\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__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)] $\sup_{0__~~0$ 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
$$
\sup_{0~~~~0$. Then, for $f\in E_1$, we set
$$
L(f)(u)=\int_0^b \Omega(u,s)f(s)ds,\quad 0~~__\varphi_j(u)
$$
where the previous series converges uniformly on $(0,b)$.
\end{proof}
\section{Existence and differentiability of $K(t,u)$}
The main objective of this section is the
resolution of the Gelfund-Levitan equation \eqref{e6} associated with
$\Omega(t,u)$. Thus by mean of the general theory of compact
self-adjoint operators (see \cite{l1,y1}) and the Lemma \ref{lem2.1}, we
conclude that, for a fixed $t>0$, \eqref{e6} is with respect to $K(t,u)$
of Fredholm's.
\subsection*{Existence of $K(t,u)$}
\begin{lemma} \label{lem3.1}
Let $t_0>0$ be fixed. Then if the rows of $\Omega(t,u)$ satisfy
the conditions of the Lemma \ref{lem2.1}, for $0____t_0
\end{cases}
$$
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 $\epsilon ____0$ 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 $0____0$, the function
$u\mapsto K(t,s)\Omega(s,u)$, $0__~~0$, 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 $0~~__0$ such that the assumptions of
Lemma \ref{lem5.1} hold, there exists a positive constant $c(R)$ such that
$$
\|2{d\over dt}\Omega(t,t)-Q(t)\|\leq
c(R)\Bigr(\int_t^{2t} G(s)ds\Bigr)^2,\quad 00$,
$$
\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}
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