\documentstyle[tr_def,amssymbo,twoside,paper]{article}
\soyad{BAIRAMOV \& \c{C}ELEB\.{I}}
\author{Elgiz Bairamov \& A. Okay \c{C}elebi}
\title{EIGENFUNCTION EXPANSIONS FOR THE
KLEIN-GORDON $S$-WAVE EQUATION WITH COMPLEX POTENTIAL}
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\begin{document}
\maketitle
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\begin{abstract}

In this article, we consider the operator
$L$ generated in 
$L_2(\Bbb{R}_+)$ by the Klein - Gordon s-wave
equation with complex potential Q. Under the conditions

\[
\db \lim_{x \rightarrow \infty}
Q(x)=0, \quad \db \sup_{0 \leq x < \infty} \{
exp (\epsilon \sqrt{x}) | Q'(x)|\} <
\infty, \epsilon>0\]

we have derived a two-fold spectral expansion of 
$L$ in terms of the principal functions.
Moreover we also investigated the convergence of the spectral expansion.

\end{abstract}
\mb \vs{0.2cm}\\

{\bf Key words:} Klein-Gordon equation, non-selfadjoint  operator, eigenfunction
expansion, spectral analysis. 

{\bf AMS Subject classification:} 47 A 10

\newpage
\baslik{1. Introduction}

Let $L_0$  denote the non-selfondjoint one 
dimensional Schr”dinger operator generated in 
$L_2(\Bbb{R_+})$, by the differential expression

$$
l(y)=-y''+V(x)y, \quad x \in \Bbb{R}_+
=[0, \infty),
$$
and the boundary condition 
$y(0)=0$, 
where
$V$ is a complex valued function. The spectral
analysis of 
$L_0$ has been started by Naimark[8],
in 1960. 
He has proved the existence of the spectral
singularities on the continuous 
spectrum of $L_0$
Assuming the condition
$e^{\epsilon x} V(x) \in L_1 (\Bbb{R}_+)$ 
hold for some $\epsilon>0$, he has obtained that, 
$L_0$ has a finite number of eigenvalues and spectral singularities with finite
 multiplicities. Let us note that, 
the spectral expansion of $L_0$
has been derived with the assumption that it has no spectral
singularities [8].

The spectral expansion of $L_0$ has been
 obtained by Pavlov [10], taking
into account the spectral singularities too. 
To arrive this result he has utilized 
the analytic expression of generalized spectral function
(in sense of Marchenko[7]) and the 
properties of the Weyl-Titchmarsh function [12] of $L_0$

It is well-known that, the spectral singularities belong to the continuous spectrum
and are the poles of the resolvent; but they are not the eigenvalues. Moreover the 
principal functions corresponding to the 
spectral singularities are not elements of $L_2(\Bbb{R}_+)$. 
Hence it is natural to expect
that, the spectral singularities play a certain critical role in the spectral expansion. 

Lyance [5] has studied the role of the spectral singularities in the spectral expansion
of $L_0$ by use of Fourier $L_0$-transforms. The Laurant expansion
of the resolvents of the abstract non-selfadjoint operators in the neighbourhood 
of the spectral singularities have been 
investigated by Gasymov-Maksudov [2] and 
Maksudov-Allakhverdiev [6]. 
They also have studied the effect of the spectral singularities
in the spectral analysis of these operators. 
The spectral analysis of some class 
of dissipative operators with spectral singularities have 
been considered by Pavlov [11]
using the theory of functional model and characteristic functions.

In the present article considering the operator $L$ generated in 
$L_2 (\Bbb{R}_+)$
by the differential expression
$$
l(y)=y''+[\lambda-Q(x)]^2 y, \quad x \in \Bbb{R}_+
$$
and the boundary condition $y(0)=0$, where $Q$ is complex
valued, bounded and absolutely continuous in each 
finite subinterval of $\Bbb{R}_{+,}$ we
have derived a two-fold spectral expansion 
(in the sense of Kaldysh [4]) in terms of the principal functions of $L$ under
the conditions 

$$
\db \lim_{x \rightarrow \infty} Q(x)=0, \qquad
\db \sup_{0 \leq x<\infty} 
\{exp
(\epsilon \sqrt{x})|Q' (x)|\} <\infty, \epsilon>0, \eqno{(1.1)}$$
taking into account the existence of the spectral singularities. Moreover, 
the convergence of the spectral expansion is studied. The role of the spectral singularities
in the spectral expansion of $L$ with respect to the principal functions is investigated
by the regularizitaon of divergent integrals using summability factors. 

Related with the operator $L$, we will consider the boundary value problem

$$y''+[\lambda-Q(x)]^2 y=0, \qquad x \in \Bbb{R}_+ 
\eqno{(1.2)}$$

$$y(0)=0. \eqno{(1.3)}$$
(1.2) is called the Klein-Gordon s-wave equation for a particle of zero mass with static
potential $Q$.

Throughout this article we will assume that the condition (1.1) holds

\baslik{2. Eigenfunction expansion}

Let $\varphi(x, \lambda)$ and $\psi(x, \lambda)$  denote
the solutions of (1.2) subject to the initial conditions
$$
\varphi(0,\lambda)=0,\qquad \varphi_x(0, \lambda)=1,
$$
$$
\psi(0,\lambda)=1,\qquad \psi_x(0, \lambda)=0,
$$
respectively. It is clear [3] that, these solutions
are entire functions of $\lambda$.
Also, the Eq. (1.2) has the solutions 


$$
f^+(x,\lambda)=e^{i \alpha(x)+i\lambda x}+
\int^{\infty}_x 
K^+(x,t)e^{i \lambda t} dt
\eqno{(2.1)}
$$
and
$$
f^-(x,\lambda)= e^{-i \alpha(x)-i \lambda x}+
\int^{\infty}_x K^-(x,t) e^{-i \lambda t} dt
\eqno{(2.2)}
$$
for $\lambda \in \ovl{\Bbb{C}}_+=\{\lambda: \lambda \in 
\Bbb{C}, \quad Im \lambda \geq 0 \}$ and 
$\lambda \in \ovl{\Bbb{C}}_-=
\{\lambda: \lambda \in \Bbb{C}, Im \lambda \leq 0\}$
respectively, where
$\alpha(x)_= \db \int^{\infty}_x  Q(t) dt$ 
and the kernels 
$K^{\pm} (x,t)$ may be expressed in terms of Q$\;\;$[3].

Using the standart technique  [9] we easily obtaine that 
$$
G(x,t;\lambda)=
\left\{
\begin{array}{ll}
G^+(x,t; \lambda),& \lambda \in \Bbb{C}_+\\
G^-(x,t; \lambda),& \lambda \in \Bbb{C}_-
\end{array}
\right.
\eqno{(2.3)}
$$
is the Green's function of $L$, where
$$
G^\pm (x,t;\lambda)=
\db \frac{f^\pm_x (0, \lambda)}{f^\pm (\lambda)}
\varphi(x, \lambda) \varphi(t, \lambda)+a(x,t,\lambda)
\eqno{(2.4)}
$$
in which
$$
\Bbb{C}_+=\{\lambda:\lambda \in \Bbb{C}, Im \lambda>0\}, \Bbb{C}_-=
\{\lambda:\lambda \in \Bbb{C}, Im \lambda<0\},$$
$$f^\pm (\lambda):= f^\pm (0, \lambda), a(x,t,\lambda)=
\left\{
\begin{array}{ll}
\varphi (t,\lambda)\psi(x,\lambda),& 0 \leq t<x\\
\varphi (x,\lambda) \psi(t,\lambda),& x \leq t<\infty.
\end{array}
\right.
$$

We know that [1], $L$ has a finite
number of eigenvalues
and spectral
singularities and each
of them is of a finite
multiplicity. Let
$\lambda_1, \cdots, \lambda_j$ and 
$\lambda_{j+1}, \cdots \lambda_k$ denote the zeros
of the functions
$f^+$ in
$\Bbb{C}_+$
and 
$f^-$
in
$\Bbb{C}_-$
(which are the eigenvalues of 
$L$)
with multiplicities 
$m_1, \cdots, m_j$
and 
$m_{j+1}, \cdots, m_k$,
respectively. 
Similarly, let
$\mu_1, \cdots,\mu_p$
and
$\mu_{p+1}, \cdots,\mu_q$
be the zeros of 
$f^+$ and
$f^-$
in $\Bbb{R}^*:=\Bbb{R} \setminus \{0\}$
(which are the spectral singularities of $L$) 
with multiplicities 
$n_1, \cdots, n_p$
and  
$n_{p+1},\cdots, \mu_q$ 
respectively.

$H_+$ and $H_-$ will denote the Hilbert spaces

$$H_+=\{f:\db \int^\infty_0 (1+x)^{2n_0}|f(x)|^2 dx<\infty\},$$
$$H_-=\{f:\db \int^\infty_0 (1+x)^{-2n_0}|f(x)|^2 dx<\infty\},
$$
where
$$n_0= max\{n_1, \cdots, n_p, n_{p+1}, \cdots, n_q\}.$$

We have obtained [1] previously  that
$$
\left\{\frac{\partial^n}{\partial \lambda^n} 
\varphi(.,\lambda) \right\}_{\lambda=\lambda_i}
\in L_2 (\Bbb{R}_+),
\;\;n=0,1,\cdots,m_{i-1}, 
i=1, \cdots, j,j+1, \cdots,k.
\eqno{(2.5)}
$$
$$
\left\{\frac{\partial^\nu}{\partial \lambda^\nu} 
\varphi(.,\lambda)\right\}_{\lambda=\mu_l}
\in H_-,
\;\;\nu=0,1,\cdots,n_{l-1}, 
l=1, \cdots, p,p+1, \cdots,q.
\eqno{(2.6)}
$$

It is clear  from the definition of the
resolvent operator that, for every 
$f \in C^\infty_0 (\Bbb{R}_+)$,

$$f(x)=\int^\infty_0 G(x,t;\lambda) \{f''(t)+[\lambda-Q(t)]^2 f(t)\} dt
\eqno{(2.7)}
$$
holds, where $C^\infty_0(\Bbb{R}_+)$,
is the set of all infinitely differentiable
functions in $\Bbb{R}_+$ with compact support. 

Let $\gamma_+$ be the contour which 
izolates the spectral singularities $\mu_i, 
i=1, \cdots, p$ of $L$ by 
semicircles with centers at $\mu_i$
having the same radius
$\delta_0$
in the upper half-plane and $\gamma_-$ be the
corresponding contour for the spectral singularities
$\mu_i, i=p+1, \cdots, q$ of 
$L$ in the lower half-plane. The radius
$\delta_0$ will be chosen so smoll that two neighbouring semicircles have no common
points (see Fig.1)
\mb\vs{5.0cm}

From (2.3),(2.4),(2.7) and using the contour integral technique we have

$$f(x)=\db \sum^k_{i=1} \left\{\left( 
\frac{\partial}{\partial \lambda} \right)^{m_i-1}
[\lambda a_i(\lambda) \varphi (x,\lambda) 
\varphi (f, \lambda)] \right\}_{\lambda=
\lambda_i}$$

$$+\frac{1}{2 \pi i} \db \int_{\gamma_+}
\db \frac{\lambda f_x^+(0, \lambda)}{f^+  
(\lambda)} \varphi(x, \lambda) \varphi(f, \lambda) d \lambda
-\frac{1}{2 \pi i} \db \int_{\gamma_-}
\db \frac{\lambda f_x^-(0, \lambda)}{f^- (\lambda)} \varphi
(x, \lambda) \varphi(f, \lambda) d \lambda \eqno{(2.8)}$$

$$0=\db \sum^k_{i=1} \left\{\left( 
\frac{\partial}{\partial \lambda} \right)^{m_i-1}
[a_i(\lambda) \varphi 
(x,\lambda) \varphi (f, \lambda)]\right\}_{\lambda=
\lambda_i}$$


$$
+\frac{1}{2 \pi i} 
\db \int_{\gamma_+}
\db \frac{f_x^+(0, \lambda)}{f^
+ (\lambda)} \varphi
(x, \lambda) \varphi (f, \lambda) d \lambda 
-\frac{1}{2 \pi i} 
\db \int_{\gamma_-}
\db \frac{f_x^- (0, \lambda)}{f^
- (\lambda)} \varphi
(x, \lambda) 
\varphi (f, \lambda) d \lambda ,
\eqno{(2.9)}$$

for every $f \in C^\infty_0 (\Bbb{R}_+)$,
where

$$a_i(\lambda)=
\left\{
\begin{array}{ll}
- \frac{(\lambda-\lambda_i)^{m_i} f_x^+ 
(0, \lambda)}{(m_i-1)! f^+(\lambda)},&
i=1, \cdots,j\\
- \frac{(\lambda-\lambda_i)^{m_i} f_x^- (0, \lambda)}
{(m_i-1)! f^- (\lambda)},&
i=j+1, \cdots,k
\end{array}
\right.
\eqno{(2.10)}
$$

and

$$\varphi (f, \lambda)= \int^\infty_0 f(t) \varphi (t, \lambda) dt.$$

\begin{claim}{Lemma 2.1.}
For any $f \in C^\infty_0 (\Bbb{R}_+)$,
there exists a constant $C>0$ so that

$$ \int^\infty_{- \infty}|\lambda \varphi (f, 
\lambda)|^2 d \lambda \leq C \int^\infty_0
|f(x)|^2 dx. \eqno{(2.11)}$$
\end{claim}

\begin{proof} It is known that
$$ 2i \lambda \varphi(x, \lambda)=
f^- (\lambda) f^+(x, \lambda)-f^+ (\lambda) f^- (x, \lambda), 
\eqno{(2.12)}
$$
holds for $\lambda \in \Bbb{R}^*$. Using (2.1), (2.2), (2.12) and Parseval's
equation for the Fourier transformation
we get (2.11). 
\end{proof}


By the preceding Lemma, for every $f \in L_2 (\Bbb{R}_+)$
the limit
$$
\varphi (f, \lambda)=\lim_{N \rightarrow \infty} 
\int^N_0 f(x) \varphi (x, \lambda) dx$$
exists in the sense of convergence 
in the mean square relative to the measure 
$\lambda^2 d \lambda$ on the real axis; that is

$\db \lim_{N \ra \infty} \int^\infty_{- \infty} |\varphi 
(f, \lambda)- \int^N_0 f(x)
\varphi(x, \lambda) dx|^2 \lambda^2 d \lambda=0$

The estimate (2.11) may be extended onto $L_2 (\Bbb{R}_+)$.
Similar to (2.11) we have 

$$\int^\infty_{- \infty} |(\frac{d}{d \lambda})^\nu 
[\lambda \varphi (f, \lambda)]|^2 d \lambda \leq C_\nu \int^\infty_0
(1+x)^{2 \nu} |f(x)|^2 dx, \nu=1, \cdots, n 
\eqno{(2.13)}
$$
where $f$ satisfies
$\int^\infty_0 (1+x)^{2n} |f(x)|^2 dx<\infty$ and $C_\nu>0$ are
constants. 

In order to transform (2.8) and (2.9) into the 
spectral expansion of $L$ we have to deform the integrals over $\gamma_+$
and $\gamma_-$ onto the real axis 
(i.e. to the continuous spectrum of $L$). For
this purpose we will use the technique of the regularization of divergent
integrals. So we will define the following summability factors: 

$$F^+_{\nu \wr} (\lambda)=
\left\{
\begin{array}{ll}
\frac{(\lambda-\mu_\nu)^\wr}{\wr !},& |\lambda-\mu_\nu|<\delta, \nu=1,2, \cdots, p\\
0,&|\lambda-\mu_\nu| \geq \delta, \nu=1,2, \cdots, p
\end{array}
\right.
$$

$$F^-_{\nu \wr} (\lambda)=
\left\{
\begin{array}{ll}
\frac{(\lambda-\mu_\nu)^\wr}{\wr !},& |\lambda-\mu_\nu|
<\delta, \nu=p+1, \cdots, q\\
0,&|\lambda-\mu_\nu| \geq \delta, \nu=p+1, \cdots, q
\end{array}
\right.
$$
with $\delta > \delta_0$. We can choose $\delta>0$ so
small that the $\delta$-neighbourhoods
of $\mu_\nu, \nu=1, \cdots, p, p+1, \cdots,q$ have
no common points. Let us define the functionals

$$F^+ \{g_1 (\lambda)\}=g_1(\lambda)-\db \sum^p_{\nu=1} 
\db \sum^{n_\nu-1}_{\wr=0} 
\{(\frac{d}{d \lambda})^\wr 
g_1 (\lambda)\}_{\lambda=\mu_\nu}  F^+_{\nu \wr} (\lambda) 
\eqno{(2.14)}$$

$$F^- \{g_2 (\lambda)\}=g_2(\lambda)-\db \sum^q_{\nu=p+1} 
\db \sum^{n_\nu-1}_{\wr=0} 
\{(\frac{d}{d \lambda})^\wr 
g_2 (\lambda)\}_{\lambda=\mu_\nu}  F^-_{\nu \wr} (\lambda) 
\eqno{(2.15)}$$
where $g_1$ and $g_2$ are chosen so that the right hand side
of the above formulas are meaning-ful. 

In the following we will use the operators $A^+$ and $A^-$ given by 

$$
A^\pm f(x)= 
\db \frac{1}{2 \pi i} 
\db \int_{\gamma_\pm}
\db \frac{\lambda f^\pm_x (0, \lambda)}
{f^\pm (\lambda)} 
\varphi(x, \lambda) \varphi (f, \lambda) d \lambda
\eqno{(2.16)}$$

Let $||.||_+$ and $||.||_-$ 
denote the norms of the spaces $H_+$ and $H_-$ respectively. 

\begin{claim}{Lemma 2.2.} For each $f \in H_+$, there exists a constant $C>0$ such that 

$$
||A^\pm f||_- \leq C|| f||_+$$

holds. 
\end{claim}

\begin{proof} From (2.14) and (2.16) we have 

$$
A^+ f(x)= \db \frac{1}{2 \pi i} 
\db \int_{-\infty}^\infty
\db \frac{f^+_x (0, \lambda)}{f^+ (\lambda)} 
F^+ \{\lambda \varphi(x, \lambda) 
\varphi (f, \lambda)\} d \lambda
$$
$$
+\db \frac{1}{2 \pi i} \db \sum^p_{\nu=1} \db \sum^{n_\nu-1}_{\wr=0}
\{(\db \frac{\partial}{\partial \lambda})^\wr 
[\lambda \varphi(x, \lambda) \varphi(f, \lambda)] \}_{\lambda=\mu_\nu}
\db \int_{\gamma_+}
\db \frac{f_x^+ (0, \lambda)}{f^+ (\lambda)}
F^+_{\nu_\wr} (\lambda) d \lambda
\eqno{(2.17)}
$$

Using the integral form of the remainder in the Taylor's formula we get  


$$
F^+ \{ \lambda \varphi (x, \lambda) \varphi(f, \lambda) \}=
\left\{
\begin{array}{ll}
\lambda \varphi(x, \lambda) \varphi(f, \lambda), \qquad \quad \lambda
\in \wedge^+_0&\\
 \frac{1}{(n_\nu-1)!}
 \int^{\lambda}_{\mu_\nu}
(\lambda-t)^{n_\nu -1}
\{(\frac{\partial}{\partial t})^{n_\nu}
[t \varphi (x,t) \varphi (f,t)]\}dt, &\\
\qquad \qquad \lambda \in \wedge^+_\nu, \nu=1, \cdots, p,\\
\end{array}
\right.
\eqno{(2.18)}
$$

where

$\wedge^+_\nu=(\mu_\nu-\delta, \mu_\nu+\delta), \nu=1, \cdots, p, \wedge_0^+=
\Bbb{R} \setminus  \cup^p_{\nu=1} \wedge_\nu^+$.

If we use the notations

$$
A^+_\nu f(x)= \frac{1}{2 \pi i} 
\int_{\wedge^+_\nu}
\db \frac{f^+_x (0, \lambda)}{f^+ (\lambda)} 
F^+ \{ \lambda \varphi(x, \lambda) 
\varphi (f, \lambda) \} d \lambda, \nu=0,1,\cdots, p
$$

$$
\tilde{A}^+ f(x)= 
\db \frac{1}{2 \pi i} 
\db \sum^p_{\nu=1} 
\db \sum^{n_\nu-1}_{\wr=0}
\{(\db \frac{\partial}{\partial \lambda})^\wr 
[\lambda \varphi(x, \lambda) 
\varphi(f, \lambda)]\}_{\lambda=\mu_\nu}
\db \int_{\gamma_+}
\db \frac{f_x^+ (0, \lambda)}{f^+ (\lambda)}
F^+_{\nu_\wr} (\lambda) d \lambda
$$

\noindent we obtain

$A^+=\db \sum^p_{\nu=0} A^+_\nu+\tilde{A}^+$

\noindent from (2.17) and (2.18).
By (2.11) for $f \in L_2 (\Bbb{R}_+)$ 
and (2.13) we easily prove that each
of the operators 
$A^+_0, A_1^+, \cdots, A_p^+$
and $\tilde{A}^+$ are continuous from $H_+$ into
$H_-$. 
So we get

$$
||A^+ f||_- \leq C ||f||_+$$

In a similar way

$$
||A^- f||_- \leq C ||f||_+$$

can be proved. 
\end{proof}

Let us introduce the following notation

$$
a_{\nu \wr}=
\left\{
\begin{array}{ll}
\db \frac{1}{2 \pi i} \int_{\gamma_+}
\db \frac{f^+_x (0, \lambda)}{f^+(\lambda)}
F^+_{\nu \wr} (\lambda) d \lambda,  & l=0,1,
\cdots,n_\nu-1, \nu=1, \cdots, p\\
\frac{1}{2 \pi i} \int_{\gamma_-}
\frac{f^-_x (0, \lambda)}{f^-(\lambda)}
F^-_{\nu \wr} (\lambda) d \lambda,  & l=0,1,
\cdots,n_\nu-1, \nu=p+1, \cdots, q
\end{array}
\right.
\eqno{(2.19)}
$$

\begin{claim}{Theorem 2.3.} Under the conditions (1.1), 
the two-fold spectral expansion

$$
f(x)=\db \sum^k_{i=1}
\{
(
\db \frac{\partial}{\partial \lambda})^{m_i-1}
[\lambda a_i (\lambda) \varphi(x, \lambda)
\varphi (f, \lambda)] \}_{\lambda=\lambda_i}$$

$$+ \db \sum^q_{\nu=1}
\sum^{n_\nu-1}_{\wr=0}
a_{\nu \wr}
\{
(\db \frac{\partial}{\partial \lambda})^\wr
[
\lambda \varphi (x, \lambda)
\varphi (f, \lambda)]\}_{\lambda=\mu_\nu}$$

$$+
\frac{1}{2 \pi i} \int_{-\infty}^\infty
\frac{f^+_x (0, \lambda)}
{f^+(\lambda)}
F^+\{\lambda \varphi (x, \lambda) \varphi(f, \lambda)\} d \lambda$$

$$-
\frac{1}{2 \pi i} \int_{-\infty}^\infty
\frac{f^-_x (0, \lambda)}
{f^-(\lambda)}
F^-\{\lambda \varphi (x, \lambda) \varphi(f, \lambda)\} d \lambda
\eqno{(2.20)}$$

$$0 =
\db \sum^k_{i=1}
\{
(
\frac{\partial}{\partial \lambda})^{m_i-1}
[a_i (\lambda) \varphi(x, \lambda)
\varphi (f, \lambda)] \}_{\lambda=\lambda_i}$$

$$+ \db \sum^q_{\nu=1}
\db \sum^{n_\nu-1}_{\wr=0}
a_{\nu \wr}
\{
\biggl(
\frac{\partial}{\partial \lambda}\biggr)^\l
[
\varphi
(x,\lambda) \varphi (f, \lambda)]\}_{\lambda=\mu_\nu}$$

$$+
\frac{1}{2 \pi i} \int_{-\infty}^\infty
\frac{f^+_x (0, \lambda)}
{f^+(\lambda)}
F^+ \{\varphi (x, \lambda) \varphi(f, \lambda)\} d \lambda$$

$$-
\frac{1}{2 \pi i} \int_{-\infty}^\infty
\frac{f^-_x (0, \lambda)}
{f^-(\lambda)}
F^- \{\varphi (x, \lambda) \varphi(f, \lambda)\} d \lambda
\eqno{(2.21)}$$
of $L$, in terms of the  principal functions, holds, 
for any $f \in H_+$ and the 
integrals in (2.20) and (2.21) converges 
in the norm of $H_-$, where $a_i, F^+, F^-$
and $a_{\nu \wr}$ are
defined by (2.10), (2.14), (2.15) and (2.19) respectively. 
\end{claim}

\begin{proof} Using (2.8), (2.9), (2.16) and (2.17) we find
(2.20) and (2.21). 
The convegence of the integrals appearing in (2.20) and (2.21) in the norm $H_-$
has been given in Lemma 2.2.
\end{proof}

{\bf Acknowledgement:}
E.Bairamov would like to thank Scientific and Technical Research Council
of Turkey for the financial support.


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87-110, Consultants Bureau, New York, 1967.

\item[11.] B.S. Pavlov, On separation conditions for the spectral components
of a dissipative operator, Math. USSR Izvestiya 9, 113-137 (1975).

\item[12.] E.C. Titchmarsh, Eigenfunction expansions associated with second order
differential equations, Oxford U.P. I, 1962.
\end{itemize}

\bc
Elgiz Bairamov\\
Ankara University\\
Department of Mathematics\\
06100 Beźevler-Ankara, TURKEY
\ec

\bc
A. Okay €elebi\\
Middle East Technical University\\
Department of Mathematics\\
06531 Ankara, TURKEY\\
\ec


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