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\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}

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\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, $0<\theta<1$, $1<p<\infty$, $1\le q\le\infty$
and $s=(1-\theta)s_0+\theta s_1$. Set $W^s_p(G):=B^s_{p,p}(G)$, where
$0<s$ is not an integer.

Let $\{ E_0,E_1 \}$ be an interpolation couple. Further, let
$\ell=1,2,\dots $, and $1\le p\le\infty$.
Then one sets
\begin{align*}
W^\ell_p((0,1);E_0,E_1):=\big\{&u(t)\mid u(t)
\text{ is  an } (E_0+E_1)\text{-valued function in } (0,1)\\
&\text{with } u(t) \in L_{p}((0,1);E_0),\; u^{(\ell)} (t)
\in L_{p}((0,1);E_1) \big\}\\
\|u\|_{W^\ell_p((0,1);E_0,E_1)}
&:=\|u(t)\|_{L_{p}((0,1);E_0)}+
\|u^{(\ell)}(t)\|_{L_{p}((0,1);E_1)},
\end{align*}
where $L_p((0,1);E):=W^0_p((0,1);E)$. It is known that $W^\ell_p((0,1);E_0,E_1)$
is a Banach space \cite[Lemma 1.8.1]{T}. One can also replace
$(0,1)$ by $(0,\infty)$.

Let $H$ be a Hilbert space. Consider a polynomial operator pencil equation in
$H$
\begin{equation}
L(\lambda)u:=\lambda^n u+\lambda^{n-1}A_1u+\cdots+A_nu=0,
\label{0.1}
\end{equation}
where $n$ is a natural number and $A_k$ are, generally speaking, unbounded
operators in $H$. Let $H_n\subset H$ be a Hilbert space, such that
operators $A_k$, $k=1,\dots,n$, from $H_n$ into $H$, are  bounded.

A number $\lambda_0$ is called an {\it eigenvalue} of equation
(\ref{0.1}), or of the operator pencil $L(\lambda)$, if 
$$
L(\lambda_0)u=0
$$
has a nontrivial solution belonging to $H_n$. The nontrivial
solution $u_0 \in H_n$ is called an {\it eigenvector} of equation
(\ref{0.1}), or of the operator pencil $L(\lambda)$ corresponding
to the eigenvalue $\lambda_0$.  A solution of the equation
$$L(\lambda_0)u_p+{1\over 1!}L'(\lambda_0)u_{p-1}+\cdots+{1\over p!}L^{(p)}(\lambda_0)u_0=0,$$
$u_p \in H_n$ is called a {\it $p$-associated vector} to the
eigenvector $u_0$ of (\ref{0.1}), or of the operator
pencil $L(\lambda)$.

Eigenvectors and associated vectors are
combined under the general name  {\it root vectors} of equation
(\ref{0.1}), or of the operator pencil $L(\lambda)$. The dimension
of the linear space of all root vectors corresponding to 
$\lambda_0$ is called {\it algebraic multiplicity} of $\lambda_0$.

A complex number $\lambda$ is called a {\it regular point} of
(\ref{0.1}), or of the operator pencil $L(\lambda): u \to
L(\lambda)u$ which  is bounded from $H_n$ into $H$, if for any $f\in H$,
$$ L(\lambda)u=f
$$
 has a unique solution $u\in H_n$ and $ \|u\|_{H_n}\le C(\lambda)\|f\|$.


The complement of the set of regular points in the complex plane is
called the {\it spectrum} of (\ref{0.1}), or of the
operator pencil $L(\lambda)$.

The spectrum of (\ref{0.1}), or of the operator pencil
$L(\lambda)$, is called {\it  discrete}, if:
\begin{enumerate}
\item[a)] All points which are not eigenvalues of
(\ref{0.1}) are regular points of (\ref{0.1})

\item[b)] The eigenvalues are isolated and have finite algebraic
multiplicities

\item[c)] Infinity is the only limit point of the set of the eigenvalues
of (\ref{0.1}).
\end{enumerate}
Consider the Cauchy problem for a differential-operator  equation
\begin{gather}
L(D)u:= u^{(n)}(t)+A_1u^{(n-1)}(t)+\cdots+A_nu(t)=0,\label{0.2}\\
u^{(k)}(0)=v_{k+1},\quad k=0,\dots,n-1,\label{0.3}
\end{gather}
where $v_{k+1}$ are given elements of $H$, $D:=\frac d{dt}$, and $t\ge 0$.

By \cite[ Lemma 1, p.56]{YY}, a function 
\begin{equation}
u(t):= \text{e}^{\lambda_0 t}\big({t^k\over
k!}u_0+{t^{k-1}\over(k-1)!}u_1+\cdots+u_k\big)\label{0.4}
\end{equation}
is a solution of (\ref{0.2}), if and only if the system
of vectors $u_0, u_1,\cdots,u_k$ is a chain of root vectors of
(\ref{0.1}), corresponding to the eigenvalue $\lambda_0$.
A solution of the form (\ref{0.4}) is called an {\it elementary
solution} of (\ref{0.2}).

The possibility of approximating solutions of (\ref{0.2})--(\ref{0.3}) 
by linear  combinations  of the elementary solutions, suggests that  
the vector $(v_1,v_2,\dots,v_n)$ should be approximated by linear
combinations of vectors of the form
\begin{equation}
(u(0),u'(0),\dots,u^{(n-1)}(0)),\label{0.5}
\end{equation}
where $u(t)$ is an elementary solution of the form (\ref{0.4}).

Let $\mathcal{H}$ be a Hilbert space, continuously embedded into the orthogonal
sum of Hilbert spaces 
$\overset n {\oplus} H=H\oplus H \oplus\cdots \oplus H$.

A system of  root vectors of (\ref{0.1}) is called {\it
$n$-fold complete in the space} $\mathcal{H}$, if the system of vectors 
(\ref{0.5}) is complete in $\mathcal{H}$, i.e., the closure of a linear 
span of vectors (\ref{0.5}) is equal to $\mathcal{H}$.

\section{Abstract results for  second order elliptic equations}

In this section we prove completeness of a system of root vectors corresponding
to a part of the spectrum of a quadratic operator pencil in a Hilbert space.
Isomorphism and the completeness of elementary solutions corresponding to the
eigenvalues $\lambda_i$ with $\mathop{\rm Re}\lambda_i<0$ for some special
cases of abstract differential equations of the second order are established.

\subsection{Completeness of a system of root vectors}
Let us consider, in a Hilbert space $H$, the unbounded operator pencil:
\begin{equation}
L(\lambda):= \lambda^2 I+B.\label{1.1}
\end{equation}

\begin{theorem} \label{thm1}
Let the following conditions be satisfied:
\begin{enumerate}
\item $B$ is a densely defined and closed operator in in a Hilbert space $H$;
\item There exists a Hilbert space $H_1$ for which the compact
embeddings $H(B)\subset H_1\subset H$ take place;
$\overline{H_1}  \big|_H=H$ and $\overline{H(B)}  \big|_{H_1}=H_1$;
\item $s_j(J_1;H(B),H_1)\le Cj^{-p}$ and $s_j(J_2;H_1,H)\le Cj^{-p}$,
 $j=1,\dots,\infty$, for some $p>0$ \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}.

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\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}