\documentclass[reqno]{amsart}
\usepackage{hyperref}


\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 16, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/16\hfil Inverse spectral analysis]
{Inverse spectral analysis for singular
  differential operators with matrix coefficients}
\author[N. H. Mahmoud, I. Yaich \hfil EJDE-2006/16\hfilneg]
{Nour el Houda Mahmoud, Imen Ya\"\i ch}  % in alphabetical order

\address{Nour el Houda Mahmoud \hfill\break
D\'epartement de Math\'ematiques,
Facult\'e des Sciences de Tunis,
Campus Universitaire, 1060 Tunis, Tunisia}
\email{houda.mahmoud@fst.rnu.tn}

\address{Imen Ya\"\i ch \hfill\break
D\'epartement de Math\'ematiques,
Facult\'e des Sciences de Tunis,
Campus Universitaire, 1060 Tunis, Tunisia}
\email{imen.maalej@fst.rnu.tn}

\date{}
\thanks{Submitted October 14, 2005. Published February 2, 2006.}
\subjclass[2000]{45Q05, 45B05, 45F15, 34A55, 35P99} 
\keywords{ Inverse problem; Fourier-Bessel transform;
 spectral measure; \hfill\break\indent
Hilbert-Schmidt operator; Fredholm's equation}

\begin{abstract}
 Let $L_\alpha$ be the Bessel operator with
 matrix coefficients defined on $(0,\infty)$ by
$$
 L_\alpha U(t) = U''(t)+  {I/4-\alpha^2\over t^2}U(t),
$$
 where $\alpha$ is a fixed diagonal matrix. The aim of this study,
 is to  determine, on the positive half axis,  a singular
 second-order differential operator of $L_\alpha+Q$  kind and its
 various properties from only its spectral characteristics.
 Here $Q$ is a matrix-valued function. Under suitable
 circumstances, the solution is constructed by means of the spectral
 function, with the help of the Gelfund-Levitan process. The
 hypothesis on the spectral function are inspired on the results
 of some direct problems. Also the resolution of Fredholm's  equations
 and properties of  Fourier-Bessel transforms are used here.
\end{abstract}


\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\section{Introduction}

By  an inverse problem, physicists mean the
derivation of forces from experimental data. A well-known solution
of an inverse problem was the discovery of the gravitation law by
Newton from the observations of Kepler. Inverse  problems receive
considerable attention in mathematics, physics, mechanics,
meteorology and other branches of science. In spectral analysis,
this consists in recovering  operators from their spectral
characteristics that means  the bounded states and the scattering
matrix or the spectral function.  A procedure for explicitly
constructing a potential for a boundary-problem without
singularity from its spectral characteristics  was formulated by
Gelfand and Levitan in  \cite{g1}, they reduced the problem to a
linear integral equation.
 The extension of the Gelfund-Levitan
 theory to  higher waves ($l>0)$ are due first to Stashevskaya \cite{s1},
  Volk \cite{v1} and also to Jost  and Kohn \cite{j1}.
In the literature, in this  direction, we have  several other studies;
see for example \cite{a1,c1,f1,g2,n1,n2}.
For example, the inverse scattering  problem for the radial
 Schr\"odinger equation with coupling between the $l^{\rm th}$ and the
 $(l+2)$ angular momentum,  which reduces to a system of two
singular second order differential equations is considered in
\cite{n1}.
 Spectral problems associated with a generalization of a such
system are studied in \cite{c2,m1,m2}. These papers deal
with the equation defined, on $]0,\infty[$, by
\begin{equation}
U'' + {I/4-\alpha^2\over t^2}U+Q(t)U=-\lambda^2U,
\label{e1}
\end{equation}
where $\lambda$ is a complex parameter, $\alpha$ is a
diagonal $n\times n$ matrix, such that
\begin{equation}
[\alpha]_{ii}=\alpha_i,\quad
\alpha_n \geq \dots \geq \alpha_1>-1/2\label{e2}
\end{equation}
 and $Q$ is a real symmetric sufficiently smooth $n\times n$
matrix-valued function. For a such potential, \eqref{e1} is solved and
its various needed solutions are determined. Associated
Fourier-Bessel transform is studied and  properties of the
spectral function  are deduced.  In the following, we make a brief
recall of useful results. Let so be given the matrix Bessel
operator $L_\alpha$ defined, for $t>0$, by
\begin{equation}
L_\alpha U(t) =
U_{tt}(t)+ {I/4-\alpha^2\over t^2}U(t)\label{e3}
\end{equation}
 for which the $n\times n$ diagonal matrix-valued function given, for
$\lambda\in\mathbb{C}$, by
\begin{equation}
\big[\mathcal{J}_{\alpha}(t,\lambda)\big]_{jj}
=(2/\lambda)^{\alpha_j}\Gamma(\alpha_j+1)\sqrt{t}J_{\alpha_j}(\lambda t),
\label{e4}
\end{equation}
is the eigenfunction associated with the eigenvalue $-\lambda^2$
such that
\begin{equation}
\lim_{t\to 0^+}t^{-\alpha-I/2}\mathcal{J}_\alpha(t,\lambda)=I,\label{e5}
\end{equation}
 where $J_\nu$ is the Bessel function of the first kind. Under conditions
on $Q$, the solution $\Phi(t,\lambda)$ of \eqref{e1} satisfying \eqref{e5}
may have the form
$$
\Phi(t,\lambda)=\mathcal{J}_\alpha(t,\lambda)
+\int_0^tK(t,u)\mathcal{J}_\alpha(u, \lambda)du.
$$
Properties of the kernel $K(t,u)$, as for example its twice
differentiability on $t$ and $u$, are deduced. Among other the
following relation  holds
$$
(L_\alpha +Q)_t
K(t,u)=\Bigr[(L_\alpha )_u K^*(t,u)\Bigr]^*,
$$
 where
$$
\big[L_\alpha U^*(t)\big]^*=U_{tt}(t)+U(t){I/4-\alpha^2\over
t^2}.
$$
 We have also the useful  relation
$$
K(t,t)=-{1\over 2}\int_0^tQ(s)ds ,\quad t>0\,.
$$
Since $Q(t)$ is usually taken
integrable at zero, so  obviously $K(t,t)$ vanish at zero. Finally
let $S_0(\lambda)$ be the spectral function associated with
$L_\alpha$ and let $S(\lambda)$ be the  portion of the spectral
function  associated with the continuous spectrum of $L_\alpha+Q$,
we show among other that for $\lambda$ large we have
$$
S(\lambda)-S_0(\lambda)=
2^{-2\alpha}{\bf\Gamma}^{-2}(\alpha+I)\lambda^{\alpha}O(1)
\lambda^{\alpha}
$$
 and that, for $\alpha_1\geq 1$ (see \cite{c2}),
$S(\lambda)$ is integrable at zero.  Here ${\bf\Gamma}(\alpha)$ is
the diagonal constant matrix defined by
$[{\bf\Gamma}(\alpha)]_{jj}=\Gamma(\alpha_j)$,~  $1\leq j\leq n$.
In \cite{m2} and for $t,u>0$, we consider the function
$$
 \Omega(t,u)=\int_0^\infty \mathcal{J}_\alpha(t,\lambda)
(S-S_0)(\lambda)\mathcal{J}_\alpha(u,\lambda) d\lambda
+\sum_{j=1}^m\mathcal{J}_\alpha(t, \lambda_j)C_j\mathcal{J}
_\alpha(u,\lambda_j),
$$
 where the $C_j$ are spectral parameters
associated with the finite discrete spectrum $\lambda_j$, $1\leq
j\leq m$, of the considered operator. The function $\Omega(t,u)$
is related to the kernel $K(t,u)$, for $0<u\leq t$, by the
Gelfand-Levitan equation: \begin{equation}
K(t,u)+\Omega(t,u)+\int_0^tK(t,s)\Omega(s,u)ds=0. \label{e6}
\end{equation}
 The previous equation is solved in \cite{m2} as a Volterra one, where
$\Omega(t,u)$ is seen as its unknown component. Properties of
differentiability and estimates on this function are thus
obtained. Among other properties, we have
\begin{gather*}
\lim_{u\to0^+}\Omega(t,u)=\lim_{u\to0^+}\Omega_u(t,u)=0,\\
(L_\alpha)_t\Omega(t,u)=\Bigr[(L_\alpha)_u\Omega^*(t,u)\Bigr]^*.
\end{gather*}
 Conversely, would it be possible to
construct a system of  singular differential  operators from only
its spectral characteristics? This question is fairly obvious
since the used functions are measurable quantities. Thing which
allows scientists to be very interested by a such subject, usually
called inverse spectral problem.

In the present paper we are concerned with  the resolution of such
problem for a singular second order differential  operator
$L_\alpha+Q$, with matrix coefficients, for which $\alpha$ and
$L_\alpha$ are given respectively by \eqref{e2}, \eqref{e3} and
where the potential $Q$ is to recover
from the measured spectral properties. Analogous processes to
those handled in the references above are used here. The main mean
is the resolution of the Gelfand-Levitan equation \eqref{e6} where
$K(t,u)$ is taken as an unknown function. Properties of symmetry
for $\Omega(t,u)$ and conditions on the spectral function  allow
to solve \eqref{e6} as a Fredholm's equation which give $K(t,u)$ and its
useful properties. This enables us to set
\begin{equation}
Q(t)=-2{d\over dt}K(t,t), \quad t>0.\label{e7}
\end{equation}
Let us give a brief outline of the plan and basic ideas of this survey.
>From the hypothesis below, we first obtain in the second section useful
properties of $\Omega(t,u)$. Then in the third we construct a
function $K(t,u)$ related to $\Omega(t,u)$  by the Gelfand-Levitan
equation \eqref{e6}. Properties of differentiability and estimates on
$K(t,u)$ deduced from those of $\Omega(t,u)$  are also obtained.
This allows to construct, in the forth section, a potential $Q$ by
the relation \eqref{e7} and a function $\Phi(t,\lambda)$ which should  be
an eigenfunction of the operator $L_\alpha+Q$. In the fifth
section, the symmetry of $Q$ is proved and its asymptotic behavior
is obtained in  special cases. The case where the spectrum differs
from which of $L_\alpha$ by a finite discrete one is finally
studied in the sixth  section.

 The final thing to be said  here is that recovering the properties
of $Q(t)$ from
those of $S(\lambda)$ turns out to be difficult. This is due to
the fact that the kernel $\Omega(t,u)$ of \eqref{e6} is a matrix-valued
one expressed in terms of Bessel functions for which there are no
simple addition formulas such as exist for the trigonometric
functions in the scalar case. These difficulties appear in solving
this equation  as well as in searching properties of its solution
and yield us to look for the asymptotic behavior of $Q(t)$ in
restricted  cases.

\section{Preliminaries}

For the case where the operator $L_\alpha$ is the matrix Bessel operator,
with $\alpha$ given by \eqref{e2} and whose  spectrum  reduces
to the continuous one associated with the spectral function
$$
S_0(\lambda)=
2^{-2\alpha}\Gamma^{-2}(\alpha+I)\lambda^{2\alpha+I},\quad \lambda>0,
$$
 we require a singular differential operator which takes the
form $L_\alpha+Q$.
 We assume  given a finite system of discrete eigenvalues
 $\lambda_j=-i\mu_j$, $\mu_j>0$ for
$1\leq j\leq m$, that these parameters  are associated with
hermitian normalizing factors $C_j$, the latest being positive
defined and hermitian matrices. We suppose also given  a
prescribed $n\times n$ matrix-valued function $S(\lambda)$,
defined for $\lambda\in \mathbb{R}^*$, seen as the portion of the
spectral function associated with the continuous spectrum
satisfying some regularity conditions. The goal of this study is
to construct a function  $K(t,u)$ which allows  to deduce, for the
required operator, the potential $Q$ as well as an associated
eigenfunction and to show some of their classical properties. The
key  of this problem is the resolution of the Gelfund-Levitan
equation \eqref{e6}, where $\Omega(t,u)$ is given, for $t,u>0$, by
\begin{equation}
\Omega(t,u)=\int_0^\infty\mathcal{J}_\alpha(t,\lambda)
(S-S_0)(\lambda)\mathcal{J}_\alpha(u,\lambda) d\lambda
+\sum_{j=1}^m\mathcal{J}_\alpha(t, \lambda_j)C_j\mathcal{J}
_\alpha(u,\lambda_j).\label{e8}
\end{equation}


 \subsection*{Notation and hypotheses}
 First we suppose obviously  that
$C_j$ and $S(\lambda)$ induce a tempered measure where especially
we have
\begin{equation}
S(\lambda)=S^*(\lambda), \quad \lambda>0\,.\label{e9}
\end{equation}
Then further notation and hypothesis are needed.
 \subsection*{Notation}
 Under the assumption \eqref{e9} a Hilbert space should be constructed.
$$ L^2_s=\bigr\{ f:]0,\infty[\rightarrow\mathbb{C}^n :
\|f\|_s^2=\int_0^\infty f^*(\lambda)S(\lambda)f(\lambda)
d\lambda<+\infty\bigr\}. $$ We set also, for $t>0$ and $u>0$,
\begin{equation}
\Omega^1(t,u)=t^{\alpha+I/2}\Omega(t,u)u^{-\alpha-I/2} \label{e10}
\end{equation} This function used  in \eqref{e6} yields  the
equation
\begin{equation}
K^1(t,u)+\Omega^1(t,u)+\int_0^tK^1(t,s)\Omega^1(s,u)ds=0.\label{e11}
\end{equation} where
\begin{equation}
 K^1(t,u)= t^{\alpha+I/2}K(t,u)u^{-\alpha-I/2}~\label{e12}
\end{equation} For any $n\times n$ matrix $A$, we denote $$
\|A\|=\max_j\sum_k|A_{jk}| $$ recall that for a such norm we have
$\|AB\|\leq \|A\|.\|B\|$.

\subsection*{Hypotheses}
 For simplicity of computations, we assume the given function
$S(\lambda)$ sufficiently regular such that $\Omega (t,u)$ is well
defined. Then further hypothesis  built from the results obtained
in \cite{m2} are assumed. Thus, for $t>0$, we suppose that:
\begin{itemize}
\item[(A0)] The function $u\to\Omega(t,u)$ is of class
${\it C}^2$ on $]0,\infty[$.

\item[(A1)] (i) $\lim_{u\to 0^+}\Omega(t,u)=0$, \quad
(ii) $\lim_{u\to 0^+}\Omega_u(t,u)=0$

\item[(A2)]  $(L_\alpha)_t\Omega (t,u)=\bigr[(L_\alpha)_u\Omega
^*(t,u)\bigr]^*$, $u>0$.
\end{itemize}
 We assume also that, for any real $R>0$ and for
$k=0,1$, there exist  functions $F_k^{^R}$, measurable and bounded
on $(0,R)$, such that:
\begin{itemize}

\item[(B0)]  $\sup_{0<s,\,u\leq t}|[\Omega^1_u(s,u)]_{ij}|\leq F^{^R}_0(t)$,
$1\leq i\leq j\leq n$

\item[(B1)] $\sup_{0<s,u\leq t}|s^{2(\alpha_j-\alpha_i)}
[\Omega^1_u(s,u)]_{ij}|\leq F^{^R}_1(t)$, $1\leq j\leq i\leq n$

\item[(B2)] There exists a function $F_2^{^R}$, integrable on $(0,R)$,
 such that
$$
\sup_{0<s,u\leq t}\|\Omega^1_{uu}(s,u) \|\leq F^{^R}_2(t),
$$
\end{itemize}

\begin{remark} \label{rmk2.1}
(i) The hypothesis above are coherent with
the results of \cite{m2}.

\noindent(ii) The function $\Omega(t,u)$ satisfies the enumerated
hypothesis in the case where there exists a some small real
$\delta>0$ such that
$$
\lambda^{-\alpha}\Bigr(S(\lambda)-S_0(\lambda)\Bigr)
\lambda^{-\alpha}={O(1)\over\lambda^{2+ \delta}},\quad
(\lambda\to +\infty)
$$
\end{remark}

\begin{remark} \label{rmk2.2}
Given an operator $L_1$ with a potential $Q_1\not\equiv 0$ and
with no discrete spectrum, such that $S_1(\lambda)$ is its
spectral function. By the technics below and under conditions
analogous to those given above, it is possible to construct an
operator $L$ for a prescribed spectral function $S(\lambda)$. Both
$L$ and $L_1$ are considered in the class of singular differential
operators of type $L_\alpha+Q$.
\end{remark}

\subsection*{Further properties  of $\Omega(t,u)$}
Additional properties of  $\Omega(t,u)$ are needed
to deduce the  existence of the solution of  \eqref{e6} and its useful
properties.

\begin{remark} \label{rmk2.3} \rm
(i) The Hypothesis  (A1) implies
$\lim_{t\to0^+}\Omega(t,t)=0$.

\noindent(ii) By means of the properties \eqref{e9} of $S(\lambda)$ and
those of $C_j$, $1\leq j\leq m$, the relation \eqref{e8} yields
$\Omega(t,u)=\Omega^*(u,t)$.

\noindent (iii) The above property  shows easily that if $\Omega(t,u)$ is derivable
on $u$, then it is also derivable on $t$. Moreover, for $R>t>0$,
we have
$$
\sup_{0<s,u\leq t}\|\Omega_u(s,u)\|=\sup_{0<s,u\leq t}\|\Omega_s(s,u)\|.
$$
(iv) For $0<s,u\leq t\leq R$ and for $i\leq j$, the Assumption
(B0) implies
$$
|[\Omega^1(s,u)]_{ij}|\leq \int_0^u
|[\Omega^1_v(s,v)]_{ij}|dv\leq C(R) tF^{^R}_0(t)
$$
 and so, using (B1), we deduce that the functions $\Omega^1(t,u)$,
$\Omega(t,u)$ as well as
$$
\omega(t,u)=t^{-\alpha-I/2}\Omega(t,u)u^{\alpha+I/2}
$$
 are
bounded on $(0,R)\times(0,R)$.
\end{remark}

Remark \ref{rmk2.3} (ii)  enables us  to manipulate
\eqref{e6} as a Fredholm's equation associated with the countable Hilbert
space $E=L^2\bigr((0,R),M_n(\mathbb{C})\bigr)$. This space is supplied
with the norm $\|.\|_2$, associated with the scalar product
$$
\langle f,g \rangle =\sum_{j=1}^n \int_0^R g^*_j(s)f_j(s)ds.
$$
where $f_j$ are the columns vectors of the $n\times n$ matrix-valued
function $f$. Let $E_1=L^2\Bigr((0,R),\mathbb{C}^n\Bigr)$ equipped with its
usual scalar product  and let $b>0$. Then, for $f\in E_1$, we set
$$
 L(f)(u)=\int_0^b \Omega(u,s)f(s)ds,\quad 0<u\leq b.
$$
For further use, we recall the following results.

\begin{lemma} \label{lem2.1}
Under the Hypothesis (A0), (A1), (B0), (B1), the operator $L$ defined
above is compact and self-adjoint on the Hilbert space $E_1$.
\end{lemma}

\begin{proof} By the Remark \ref{rmk2.3} (ii), we see easily that
$L$ is self-adjoint in the countable space $E_1$. Therefore since
by the Remark \ref{rmk2.3} (vi), we have
$$
\int_{(0,b)\times(0,b)}\|\Omega(t,u)\|^2dtdu<+\infty,
$$
so the components of $L(f)$  given by
$$
[L(f)]_i(u)=\sum_{k=1}^n\int_0^b\Omega_{ik}(u,s)f_k(s)ds,~~~~~~~1\leq
i\leq n
$$
  are compact on $E_1$ and so is  $L$. This allows to
deduce that it's possible to construct a Hilbert basis
$\varphi_j$, $j=1,2,\dots $, of $E_1$ which are eigenfunctions of $L$
whose eingenvalues denoted $\lambda_j$ are real. Furthermore, for
$u\in(0,b)$ and $f\in E_1$,
$$
L(f)(u)=\sum_{j=1}^{+\infty}\lambda_j<f,\varphi_j>\varphi_j(u)
$$
where the previous series converges uniformly on $(0,b)$.
\end{proof}

\section{Existence and differentiability of $K(t,u)$}

 The main objective of this section is the
resolution of the Gelfund-Levitan  equation \eqref{e6} associated with
$\Omega(t,u)$. Thus by mean of the general theory of compact
self-adjoint operators (see \cite{l1,y1}) and the Lemma \ref{lem2.1}, we
conclude that, for a fixed $t>0$, \eqref{e6} is with respect to $K(t,u)$
of Fredholm's.

\subsection*{Existence of $K(t,u)$}

\begin{lemma} \label{lem3.1}
Let $t_0>0$ be fixed. Then if the rows of $\Omega(t,u)$ satisfy
the conditions of the Lemma \ref{lem2.1}, for $0<u\leq t_0$, the only solution of
\begin{equation}
h_0(u)+\int_0^{t_0}h_0(s)\Omega(s,u)ds=0\label{e13}
\end{equation}
 in $L^2(0,t_0)$ is the trivial solution.
\end{lemma}

\begin{proof}
To solve this problem ,we shall use properties of Fourier-Bessel
transforms. We assume  that the \eqref{e13} has a solution $h_0$
in $L^2(0,t_0)$ and we denote $$ h(u)=\begin{cases} h_0(u) &
\text{for } u\leq t_0 \\ 0 &\text{for }u>t_0
\end{cases}
$$
By construction this function is square
integrable on $(0,\infty)$. Multiplying the equation \eqref{e13} by
$h^*(u)$ at right and  integrating with respect to $u$ then
substituting  to $\Omega(s,u)$ its expression given by \eqref{e8}, we
obtain
\begin{align*}
\int_0^\infty h(u)h^*(u) du
+\sum_{j=1}^m\Bigr( \int_0^\infty\mathcal{
J}_\alpha(u,\lambda_j)h^*(u)du\Bigr)^*C_j\Bigr(\int_0^\infty
\mathcal{J}_\alpha(u,\lambda_j)h^*(u)du\Bigr)&\\
+\int_0^\infty\Bigr(\int_0^\infty
\mathcal{J}_\alpha(u,\lambda)h^*(u)du\Bigr)^*
(S(\lambda)-S_0(\lambda))\Bigr( \int_0^\infty
\mathcal{J}_\alpha(u,\lambda)h^*(u)du\Bigr)d\lambda&=0.
\end{align*}
By  the Plancherel's Formula for $L_\alpha$, this may be simplified to
\begin{align*}
\int_0^\infty\Bigr(\int_0^\infty \mathcal{
J}_\alpha(u,\lambda)h^*(u)du\Bigr)^* S(\lambda)\Bigr(
\int_0^\infty\mathcal{J}_\alpha(u,\lambda)h^*(u)du\Bigr)d\lambda &\\
+\sum_{j=1}^m\Bigr(\int_0^\infty\mathcal{
J}_\alpha(u,\lambda_j)h^*(u)du\Bigr)^*C_j\Bigr(\int_0^\infty \mathcal{
J} _\alpha(u,\lambda_j)h^*(u)du\Bigr)&=0.
\end{align*}
The hypothesis on $S(\lambda)$ and $C_j$ yield  that the latest
expression  can be seen as a scalar product of
$\mathcal{F}_\alpha\bigr(u^{-\alpha-I/2}h^*(u)\bigr)$ by it self in the
Hilbert space $L^2_S\bigoplus (\mathbb{C}^n)^m$ (see \cite{d1,m1}) and
so it vanishes. Since $\mathcal{F}_\alpha$ is a bijection on the
space $L^2_G$ where, for $t>0$, $G(t)=t^{2\alpha+I}$. The proof
complete.
\end{proof}

\begin{theorem} \label{thm1}
 Let   $R>0$ and let  $t\in]0,R]$ be fixed, then   under the
hypothesis of the Lemma \ref{lem2.1}, the Gelfund-Levitan equation \eqref{e6}
has  a unique solution square integrable   on $(0,t)$.
Furthermore, there exists  a measurable function $\mu_1(t)$,
bounded  on ($0,R$), such that
$$
\|K(t,u)\|\leq c(R)\mu_1(t).
$$
\end{theorem}

\begin{proof} To have the unicity of the solution $K(t,u)$ of the
equation \eqref{e6}, it suffices to recall that, by mean of the
Lemma \ref{lem3.1}, the associated homogeneous equation has for any fixed $t>0$,
a trivial solution, square integrable on $(0,t)$. To prove its
existence, we construct first by mean of \eqref{e6} and the
Remark \ref{rmk2.3}
(ii)  a new Gelfund-Levitan equation, given by
\begin{equation}
K^*(t,u)+\Omega(u,t)+\int_0^t\Omega(u,s)K^*(t,s)ds=0.\label{e14}
\end{equation}
The Lemma \ref{lem2.1} says that, for $0\leq u\leq t$, each column of \eqref{e14}
is of Fredholm's, explicitly given by
$$
[K^*(t,u)]_k+[\Omega(u,t)]_k+\int_0^t\Omega(u,s)[K^*(t,s)]_kds=0,\quad
1\leq k\leq n.
$$
 Then using results  of Lemma \ref{lem2.1}, we deduce that
their solutions in $E_1$ exist and that in the case where $(-1)$
is not an eigenvalue, we have
\begin{equation}
[K^*(t,u)]_k=-[\Omega(u,t)]_k-\sum_{j=1}^\infty
{\lambda_j(t)\over
\lambda_j(t)+1}\langle [\Omega(u,.)]_k,\varphi_j(t,.)\rangle
\varphi_j(t,u).\label{e15}
\end{equation}
where  $\varphi_j$ is  a particular eigenfunction associated with
the eigenvalue  $\lambda_j,~j=1,2,\dots $ defined in the previous
Lemma. We recall that
$$
\langle [\Omega(.,u)]_k,\varphi_j(t,.)\rangle
=\int_0^t\varphi_j^*(t,s)[\Omega(s,u)]_k\,ds.
$$
and that the series in  expression  \eqref{e15} converges uniformly on
$[0,t]$. Estimates on  the solution are obtained by use of this
relation, the Remark \ref{rmk2.3} (iv)  and Cauchy-Schawrz's inequality.
Furthermore the unicity of the solution deduced from the Lemma \ref{lem3.1}
yields  inevitably that $(-1)$ is not an eigenvalue of the
operator in question and so the results above are sufficient to
conclude. \end{proof}


\begin{corollary} \label{coro1}
Under assumptions of the Lemma \ref{lem2.1}, we have
$$
\lim_{t\to 0^+}K(t,t)=0.
$$
\end{corollary}

For the proof of the above corollary, we use the previous proposition,
 Remark \ref{rmk2.3} (i), and  \eqref{e6}.

 \begin{proposition} \label{prop3.1}
 Under assumptions of the Lemma \ref{lem2.1},  the function $K^1(t,u)$ given
by \eqref{e12} is  bounded on $0< u\leq t\leq R$.
\end{proposition}

\begin{proof}  The  Theorem \ref{thm1} and the relation \eqref{e12} yield
that the function $K^1(t,u)$  is well defined and that it's a
solution of the equation \eqref{e11}, moreover for a fixed
$\epsilon>0$, it's bounded on $\epsilon <u\leq t\leq R$.
Since the kernel $\Omega^1(t,u)$ is not hermitian, we can not use
the technics of the previous proposition to  study the behavior
 of this solution elsewhere. At zero and since $\Omega^1(t,u)$
is square integrable on $(0,R)\times(0,R)$, there exists
an $\epsilon>0$ such that
\begin{equation}
\int_0^\epsilon\int_0^\epsilon\|\Omega^1(t,u)\|^2dtdu<1.\label{e16}
\end{equation} Thus for a fixed $t=b\leq\epsilon$ and for $0<u\leq
b$, we set $K^1(b,u)=\varphi(u)$ and $\Omega^1(b,u)=-f(u)$, the
equation \eqref{e11} becomes
\begin{equation} \varphi(u)
+\int_0^b\varphi(s)\Omega^1(s,u)ds=f(u).\label{e17}
\end{equation}
By analogy with the scalar case (see  \cite[p.
121]{y1}), we solve this equation by means of the resolvent
method.  For this we set
\begin{equation}
\begin{gathered}
\gamma_1(t,u)=-\Omega^1(t,u),\\
\gamma_n(t,u)=-\int_0^b\gamma_{n-1}(t,s)\Omega^1(s,u)ds,\quad
n\geq 2
\end{gathered}\label{e18}
\end{equation}
 and we show recursively that
\begin{equation}
\gamma_{n+m}(t,u)=\int_0^b\gamma_{n}(t,s)\gamma_m(s,u)ds,
\quad n,m\geq 1.\label{e19}
\end{equation}
The relation \eqref{e18} and  Cauchy's inequality show that,
for $0<t$, $u\leq b$ and  by iteration, we have
$$
\int_0^b\int_0^b\|\gamma_n(s,u)\|^2dsdu\leq
\Bigr[\int_0^b\int_0^b\|\Omega^1(s,u)\|^2 dsdu\Bigr]^n.
$$
 Therefore, for $n\geq 3$, we deduce
that
$$
\|\gamma_n(t,u)\|^2\leq\Bigr[
\int_0^b\int_0^b\|\gamma_{n-2}(s,u)\|^2 ds\,du\Bigr]
\int_0^b \int_0^b\|\Omega^1(t,s)\Omega^1(r,u)\|^2 ds\,dr.
$$
Accordingly, it follows that
\begin{align*}
&\|\gamma_n(t,u)\|^2\\
&\leq\Bigr[\int_0^b\int_0^b\|\Omega^1(s,u)\|^2dsdu\Bigr]^{n-2}
\biggr\{\int_0^b\|\Omega^1(t,s)\|^2ds
\int_0^b\|\Omega^1(r,u)\|^2dr\biggr\}.
\end{align*}
The term between braces is bounded  by \eqref{e16}, the series
$\Gamma(t,u)=\sum_{n\geq 1}\gamma_n(t,u)$ converges uniformly on
the domain in question. Using \eqref{e19}, we deduce that $$
\Gamma(t,u)=-\Omega^1(t,u)-\int_0^b\Gamma(t,r)\Omega^1(r,u)dr $$
and so that the solution of \eqref{e17} is given by $$
\varphi(u)=f(u)+\int_0^bf(r)\Gamma(r,u)dr. $$ Estimates on
$\Gamma(t,u)$ and properties  of $\Omega^1(t,u)$ show that the
solution $K^1(t,u)$ of \eqref{e11} behaves regularly at zero. For
the case where $u\to 0^+$, and where  $t$ is sufficiently large so
that the assumption \eqref{e16} is not satisfied,  we use further
results on Fredholm's equations  which we do not detail here.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
By analogous computations to those done in the Proposition \ref{prop3.1},
we show that the function
$k(t,u)=t^{-\alpha-I/2}K(t,u)u^{\alpha+I/2}$ is bounded on the
domain $0<u\leq t\leq R$.
\end{remark}

\subsection*{Differentiability of $K(t,u)$}

\begin{lemma} \label{lem3.2}
Under the Hypothesis (A0), (B0), (B1), (B2), the function
$u\mapsto K(t,u)$ is of class $C^2$ on  $]0, t]$. By
differentiation of \eqref{e6} with respect to $u$, integral
equations associated with $K_u$ and $K_{uu}$ are determined; i.e.,
\begin{gather}
K_u(t,u)=-\Omega_u(t,u)- \int_0^tK(t,s)\Omega_u(s,u)ds  \label{lem3.2i}\\
K_{uu}(t,u)=-\Omega_{uu}(t,u)-\int_0^tK(t,s)\Omega_{uu}(s,u)ds. \label{lem3.2ii}
\end{gather}
For which we have the estimates
\begin{gather*}
\|{ K_u} (t,u)\|\leq c_1(R)\theta_1(t) \\
\|K^1_{uu}(t,u)\|\leq c_2(R)\theta_2(t)
\end{gather*}
where the $\theta_k$, $k=1,2$ are
integrable functions on $(0,R)$.
\end{lemma}

\begin{proof}  The continuity of $K(t,u)$, with respect to
$u$, is obtained by means of the properties of its expression \eqref{e15}.
To have \eqref{lem3.2i}, we use results of Theorem \ref{thm1} which yield that, for a
fixed $t$ such that $R\geq t>0$, the function
$u\mapsto K(t,s)\Omega(s,u)$, $0<s,u\leq t,$ satisfies the Derivation
Theorem hypothesis at the first order. Furthermore,
 the relation \eqref{lem3.2i} and the hypothesis gives the estimate on
$K_u$. Analogous equation to \eqref{lem3.2i} is obtained by means of
\eqref{e10}-\eqref{e12};
we have
$$
 K^1_u(t,u)=-\Omega^1_u(t,u)-\int_0^tK^1(t,s)\Omega^1_u(s,u)ds
$$ The  process above and the results of the Proposition
\ref{prop3.1}, applied to the previous equation, are then used to
obtain an equation on $K^1_{uu}(t,u)$, analogous to
\eqref{lem3.2ii}, estimates about are then deduced.
 \end{proof}

\begin{lemma} \label{lem3.3}
Under the assumptions of Lemma \ref{lem3.2} and for  $u>0$, the
function $t\mapsto K(t,u)$ is of class $C^2$ on $[u,R]$ and by
differentiation of \eqref{e6} with respect to $t$, we obtain :
\begin{gather}
K_t(t,u)=-\Omega_t(t,u)-K(t,t)\Omega(t,u)-\int_0^tK_t(t,s)
\Omega(s,u)ds, \label{lem3.3i}\\
\begin{aligned}
K_{tt}(t,u)&=-\Omega_{tt}(t,u)-\big[{d\over
dt}K(t,t)\big]\Omega(t,u)-K(t,t)\Omega_t(t,u)\\
&\quad -K_t(t,t)\Omega(t,u)-\int_0^tK_{tt}(t,s)\Omega(s,u)ds.
\end{aligned} \label{lem3.3ii}
\end{gather}
Furthermore there exist  functions $\nu_1$ and $\nu_2$,
integrable on $(0,R)$, such that:
$$
\|K_t(t,u)\|\leq\nu_1(t) \quad\text{and}\quad
\|K^1_{tt}(t,u)\|\leq\nu_2(t)\,.
$$
\end{lemma}

\begin{proof}
 For $t\geq u>0$ some fixed parameters and for $h$
sufficiently small, we consider the difference quantity
${\delta^t_hK(t,u)}= {K(t+h,u)-K(t,u)}$. Used in \eqref{e6}
it yields
$$
\delta^t_hK(t,u)+\int_0^t \delta^t_h K(t,s)\Omega(s,u)ds=-
\delta^t_h\Omega(t,u)-\int_t^{t+h}{K(t+h,s)}\Omega(s,u)ds.
$$
 We obtain hence a  Fredholm's equation with a second member
uniformly estimated in $h$, vanishing  when $h\to 0$. The technics
of Theorem \ref{thm1} allow to have the same behavior for
$\delta^t_hK(t,u)$ and so the continuity of $t\to K(t,u)$ is
deduced. Its twice differentiability will be proved  by similar
arguments. Indeed for the first derivatives, the difference
quotient ${\Delta^t_hK(t,u)}= (\delta^t_hK(t,u)/h)$
 and \eqref{e6} again give the relation
$$
{\Delta^t_hK(t,u)}+\int_0^t{\Delta^t_hK(t,s)} \Omega(s,u)ds
=-{\Delta^t_h\Omega(t,u)}
-\int_t^{t+h}{K(t+h,s)\over h}\Omega(s,u)ds.
$$
 Then since the free term
${\Delta^t_h\Omega(t,u)}+\int_t^{t+h}{K(t+h,s)\over h}\Omega(s,u)ds$
 is also  estimated uniformly in $h$  because of
the differentiability of $\Omega(t,u)$ with respect to $t$ and
since, the last equation  is of Fredholm's kind, we can so
estimate ${\Delta^t_hK(t,u)}$. As $h\to 0$, the result
\eqref{lem3.3i} follows and estimates on $K_t(t,u)$  are obtained.
An analogous equation to \eqref{lem3.3i} is deduced for
$K^1_t(t,u)$ for which we apply the above process. Finally a
similar result to \eqref{lem3.3ii} is deduced for $K^1_{tt}(t,u)$.
\end{proof}

\subsection*{Further properties of  $K(t,u)$}
 Lemmas \ref{lem3.2} and \ref{lem3.3}, imply  that the function
$K(t,t)$ is differentiable for $t>0$; therefore,  we  can set
\begin{equation}
Q(t)=-2{d\over dt}K(t,t),\quad t>0.\label{e20}
\end{equation}


\begin{remark} \label{rmk3.2} \rm
For  a class $U$, of $\mathcal{C}^2$ function defined on $]0,+\infty[$,
let
\begin{gather*}
 {\Delta}_\alpha U=U_{tt}+{2\alpha+I\over t}U_t,\\
 \tilde{\Delta}_\alpha U=U_{tt}-{2\alpha+I\over t}U_t
+{2\alpha+I\over t^2}U\,.
\end{gather*}
Then simple  computations yield
\begin{gather*}
(L_\alpha)_t\Omega(t,u)=t^{-\alpha-I/2}\big[(\tilde{\Delta}_\alpha
 )_t\Omega^1(t,u)\big]u^{\alpha+I/2},\\
\big[(L_\alpha)_u\Omega^*(t,u)\big]^*
=  t^{-\alpha-I/2}\big[(\Delta_\alpha)_u(\Omega^1)^*(t,u)\big]^*
u^{\alpha+I/2}.
\end{gather*}
\end{remark}

\begin{proposition} \label{prop3.2}
 Under the hypotheses (A0), (A1), (A2), (B0), (B1), (B2),
the function $K(t,u)$ satisfies the following two assertions:
\begin{gather}
\lim_{u\to 0^+}K(t,u) = \lim_{u\to 0^+}K_u(t,u)=0,\label{prop3.2i}\\
(L_\alpha+Q)_tK(t,u)=\big[(L_\alpha)_uK^*(t,u)\big]^*. \label{prop3.2ii}
\end{gather}
\end{proposition}

\begin{proof} The hypothesis  (A1)(i),  the relation \eqref{e6}
as well as  the Remark \ref{rmk2.3} (iv) and the Theorem \ref{thm1} imply
 that $K(t,u)$ vanish as $u\to 0^+$.
The same arguments and \eqref{lem3.2i}  complete the
proof of the first assertion.
To have \eqref{prop3.2ii} we show  that for a
fixed $\epsilon>0$, (A2) and integrations by parts yield
\begin{align*}
&t^{-\alpha-I/2}\Big(\int_\epsilon^t\big[(\tilde\Delta_\alpha)_s(K^1)^*(t,s)
\big]^*\Omega^1(s,u)ds\Big)u^{\alpha+I/2}\\
&= -K(t,t)\Omega_t(t,u)+K_u(t,t)\Omega(t,u)+K(t,\epsilon)\Omega_t(\epsilon,u)
-K_u(t,\epsilon) \Omega(\epsilon,u)\\
&\quad +t^{-\alpha-I/2}\Big(
\int_\epsilon^t K^1(t,s)\big[(\Delta_\alpha)_u(\Omega^1)^*
(s,u)\big]^*ds\Big)u^{\alpha+I/2} .
\end{align*}
Thanks to (A1) and  \eqref{prop3.2i}, the second member of the last
identity converges as $\epsilon \to 0^+$, so we have
\begin{equation}
\begin{aligned}
&\int_0^tK(t,s)\big[(L_\alpha)_u\Omega^*(s,u)\big]^*ds\\
&= \int_0^t\big[(L_\alpha)_sK^* (t,s)\big]^*\Omega(s,u)ds
  +K(t,t)\Omega_t(t,u)-K_u(t,t)\Omega(t,u).
\end{aligned}\label{e21}
\end{equation}
Then  Lemmas \ref{lem3.2}, \ref{lem3.3} and the relation \eqref{e6}, yield
\begin{align*}
& (L_\alpha+Q)_tK(t,u)-\big[(L_\alpha)_uK^*(t,u)\big]^*\\
&=-(L_\alpha)_t\Omega(t,u)+\big[(L_\alpha)_u\Omega^*(t,u)\big]^*\\
&-\int_0^t\Big\{(L_\alpha+Q)_tK(t,s)
\Omega(s,u)-K(t,s)\big[(L_\alpha)_u\Omega^*(s,u)\big]^*\Big\}ds\\
&\quad  -\Bigr[{d\over dt}K(t,t)+K_t(t,t)+Q(t)\Bigr]\Omega(t,u)
- K(t,t)\Omega_t(t,u).
\end{align*}
Finally the condition (A2), the relations \eqref{e20} and \eqref{e21}
assert that
\begin{align*}
\Bigr[(L_\alpha
+Q)_tK(t,u)-\bigr[(L_\alpha)_uK^*(t,u)\bigr]^*\Bigr]&\\
+\int_0^t\Bigr[(L_\alpha +Q)_tK(t,s)-\bigr[(L_\alpha)_sK^*(t,s)\bigr]^*\Bigr]
\Omega(s,u)ds&=0\,.
\end{align*}
Remark \ref{rmk3.2} yields
\begin{align*}
\Bigr[(\tilde\Delta_\alpha
+Q^1)_tK^1(t,u)-\bigr[(\Delta_\alpha)_u(K^1)^*(t,u)\bigr]^*\Bigr]
u^{\alpha+I/2}&\\
+\int_0^t\Bigr[(\tilde\Delta_\alpha
+Q^1)_tK^1(t,s)-\bigr[(\Delta_\alpha)_s(K^1)^*(t,s)\bigr]^*\Bigr]
s^{\alpha+I/2}\Omega(s,u)ds&=0
\end{align*}
where
$Q^1(t)=t^{\alpha+I/2}Q(t)t^{-\alpha-I/2}$, $t>0$. Since
$\alpha_1>1,$  the properties of  the kernel $\Omega(t,u)$ and
those of the solution $K^1(t,u)$, obtained in the
Lemmas \ref{lem3.2}  and \ref{lem3.3} show that the mapping
$$
s\mapsto \Bigr[(\tilde\Delta_\alpha
+Q^1)_tK^1(t,s)-\bigr[(\Delta_\alpha)_s(K^1)^*(t,s)\bigr]^*\Bigr]
s^{\alpha+I/2}
$$
is in $L^2(0,t)$, then by the Lemma \ref{lem3.1} the proof is complete.
\end{proof}

\section{Derivation of the differential operator}

 For $t>0$ and $\lambda\in\mathbb{C}$, we set
\begin{equation}
\Phi(t,\lambda)=\mathcal{J}
_\alpha(t,\lambda)+\int_0^tK(t,u)\mathcal{J}_\alpha(u,
\lambda)du,\label{e22}
\end{equation}
 where $\mathcal{J}_\alpha$ is given  by \eqref{e3}
and $K(t,u)$ is the solution of Fredholm's equation \eqref{e6}. In this
section we plan to show important properties of $\Phi(t,\lambda)$.
First we remark
that the regularity of
$K(t,u)$ yields that this function  is well defined. Then, by the
Remark \ref{rmk3.1} and the relation \eqref{e22} we deduce that
\begin{equation}
t^{-\alpha-I/2}\Phi(t,\lambda)=t^{-\alpha-I/2}\mathcal{J}
_\alpha(t,\lambda)+\int_0^tk(t,u)u^{-\alpha-I/2}\mathcal{J}_\alpha(u,
\lambda)du.\label{e23}
\end{equation}
 We have so, for the potential $Q$ defined
by \eqref{e20}, the results below.


\begin{theorem} \label{thm2}
 For $\lambda\in\mathbb{C}$ and under the hypothesis
 (A0), (A1), (A2), (B0), (B1), (B2),
the function $\Phi(.,\lambda)$  is, on $]0,\infty[$,  the solution of the
singular second order differential equation with matrix coefficients
 given by
$$
U''+ {I/4-\alpha^2\over
t^2}U+Q(t)U=-\lambda^2U
$$
 such that
$$
\lim_{t\to 0^+}
t^{-\alpha-I/2}\Phi(t,\lambda)=I.
$$
\end{theorem}

\begin{proof}  The second derivatives of $K(t,u)$ with respect to $t$
obtained in Lemma \ref{lem3.3} and its estimates imply  that, for a fixed
$\lambda\in\mathbb{C}$, the mapping $t\mapsto \Phi(t,\lambda)$ is twice
differentiable on $]0,+\infty[ $. By the expressions of these
derivatives and since $\mathcal{{J}}_\alpha(\lambda ,.)$ is an
eigenfunction of the operator $L_\alpha$ associated with the
eigenvalue $-\lambda^2$. Then justified integrations by parts and
technics, used in the proof of the Proposition \ref{prop3.2}, show that
\begin{align*}
& \big[L_\alpha +Q+\lambda ^2I\big]\Phi (t,\lambda )\\
&= \int_0^t\Bigr[(L_\alpha
+Q)_tK(t,u)-\bigr \{(L_\alpha)_uK^*(t,u)\bigr \}^*\Bigr]
\mathcal{{J}}_\alpha(\lambda ,u)du\\
&\quad +\Bigr[{d\over dt}K(t,t)+\Bigr( {\partial\over
\partial t}K(t,u)\Bigr)_{u=t}+\Bigr({\partial\over
\partial u}K(t,u)\Bigr)_{u=t}+Q(t)\Bigr]\mathcal{{J}}_\alpha(\lambda ,t)\\
&\quad  +\lim_{u\to 0^+}\Bigr[K(t,u){\partial\over \partial u} \mathcal{{J}}_\alpha(\lambda
,u)-K_u(t,u) \mathcal{{J}}_\alpha(\lambda ,u)\Bigr].
\end{align*}
Proposition \ref{prop3.2} again and  the relation \eqref{e20}  allow us to deduce that
the last expression vanishes. Then  relations \eqref{e5} and \eqref{e23} as
well as the Remark \ref{rmk3.1} show
 that  $\lim_{t\to 0^+}t^{-\alpha-I/2}\Phi(t,\lambda)=I$.
 \end{proof}

 \begin{corollary} \label{coro2}
Under the hypothesis of the Theorem \ref{thm2}, the mapping
$\lambda\mapsto \Phi(t,\lambda)$ is even and analytic on
$\mathbb{C}$.
\end{corollary}

\begin{proof} For this we recall only that the mapping
$\lambda\mapsto\mathcal{J}_\alpha(t,\lambda)$ is even and analytic
on $\mathbb{C}$. Then by the relation \eqref{e23} and the
properties of $k(t,u)$ given in the Remark \ref{rmk3.1}, the
result is easily deduced.
\end{proof}

\section{Properties of the potential  $Q$}

We look in this section for common  properties
of the potential $Q$. We recall for example that from the
properties of $K(t,u)$, the function $Q(t)$, $t>0$, is well
defined by the relation \eqref{e20}. Furthermore, by means of  the
Corollary \ref{coro1}, we can set
$$
K(t,t)=-{1\over 2}\int_0^tQ(s)ds,\quad t>0
$$
so the locally integrability of
$Q(t)$  is simply deduced.  Next we will look
 for its further classical properties as symmetry and integrability at infinity.


\subsection*{Symmetry  of $Q(t)$}

\begin{theorem} \label{thm3}
 Under the hypothesis (B1), (B2),  and since $S(\lambda)$, $\lambda>0$, is an
hermitian matrix-valued function, then so is the  potential
$Q(t)$, for $t>0$.
\end{theorem}

\begin{proof}
The Gelfand-Levitan equation $\eqref{e6}$ and the
Remark \ref{rmk2.3} (ii), yield that the both  relations below hold
\begin{gather*}
\Omega(t,t)+K(t,t)+\int_0^tK(t,s)\Omega(s,t)ds=0,\\
\Omega(t,t)+K^*(t,t)+\int_0^t\Omega(t,s)K^*(t,s)ds=0.
\end{gather*}
 To have $Q^*=Q$ it is sufficient  to show that the integrals
in the two preceding expressions are equal. Now by the same
arguments as before, we have
$$
\Omega(u,t)=\begin{cases}
-K^*(t,u)-\int_0^t\Omega(u,s)K^*(t,s)ds, & t\geq u>0\\
-K(u,t)-\int_0^uK(u,s)\Omega(s,t)ds, & u\geq t>0.
\end{cases}
$$
 Therefore, we deduce that
$$
\int_0^t\Omega(t,s)K^*(t,s)ds=
-\int_0^tK(t,s)K^*(t,s)ds-\int_0^t\int_0^tK(t,s)\Omega(s,u)K^*(t,u)dsdu
$$
and that also
$$
 \int_0^tK(t,s)\Omega(s,t)ds=
-\int_0^tK(t,s)K^*(t,s)ds-\int_0^t\int_0^tK(t,s)\Omega(s,u)K^*(t,u)
dsdu.
$$
 This suffices to prove the required result.
\end{proof}

\begin{remark} \label{rmk5.1}
 In order that the potential $Q(t)$, $t>0$, to be real, it suffices
to assume that the matrix-valued function $S(\lambda)$,
$\lambda\in\mathbb{R}^*$,
and the matrices $C_j$, $1\leq j\leq m$, are so.
\end{remark}

\subsection*{Behavior of $Q(t)$}

 Recall that ${d\over dt}\Omega(t,t)$ is a well defined function,
on the positive half axis. In the following, the behavior of
$Q(t)$ at zero and at infinity will be studied by mean of
its relation with this function. Because of all the difficulties
mentioned in the introduction, the relation between the  both as well as the
behavior of the latest at infinity will be obtained under strong
conditions some of them are satisfied in the regular case
(see \cite{a1}). In this aim we introduce the following assumptions.
\begin{itemize}
\item[(H1)]  For a fixed $R>0$, there exists a function
$G$ which is integrable on $]0,2R[$ and such that
$\|\Omega_u(t,u)\|\leq G(t+u)$.

\item[(H2)] Moreover, we suppose that
$\int_0^{2R} s G(s)ds<1$.
\end{itemize}

\begin{remark} \label{rmk5.2} \rm
 By  Remark \ref{rmk2.2}, we deduce that the
second assumption is not as restrictive as it appears.
\end{remark}

For the following results, we denote
\begin{equation}
\sigma(t)=\int_t^{2t}G(s)ds,\quad
\sigma_1(t)=\int_0^tsG(s)ds,\quad
\tilde\sigma_1(t)=[1-\sigma_1(t)]^{-1} \label{e24}
\end{equation}


 \begin{lemma} \label{lem5.1}
 For $0<u,t\leq R$ and under the hypothesis of
 the Lemma \ref{lem3.2}, (H1) and (H2),  we have
\begin{equation}
\|\Omega(t,u)\|\leq \int_{t\vee u}^{t+u}G(s)ds. \label{lem5.1i}
\end{equation}
Moreover, for $0<u\leq t\leq R$, we have
\begin{gather}
\|K(t,u)\|\leq \sigma(t)\Bigr[1+\tilde\sigma_1(t)\int_u^{t+u}wG(w)dw\Bigr],
\label{lem5.1ii} \\
\|K_t(t,u)\|\leq  G(t+u)+\sigma^2(t)\delta_0(t)
+\sigma(t)\tilde\sigma_1(2t)\int_u^{t+u}G(s)ds. \label{lem5.1iii}
\end{gather}
where $t_\vee u=sup(t,u)$ and  where $\delta_0$
is a function of $\sigma_1$.
\end{lemma}

\begin{proof}
 From Remark \ref{rmk2.3} (ii) and  (H2), we deduce easily that
$\|\Omega_t(t,u)\|\leq G(t+u)$, hence the result \ref{lem5.1i} is
obtained. To obtain \ref{lem5.1ii}, we use successive
approximations on \eqref{e6}. Thus we set
\begin{gather*}
 K^{(0)}(t,u)=-\Omega(t,u), \\
 K^{(n)}(t,u)=-\int_0^tK^{(n-1)}(t,s)\Omega(s,u)ds,
\end{gather*}
 and we show recursively that
$$
\|K^{(n)}(t,u)\|\leq\sigma(t){\sigma_1^{n-1}(2t)\int_u^{t+u}wG(w)dw},
\quad n\geq  1.
$$
 This result is justified by means of \ref{lem5.1i} and  simple
permutation of integrals.
To have estimates on $K_t(t,u)$, we use the same process as above
applied to the relation (i) of the Lemma \ref{lem3.3}. We set
\begin{gather*}
K^{(0)}_t(t,u)=-\Omega_t(t,u)-K(t,t)\Omega(t,u), \\
  K^{(n)}_t(t,u)=-\int_0^tK^{(n-1)}_t(t,s)\Omega(s,u)ds
\end{gather*}
then by  (H1), the results \ref{lem5.1i} and \ref{lem5.1ii},
we have
$$
\|K^{(0)}_t(t,u)\|\leq G(t+u)+\sigma^2(t)\Bigr[1+\tilde
 \sigma_1(2t)\int_t^{2t}wG(w)dw\Bigr]
$$
and recursively  again this yields that, for $n\geq 1$,
\begin{align*}
&\|K^{(n)}_t(t,u)\|\\
&\leq\sigma(t)\sigma_1^{n-1}(2t)\int_u^{t+u}G(w)dw+
\sigma^2(t)\Bigr(1+\tilde\sigma_1(2t)\sigma_1(2t)\Bigr)
\sigma^{n}_1(2t)\int_u^{t+u}wG(w)dw,
\end{align*}
 so the last estimate follows.
\end{proof}

We have then  the following useful results.

\begin{corollary} \label{coro5.1}
Under the hypothesis of Lemma \ref{lem5.1}, we have
\begin{gather*}
\int_0^t\|K(t,s)\Omega_u(s,t)\|ds
\leq \sigma^2(t)\Bigr[1+\tilde\sigma_1(2t)\sigma_1(2t)\Bigr],\\
\int_0^t\|K_t(t,s)\Omega(s,t)\|ds\leq
\sigma^2(t)\delta_1(t),
\end{gather*}
 where $\delta_1$ is a bounded
function expressed by mean of $\sigma_1$.
\end{corollary}

 \begin{theorem} \label{thm4}
For any fixed $R>0$ such that the assumptions of
Lemma \ref{lem5.1} hold, there exists a positive constant $c(R)$ such that
$$
\|2{d\over dt}\Omega(t,t)-Q(t)\|\leq
c(R)\Bigr(\int_t^{2t} G(s)ds\Bigr)^2,\quad 0<t<R.
$$
 In particular if these assumptions are satisfied for
$R=+\infty$, then
$$
\int_0^\infty (1+t)\|Q(t)\|dt<\infty\,.
$$
Moreover the function $Q(t)$  has the same asymptotic
behavior as $ 2{d\over dt}\Omega(t,t)$.
\end{theorem}

\begin{proof} By the \eqref{lem3.2i} and  \eqref{lem3.3i},
we have
\begin{align*}
&\Omega_t(t,u)+\Omega_u(t,u)+K_t(t,u)+K_u(t,u)\\
&=-K(t,t)\Omega(t,u)-\int_0^tK_t(t,s)\Omega(s,u)ds
-\int_0^tK(t,s)\Omega_u(s,u)ds.
\end{align*}
Therefore, the relation \eqref{e20} and properties
of derivatives  allow us  to have
\begin{align*}
&{d\over dt}\Omega(t,t)-{1\over 2}Q(t)\\
&=-K(t,t)\Omega(t,t)-\int_0^tK_t(t,s)\Omega(s,t)ds
-\int_0^tK(t,s)\Omega_u(s,t)ds.
\end{align*}
By this relation, Corollary \ref{coro5.1}, and since Lemma \ref{lem5.1}
implies
$$
\|K(t,t)\Omega(t,t)\|\leq
\sigma^2(t)\Bigr[1+\tilde \sigma_1(2t) \sigma_1(2t)],
$$
 we deduce easily that
\begin{equation}
\|{d\over  dt}\Omega(t,t)-{1\over 2}Q(t)\|
\leq  2\big(1+\tilde\sigma_1(2t)\sigma_1(2t)+\delta_1(t)\big)
 \sigma^2(t).\label{e25}
\end{equation}
 By noticing  that the function  $\tilde\sigma_1(2t)\sigma_1(2t)+\delta_1(t)$
 is bounded on $(0,R)$, the first assertion of the theorem is proved.
 Furthermore since
$$
t\sigma(t)\leq\int_t^{2t}sG(s)ds,
$$
 it follows that, for any
$R>0$,
$$
\int_0^R t\sigma^2(t)dt\leq
\int_0^R\Bigr(\int_t^{2t}sG(s)ds\Bigr)\sigma(t)dt\leq
\sigma_1(2R)\int_0^R\sigma(t)dt\leq \sigma_1^2(2R)<+\infty.
$$
Therefore,  if  (H2) is satisfied for $R=+\infty$, we deduce
that
$$
\int_0^\infty (1+t)\|Q(t)\|dt <\infty.
$$
By these assumptions, we can remark also that
$$
\int_t^{2t} G(s) ds =o(1)
$$
 as $t\to 0^+$, or $t\to+\infty$. Therefore,  the relation
\eqref{e25} yields that the functions $ 2{d\over dt}\Omega(t,t)$
and $Q(t)$ are equivalent in this sense and   the
proof is complete.
\end{proof}

 \section{Inverse problem and discrete spectrum}

 We consider here the simplest case where the
required operator $L$ has, associated with the continuous
spectrum, the same spectral function $S_0(\lambda)$ as $L_\alpha$.
We assume also that the discrete spectrum reduces to an only one
eigenvalue $\lambda_0=-i\mu_0,$ $\mu_0>0$ with a corresponding
normalizing factor $C_0$, which is a positive definite hermitian
constant matrix not necessary diagonal. We remark that  in this
case, for $t>0$ and $u>0$, we have
$$
\Omega(t,u)=\mathcal{Y}_\alpha^*(t)C_0\mathcal{
Y}_\alpha(t),
$$
where $\mathcal{Y}_\alpha(t) =\mathcal{J}_\alpha^*(t,-i\mu_0)$,
is the real valued function deduced from \eqref{e4}.
Our purpose in this section is to study the behavior at zero
and at infinity of the potential $\Delta Q$, associated with this
problem. We recall that  in the third section, we have shown the
existence and the unicity of a square integrable solution of \eqref{e6}.
In this special  case we will solve it rather algebraically. We
try to look for its solution in the form
$$
K(t,u)=K(t)\mathcal{Y}_\alpha(u).
$$
This allows to replace \eqref{e6} by
$$
\Bigr[K(t)+\mathcal{Y}_\alpha^*(t)C_0+K(t)\Bigr( \int_0^t\mathcal{
Y}_\alpha(s)\mathcal{Y}_\alpha^*(s)ds\Bigr)C_0\Bigr]\mathcal{
Y}_\alpha(u)=0.
$$
 The location of the zeros for the Bessel
function of the first kind yields that necessarily that
$$
K(t)\Bigr[I+R(t)C_0\Bigr]=-\mathcal{Y}_\alpha^*(t)C_0,
$$
where
\begin{equation}
R(t)=\int_0^t\mathcal{Y}_\alpha(s)\mathcal{
Y}_\alpha^*(s)ds.\label{e26}
\end{equation}
 To obtain $K(t)$, we need the following result.

 \begin{lemma} \label{lem6.1}
For a fixed $t>0$, the $n\times n$ matrix valued function $I+R(t)C_0$,
$t>0$ is positive defined and so it is invertible.
\end{lemma}

\begin{proof} For $X\in \mathbb{C}^n$ and $t>0$,  we have
$$
 X^* R(t)X=\int_0^t[\mathcal{Y}_\alpha^*(s)X]^*
[\mathcal{Y}_\alpha^*(s)X]ds\geq 0
$$
and if this quantity vanish then $X=0$. It results that for any
 $t>0$, $R(t)$ is a positive defined matrix and since $C_0$
satisfies yet this property, then $I+R(t)C_0$ is
 positive defined too and so it is invertible.
\end{proof}

 From the result above, we deduce that the
$n\times n$ matrix valued function
$$
V(t)=C_0^{-1}+R(t)
$$
 is invertible and so that  $K(t)=-\mathcal{Y}_\alpha^*(t)V^{-1}(t)$.
Consequently, for $0< u\leq t$, the function
$$
K(t,u)=-\mathcal{Y}_\alpha^*(t)V^{-1}(t)\mathcal{Y}_\alpha(u)
$$
is a solution of \eqref{e6}.
In particular, we have
\begin{equation}
\Delta Q(t)=2{d\over dt}\big[ \mathcal{Y}_\alpha^*(t)V^{-1}(t)
\mathcal{Y}_\alpha(t)\big]\label{e27}
\end{equation}
 and the relation \eqref{e22} above takes the form
\begin{equation}
\Phi(t,\lambda)=\mathcal{J}_\alpha(t,\lambda)
-\mathcal{Y}_\alpha^*(t)V^{-1}(t)\int_0^t\mathcal{Y}_\alpha(u)
\mathcal{J}_\alpha(u,\lambda)du.\label{e28}
\end{equation}
 The study of the asymptotic behavior of $\Phi(t,\lambda)$ is possible
from the estimates below, but our main interest will be the asymptotic
behavior of $\Delta Q$. The relation \eqref{e27} shows that it suffices
to have those  of $\mathcal{Y}_\alpha(t)$, $\mathcal{Y}'_\alpha(t)$ and
$V^{-1}(t)$ there. In this aim, we set
\begin{equation}
N_\alpha(t)={\Gamma(\alpha+I)\over2\sqrt\pi}
\Bigr({2\over\mu_0}\Bigr)^{\alpha+I/2}e^{\mu_0t} \label{e29}
\end{equation}
 and
$$
(\alpha,k)={1\over k!}
\big(\alpha^2-I/4\big)\dots \big(\alpha^2-I(k-1/2)^2\big),
\quad k=1,2,\dots
$$


\begin{remark} \label{rmk6.1} \rm
The asymptotic behavior of the Bessel functions (see \cite{o1,w1})
yield that as $t\to 0^+$,
\begin{gather*}
\mathcal{Y}_\alpha(t)=
t^{\alpha+I/2}\Bigr[I+(\alpha+I)^{-1}({\mu_0t\over 2})^2+O(t^4)\Bigr],\\
\mathcal{Y}_\alpha'(t)= t^{\alpha-I/2}\Bigr[(\alpha+I/2)
+(\alpha+{5I/2})(\alpha+I)^{-1}({\mu_0t\over 2})^2+O(t^4)\Bigr]\,.
\end{gather*}
As $t$ approaches infinity, we have
\begin{gather*}
\mathcal{Y}_\alpha(t)=N_\alpha(t)\Bigr[I -{(\alpha,1)\over 2\mu_0
t}+{(\alpha,2)\over (2\mu_0t)^2}-{(\alpha,3)\over (2\mu_0t)^3}
+O({1\over t^4})\Bigr], \\
\mathcal{Y}_\alpha'(t)= \mu_0N_\alpha(t)\Bigr[I-{(\alpha,1)\over
2\mu_0t}+{(\alpha,2)+2(\alpha,1)\over (2\mu_0t)^2}
 -{(\alpha,3)+4(\alpha,2)\over (2\mu_0 t)^3}+O({1\over t^4})\Bigr].
\end{gather*}
\end{remark}
 The study of the asymptotic behavior of the function $R(t)$
must be done too.

\begin{lemma} \label{lem6.2}
For $t>0$, $R(t)$ is a diagonal matrix-valued  function and
it can be expressed as
\begin{equation}
R(t)={1\over 2\mu_0^2}\Big\{t\Bigr(\mu_0^2
\mathcal{Y}_\alpha^2(t)-\mathcal{Y}_\alpha'^2(t)\Bigr)
+\mathcal{Y}_\alpha(t)\mathcal{Y}_\alpha'(t)
+ {(\alpha,1)\over t}\mathcal{Y}_\alpha^2(t)\Big\} \label{lem6.2i}
\end{equation}
Its asymptotic behavior,  at zero and at infinity, are
respectively
\begin{gather}
R(t)={1\over 2} (\alpha+I)^{-1} t^{2\alpha+2I}\big[I+O(t^2)\big],
\label{lem6.2ii}\\
R(t)={N_\alpha^2(t)\over 2\mu_0}\Bigr[I-{ (\alpha,1)\over \mu_0t}
+{2(\alpha,2)\over (2\mu_0t)^2}+O({1\over t^3})\Bigr],
\label{lem6.2iii}
\end{gather}
 where $N_\alpha$ is defined by \eqref{e29}.
\end{lemma}

\begin{proof}
It is easy to see that by \eqref{e4} and \eqref{e26},
\begin{equation}
 R(t)= \big({2\over \mu_0}\big)^{2\alpha}e^{i\alpha\pi}
{\bf \Gamma}^2(\alpha+I)\int_0^tsJ_\alpha^2(s\lambda_0)ds. \label{e30}
\end{equation}
Manipulating  Bessel equations we show, for $\lambda$ and $\nu$
in $\mathbb{C}$ distinct complex parameters, that (see  \cite[p. 128]{l2})
$$
\int_0^a tJ_\mu (\lambda t)J_\mu(\nu t)dt={a\nu
J_\mu(\lambda a)J'_\mu(\nu a)-a\lambda J'_\mu(\lambda a)J_\mu(\nu
a)\over \lambda^2-\nu^2},\quad a>0.
$$
 Taking the limit of this
quantity as $\nu \to \lambda$, we obtain
$$
\int_0^a tJ_\mu^2(\lambda t)dt={a^2\over 2}\Bigr[(J'_\mu)^2(\lambda
a)+(1-{\mu^2\over \lambda^2a^2} )J^2_\mu(\lambda a)\Bigr].
$$
This result  yields that
$$
R(t)=\Bigr({2\over \mu_0}\Bigr)^{2\alpha}
e^{i\alpha\pi}{\bf \Gamma}^2(\alpha+I){t^2\over 2}
\Bigr[(J'_\alpha)^2(-i\mu_0t)+(1+{\alpha^2\over t^2\mu_0^2})
J^2_\alpha(-i\mu_0t)\Bigr].
$$
 The relation between $J_\alpha(t)$
and $\mathcal{Y}_\alpha(t)$ completes the proof of  assertion
\ref{lem6.2i}. To prove \ref{lem6.2ii} and \ref{lem6.2iii},
 we use \ref{lem6.2i} and Remark \ref{rmk6.1}.
\end{proof}

\begin{proposition} \label{prop6.1}
The function $\Delta Q(t)$ has the  behavior
$$
\Delta Q(t)=\begin{cases}
2t^\alpha\Bigr[C_0(\alpha+I/2)+(\alpha+I/2)C_0+O(t^2)\Bigr]t^{\alpha},
&\text{as }t\to 0^+\\
{(\alpha,2)\over2t^2} [I+O({1\over t^2})] &\text{as } t\to+\infty
\end{cases}
$$
\end{proposition}

\begin{proof}  On the one hand, by definition and from the Lemma \ref{lem6.2},
we show that  at infinity,
$$
V(t)={N_\alpha^2(t)\over 2\mu_0}\Bigr[I-{
(\alpha,1)\over\mu_0t}+{2(\alpha,2)\over (2\mu_0t)^2}+O({1\over
t^3})\Bigr]\,.
$$
So that, when $t\to+\infty$,
$$
V^{-1}(t)=2\mu_0N_\alpha^{-2}(t)\Bigr[I+{ (\alpha,1)\over
\mu_0t}+{2(\alpha,1)^2-(\alpha,2)\over 2(\mu_0t)^2}+O({1\over
t^3})\Bigr].
$$
On the other hand and by means of \eqref{e27},
the potential $\Delta Q$, defined by  \eqref{e20},  takes the form
\begin{align*}
\Delta Q(t)&=2\Bigr[(\mathcal{Y}_\alpha^*)'(t)V^{-1}(t)\mathcal{
Y}_\alpha(t)+\mathcal{Y}_\alpha^*(t)V^{-1}(t)\mathcal{Y}_\alpha'(t)\\
&\quad -\mathcal{Y}_\alpha^*(t)V^{-1}(t)\mathcal{Y}_\alpha(t)
\mathcal{Y}_\alpha^*(t) V^{-1}(t)\mathcal{Y}_\alpha(t)\Bigr].
\end{align*}
 Then, by the behavior at
infinity of $\mathcal{Y}_\alpha(t)$, $\mathcal{Y}'_\alpha(t)$, and
$V^{-1}(t)$, the result is deduced. For the behavior at zero, we
use an analogous approach.
\end{proof}

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\end{document}