism, where $m=0$ if $\delta=0$; $m=1$ if $\delta\neq 0$. \end{theorem} \paragraph{Proof} Problem (\ref{2.9})--(\ref{2.11}) can be rewritten in the form \begin{gather} u''(x) + Bu(x)=f(x),\quad x>0,\label{2.17}\\ \gamma u(0)+\delta u'(0)=\varphi,\label{2.18} \end{gather} where $u(x):= u(x,\cdot ), f(x):= f(x,\cdot )$ are functions with values in the Hilbert space $H := L_{2}(G)$, $\varphi:= \varphi(\cdot )$ is an element of $H$, the operator $B$ is defined by the equalities (\ref{2.14}). Apply Theorem 2 to problem (\ref{2.17})--(\ref{2.18}). From Theorem 1 in \cite[p.207]{YY} it follows that the operator $(\lambda^2 I+B)^{-1}$ is bounded in $L_2(0,1)$. A bounded operator is closed. The inverse operator to a closed operator is also closed. Therefore, $\lambda^2 I+B$ is a closed operator. This implies that the operator $B$ is closed, i.e., condition (1) of Theorem 2 is fulfilled. On the other hand, from Theorem 1 in \cite[p.207]{YY} and condition (6), condition (2) of Theorem 2 follows. The last part of the proof is similar to that in the proof of Theorem 5. \hfill $\Box$ \subsection{Completeness of elementary solutions } Let us consider, in the semi-infinite strip $\Omega:=[0,\infty)\times [0,1]$, a principally boundary-value problem for an elliptic equation of the second order, \begin{gather} D^{2}_{x}u(x,y) + b(y)D^{2}_{y}u(x,y)+M u(x,\cdot)\big|_y= 0, \label{2.19} \\ \gamma u(0,y)+\delta D_x u(0,y)=\varphi(y),\quad y\in [0,1],\label{2.20}\\ \begin{gathered} L_{1}u := \alpha_{1}D_{y}u(x,0) + \alpha_{0}u(x,0)= 0,\ x \in [0,\infty),\\ L_{2}u := \beta_{1}D_{y}u(x,1) + \beta_{0}u(x,1)= 0,\ x \in [0,\infty), \end{gathered} \label{2.21} \end{gather} and the corresponding spectral problem (\ref{2.4})--(\ref{2.5}), where $\alpha_\nu$, $ \beta_\nu$ are complex numbers, $\ D_x:={\partial\over{\partial x}},\ D_y:={\partial\over{\partial y}}$; $m_\nu:=\mathop{\rm ord}L_\nu$ and $W^{\ell,s}_{p,q} (\Omega):=W^\ell_p((0,\infty);$ $W^s_q(0,1),L_q(0,1))$. As it was mentioned in the introduction, a function of the form \begin{equation} u_i(x,y):=\text{e}^{\lambda_i x} \big({x^{k_i}\over k_i!}u_{i0}(y)+{x^{k_i-1}\over(k_i-1)!}u_{i1}(y)+\cdots+u_{ik_i}(y) \big) \label{2.22} \end{equation} becomes an elementary solution of problem (\ref{2.19}), (\ref{2.21}) if and only if a system of functions $u_{i0}(y),u_{i1}(y),\dots,u_{ik_i}(y)$ is a chain of root functions of problem (\ref{2.4})--(\ref{2.5}) corresponding to the eigenvalue $\lambda_i$. See the corresponding remark in subsection 1.3. \begin{theorem} \label{thm7} Let the following conditions be satisfied: \begin{enumerate} \item The conditions of Theorem 5 are fulfilled; \item $\varphi \in B^{2-m-\frac 1p}_{2,p}((0,1);{\tilde L}_\nu u=0,m_\nu<\frac 32-m-\frac 1p)$ if $p>1$ and $p\ne2$, or $p=2$ and $m_\nu\neq 1-m$\footnote {See the corresponding remark of Theorem 5.}, where $m=0$ if $\delta=0$; $m=1$ if $\delta\neq 0$. \end{enumerate} Then problem (\ref{2.19})--(\ref{2.21}) has a unique solution $u\in W^{2,2}_{p,2}(\Omega)$, and there exist numbers $C_{in}$ such that \begin{align*} \lim_{n\to \infty} \int^{\infty}_0 \Bigl(&\|D^2_x u(x,\cdot)- \sum^n_{i=1}C_{in}D^2_x u_i(x,\cdot)\|^p_{L_2(0,1)}\\ &+\|u(x,\cdot)-\sum^n_{i=1}C_{in} u_i(x,\cdot)\|^p_{W_2^2(0,1)} \Bigr)\, dx=0, \end{align*} where $u(x,y)$ is a solution of problem (\ref{2.19})--(\ref{2.21}) and $u_i(x,y)$ is the elementary solution (\ref{2.22}) of problem (\ref{2.19}), (\ref{2.21}) corresponding to the eigenvalue $\lambda_i$ with $\mathop{\rm Re}\lambda_i<0$. \end{theorem} \paragraph{Proof} Apply Theorem 4 to problem (\ref{2.19})--(\ref{2.21}). In $H:=L_2(0,1)$, consider an operator $B$ which is defined by equality (\ref{2.6}). Then, problem (\ref{2.19})--(\ref{2.21}) can be rewritten in the form \begin{gather} u''(x)+B u(x)=0,\quad x>0,\label{2.23} \\ \gamma u(0)+\delta u'(0)=\varphi,\label{2.24} \end{gather} where $u(x):=u(x,\cdot) $ is a function with values in the Hilbert space $H:=L_2(0,1)$ and $\varphi:=\varphi(\cdot)$ is an element of $H$. Conditions (1) and (3) of Theorem 4 have been checked in the proof of Theorem 5. By virtue of Triebel \cite[formula 4.10.2/14]{T}, \begin{equation} s_j (J;W^2_2(0,1),L_2(0,1))\sim j^{-2}. \label{2.25} \end{equation} Since $W^2_2((0,1);{\tilde L}_\nu u=0,\nu=1,2)$ is a subspace of $W^2_2(0,1)$ then, by Lemma 3 in \cite[p.17]{YY}, from (\ref{2.15}) it follows that $$ s_j (J;H(B),H)\le Cs_j(J;W^2_2(0,1),L_2(0,1))\le C j^{-2}, $$ i.e., condition (2) of Theorem 4 is fulfilled for $q=2$. We have shown in the proof of Theorem 5 that for $\varepsilon<\arg\lambda<\pi-\varepsilon$ or $\pi+\varepsilon<\arg\lambda<2\pi-\varepsilon$ and $|\lambda|\to\infty$, $$ |\lambda|^2\|u\|_{L_2(0,1)}+\|u\|_{W^2_2(0,1)}\le C\|f\|_{L_2(0,1)},\quad f\in L_2(0,1).$$ This gives us condition (5) of Theorem 4 for $q=2$ and $\eta=0$. Condition (6) of Theorem~4 one can see in the proof of Theorem 5. So, for problem (\ref{2.23})--(\ref{2.24}) all conditions of Theorem 4 have been checked and the statement of Theorem 7 follows. \hfill $\Box$ \smallskip In the semi-infinite domain $\Omega:=[0,\infty)\times G$, where $G \subset \mathbb{R}^{r}$, $r\ge 2$, is a bounded domain with an $(r-1)$-dimensional smooth boundary $\partial G$, consider a principally boundary-value problem for an elliptic equation of the second order \begin{gather} D^{2}_{x}u(x,y)+\sum^{r}_{j,k=1}b_{jk}(y)D_{j} D_{k}u(x,y) +M u(x,\cdot) \big|_y=0,\label{2.26} \\ \gamma u(0,y)+\delta D_x u(0,y)=\varphi(y),\quad y\in G,\label{2.27} \\ L_{1}u :=\sum_{| \alpha | \leq m_{1}}b_{1\alpha } (y')D^{\alpha }_{y}u(x,y' ) = 0,\ (x,y' ) \in [0,\infty)\times \partial G,\label{2.28} \end{gather} and the corresponding spectral problem \begin{gather} \lambda^2 u(y)+\sum^{r}_{j,k=1}b_{jk}(y)D_{j}D_{k}u(y)+ M u \big|_y=0,\ \ y\in G,\label{2.29}\\ {\tilde L}_{1}u :=\sum_{| \alpha | \leq m_{1}}b_{1 \alpha } (y')D^{\alpha }_{y}u(y') = 0,\quad y' \in \partial G ,\label{2.30} \end{gather} where $m_{1}\le 1$, $y:=(y_1,\dots,y_r)$, $D_{x}:={\partial\over\partial x}$, $D^{\alpha }_{y} := D^{\alpha _{1}}_{1}\cdots D^{\alpha _{r}}_{r}$, $D_{j}={\partial\over\partial y_{j}}$. As above, $W^{\ell,s}_{p,q} (\Omega):=W^\ell_p((0,\infty);W^s_q(G),L_q(G))$. A function of the form \begin{equation} u_i(x,y):=\text{e}^{\lambda_i x} \big({x^{k_i}\over k_i!}u_{i0}(y)+{x^{k_i-1}\over(k_i-1)!}u_{i1}(y)+\cdots+u_{ik_i}(y) \big)\label{2.31} \end{equation} becomes an elementary solution of problem (\ref{2.26}), (\ref{2.28}) (see the introduction) if and only if a system of functions $u_{i0}(y),u_{i1}(y),\dots$, $u_{ik_i}(y)$ is a chain of root functions of the spectral problem (\ref{2.29})--(\ref{2.30}) corresponding to the eigenvalue $\lambda_i$. See the corresponding remark in subsection 1.3. Consider in $H:=L_2(G)$ the operator $B$ which is defined by the equalities in (\ref{2.14}). \begin{theorem} \label{thm8} Let the following conditions be satisfied: \begin{enumerate} \item The conditions of Theorem 6 are fulfilled; \item There exist rays $\ell_k$ with angles between neighbouring rays less than $\frac {\pi}{2r}$ such that for $y\in{\overline G}$, $\sigma\in\mathbb{R}^r$, $|\sigma|+|\lambda|\neq 0$, $\lambda\in\ell_k$, the following is true: $$ \lambda^2+\sum^r_{j,k=1}b_{jk}(y)\sigma_j\sigma_k\ne 0; $$ \item Let $y' $ be any point on $\partial G $, the vector $\sigma ' $ tangent and the vector $\sigma $ normal to $\partial G $ at the point $y' \in \partial G $. Consider the following ordinary differential problem \begin{gather} \Bigl [\lambda^2+\sum^{r}_{j,k=1}b_{jk}(y') \big(\sigma'_j+\sigma_j \frac d{dt} \big)\big(\sigma'_k +\sigma _{k}\frac d{dt} \big)\Bigr ]u(t) = 0,\quad t\geq 0,\ \lambda\in\ell_k, \label{2.32}\\ \sum_{| \alpha | =m_{1}} b_{1\alpha }(y') \big(\sigma ' +\sigma \frac d{dt} \big)^{\alpha }u(t) \big|_{t=0}= h_{1}; \label{2.33} \end{gather} problem (\ref{2.32})--(\ref{2.33}) should have only one solution that with all its derivatives tend to zero as $t\to \infty $ for any number $h_{1} \in \mathbb{C};$ \item $\varphi \in B^{2-m-\frac 1p}_{2,p}(G;{\tilde L}_1 u=0,m_1<\frac 32-m-\frac 1p)$ if $p>1$ and $p\ne2$, or $p=2$ and $m_1\neq 1-m$\footnote {See the corresponding remark of Theorem 5.}, where $m=0$ if $\delta=0$; $m=1$ if $\delta\neq 0$. \end{enumerate} Then problem (\ref{2.26})--(\ref{2.28}) has a unique solution $u\in W^{2,2}_{p,2}(\Omega)$, and there exist numbers $C_{in}$ such that \begin{align*} \lim_{n\to \infty} \int^{\infty}_0 \Bigl (&\|D^2_x u(x,\cdot)- \sum^n_{i=1}C_{in}D^2_xu_i(x,\cdot)\|^p_{L_2(G)}\\ &+\|u(x,\cdot)-\sum^n_{i=1}C_{in}u_i(x,\cdot)\|^p_{W^2_2(G)} \Bigr) \ dx=0, \end{align*} where $u(x,y)$ is a solution of problem (\ref{2.26})--(\ref{2.28}) and $u_{i}(x,y)$ is the elementary solution (\ref{2.31}) of problem (\ref{2.26}), (\ref{2.28}) corresponding to the eigenvalue $\lambda_i$ with $\mathop{\rm Re} \lambda_i<0$. \end{theorem} \paragraph{Proof} Apply Theorem 4 to problem (\ref{2.26})--(\ref{2.28}). Problem (\ref{2.26})--(\ref{2.28}) can be rewritten in the form \begin{gather} u''(x)+B u(x)=0,\quad x>0,\label{2.34} \\ \gamma u(0)+\delta u'(0)=\varphi,\label{2.35} \end{gather} where $u(x):=u(x,\cdot)$ is a function with values in the Hilbert space $H:=L_2(G)$ and $\varphi:=\varphi(\cdot)$ is an element of $H$. Conditions (1) and (3) of Theorem 4 have been checked in Theorem 6. By formula 4.10.2/14 in \cite{T}, \begin{equation} s_j (J;W^2_2(G),L_2(G))\sim j^{-\frac 2r}. \label{2.36} \end{equation} Since $W^2_2(G;{\tilde L}_1 u=0)$ is a subspace of $W^2_2(G)$ then, by Lemma 3 in \cite[p.17]{YY}, from (\ref{2.36}) it follows that $$ s_j (J;H(B),H)\le Cs_j(J;W^2_2(G),L_2(G))\le C j^{-\frac 2r}, $$ i.e., condition (2) of Theorem 4 is fulfilled for $q=\frac 2r$. By Theorem 1 in \cite[p.207]{YY}, from conditions (2) and (3), condition (5) of Theorem 4, for $q=\frac 2r$, follows. Condition (6) of Theorem 4 one can see in the proof of Theorem 5. So, for problem (\ref{2.34})--(\ref{2.35}) all conditions of Theorem 4 have been checked and the statement of the theorem follows. \hfill $\Box$ \smallskip The results of this paper can be applied to the thermal conduction problem from \cite{TY1} in the case when there are not mixed derivatives in the equation. We get completeness of a system of root functions of the corresponding spectral problem and completeness of elementary solutions of the original problem for eigenvalues $\lambda_i$ with $\mathop{\rm Re}\lambda_i<0$. Moreover, since the corresponding operator $B$ of the thermal conduction problem is selfadjoint then one can get a basis property theorem instead of completeness Theorem 7 (for $p=2$). But the latter needs some additional considerations. A few examples of the operator $M$ which satisfies conditions of Theorems 5 and 6 and, therefore, Theorems 7 and 8 are the following. Let $G$ denote the interval $(0,1)$ or a bounded domain in $\mathbb{R}^r$, $r\ge 2$, with an $(r-1)$-dimensional smooth boundary. \begin{enumerate} \item If $b_{j}\in L_2(G)$, then the operator $$Mu:=\sum_{j=0}^{1}b_{j}(x)u^{(j)}(x)$$ from $W_2^{2}(G)$ into $L_2(G)$ is compact. \item If $b_{ji}\in L_2(G)$ and $\varphi_{ji}(x)$ are functions mapping $\overline G$ into itself and belong to $C(\overline G)$, then the operator $$Mu:=\sum_{j=0}^{1}\sum_{i=1}^{N_{j}}b_{ji}(x)u^{(j)}(\varphi_{ji}(x)),$$ from $W_2^{2}(G)$ into $L_2(G)$ is compact. \item If $B_{j}(x,y)$ are kernels such that for some $\sigma>1$ $$\int_G|B_{j}(x,y)|^\sigma dy+\int_G|B_{j}(x,y)|^\sigma dx\le C$$ then the operator $$Mu:=\sum_{j=0}^{2}\int_G B_{j}(x,y)u^{(j)}(y)\ dy$$ from $W_2^{2}(G)$ into $L_2(G)$ is compact. \end{enumerate} The proofs can be found in \cite[p.201]{YY}. \begin{thebibliography} \frenchspacing \bibitem{B} Balakrishnan, A. V., Fractional powers of closed operators and the semigroups generated by them, {\it Pacific J. Math.}, {\bf 10} (1960), 419--437. % \bibitem{KO} Kondrat'ev, V. A. and Oleinik, O. A., Boundary-value problems for partial differential equations in non-smooth domains, {\it Russian Math. Surveys}, {\bf 38} (1983), 1--86 (translated from Russian). % \bibitem{KM} Kozlov, V. A. and Maz'ya, V. G., ``{\it Differential Equations with Operator Coefficients}, Springer, 1999. % \bibitem{KMR} Kozlov, V. A., Maz'ya, V. G. and Rossmann, J., ``{\it Elliptic Boundary Value Problems in Domains with Point Singularities"}, AMS, Math. Surv. and Monogr., v.52, 1997. % \bibitem{K} Krein, S. G., ``{\it Linear Differential Equations in Banach Space,}" Providence, 1971. % \bibitem{LM} Lions, J. L. and Magenes, E., ``{\it Non-Homogeneous Boundary Value Problems and Applications,}" v.I, Springer-Verlag, Berlin, 1972. % \bibitem{NP} Nazarov, S. A. and Plamenevskii, B. A., {\it ``Elliptic Problems in Domains with Piecewise Smooth Boundaries"}, Walter de Gruyter, Berlin, 1994. % \bibitem{S} Shkalikov, A.A., Elliptic equations in a Hilbert space and spectral problems connected with them, {\it Trudy Seminara imeny I. G. Petrovskovo}, {\bf 14} (1989), 140--224 (in Russian; translated into English: {\it J. Soviet Math.}, {\bf 51}, 4 (1990), 2399--2467). % \bibitem {TY1} Titeux, I. and Yakubov, Ya., Completeness of root functions for thermal conduction in a strip and a sector, {\it C.~R.~Acad.~Sci., Paris}, {\bf 319}, S\'erie~I (1994),~1133--1139. % \bibitem {TY2} Titeux, I. and Yakubov, Ya., Completeness of root functions for thermal conduction in a strip with piecewise continuous coefficients, {\it Math. Models and Methods in Applied Sciences}, {\bf 7}, 7 (1997), 1035--1050. % \bibitem{TY3} Titeux, I. and Yakubov, Ya., Completeness of root functions for elasticity problems in a strip, {\it Math. Models and Methods in Applied Sciences}, {\bf 8}, 5 (1998), 761--786. % \bibitem{T} Triebel, H., {\it ``Interpolation Theory. Function Spaces. Differential Operators"}, North-Holland, Amsterdam, 1978. % \bibitem{Y} Yakubov, S., {\it ``Completeness of Root Functions of Regular Differential Operators"}, Longman Scientific and Technical, New York, 1994. % \bibitem{YY} Yakubov, S. and Yakubov, Ya., {\it ``Differential-Operator Equations. Ordinary and Partial Differential Equations"}, Chapman and Hall/CRC, Boca Raton, 2000. % \end{thebibliography} \noindent\textsc{Yakov Yakubov} \\ Raymond and Beverly Sackler Faculty of Exact Sciences\\ School of Mathematical Sciences, Tel-Aviv University\\ Ramat-Aviv 69978, Israel\\ e-mail: yakubov@post.tau.ac.il \end{document}