\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 69, pp. 1--33.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/69\hfil Legendre equation]
{The Legendre equation and its self-adjoint operators}

\author[L. L. Littlejohn, A. Zettl\hfil EJDE-2011/69\hfilneg]
{Lance L. Littlejohn, Anton Zettl}  % in alphabetical order

\address{Lance L. Littlejohn \newline
Department of Mathematics, Baylor University,
One Bear Place \# 97328, Waco, TX 76798-7328, USA}
\email{lance\_littlejohn@baylor.edu}

\address{Anton Zettl \newline
Department of Mathematical Sciences,
Northern Illinois University, DeKalb, IL 60115-2888, USA}
\email{zettl@math.niu.edu}

\thanks{Submitted April 17, 2011. Published May 25, 2011.}
\subjclass[2000]{05C38, 15A15, 05A15, 15A18}
\keywords{Legendre equation; self-adjoint operators;
 spectrum; \hfill\break\indent three-interval problem}

\begin{abstract}
 The Legendre equation has interior singularities at $-1$ and $+1$.
 The celebrated classical Legendre polynomials are the eigenfunctions
 of a particular self-adjoint operator in $L^2(-1,1)$.
 We characterize all self-adjoint Legendre operators in $L^2(-1,1)$
 as well as those in $L^2(-\infty,-1)$ and in $L^2(1,\infty)$
 and discuss their spectral properties. Then, using the
 `three-interval theory', we find all self-adjoint Legendre operators
 in $L^2(-\infty,\infty)$. These include operators which are not
 direct sums of operators from the three separate intervals and thus
 are determined by interactions through the singularities at $-1$
 and $+1$.
\end{abstract}

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


\section{Introduction}

The Legendre equation
\begin{equation}
-(py')'=\lambda y,\quad p(t)=1-t^2, \label{eq0.1}
\end{equation}
is one of the simplest singular Sturm-Liouville differential
equations.  Its potential function $q$ is zero, its weight
function $w$ is the constant $1$, and its leading coefficient $p$
is a simple quadratic. It has regular singularities at the points
$\pm1$ and at $\pm\infty$. The singularities at $\pm1$ are due to
the fact that $1/p$ is not Lebesgue integrable in left and right
neighborhoods of these points; the singularities at $-\infty$ and
at $+\infty$ are due to the fact that the weight function $w(t)=1$
is not integrable at these points.

The equation \eqref{eq0.1} and its associated self-adjoint
operators exhibit a surprisingly wide variety of interesting
phenomena. In this paper we survey these important points. Of
course, one of the main reasons this equation is important in many
areas of pure and applied mathematics stems from the fact that it
has interesting solutions. Indeed, the Legendre polynomials
$\{P_{n}\}_{n=0}^{\infty}$ form a complete orthogonal set of
functions in $L^2(0,\infty)$ and, for $n\in\mathbb{N}_{0}$,
$y=P_{n}(t)$ is a solution of \eqref{eq0.1} when
$\lambda=\lambda_{n}=n(n+1)$. Properties of the Legendre
polynomials can be found in several textbooks including the
remarkable book of \cite{Szego}. Most of our results can be
inferred directly from known results scattered widely in the
literature, others require some additional work. A few are new. It
is remarkable that one can find some new results on this equation
which has such a voluminous literature and a history of more than
200 years.

The equation \eqref{eq0.1} and its associated self-adjoint
operators are studied on each of the three intervals
\begin{equation}
J_1=(-\infty,-1),\quad J_2=(-1,1),\quad J_3=(1,\infty),
\label{eq0.2}
\end{equation}
and on the whole real line $J_4=\mathbb{R}=(-\infty,\infty)$.
The latter is based on some minor modifications of the
`two-interval' theory developed by Everitt and Zettl \cite{evze86}
in which the equation \eqref{eq0.1} is considered on the whole
line $\mathbb{R}$ with singularities at the interior points $-1$
and $+1$. For each interval the corresponding operator setting is
the Hilbert space $H_i=$ $L^2(J_i)$, $i=1,2,3,4$ consisting
of complex valued functions $f\in AC_{\rm loc}(J_i)$ such that
\begin{equation}
{\int_{J_i}} |f|^2<\infty. \label{eq0.3}
\end{equation}

Since $p(t)$ is negative when $|t|>1$ we let
\begin{equation}
r(t)=t^2-1. \label{eq0.4}
\end{equation}
Then \eqref{eq0.1} is equivalent to
\begin{equation}
-(ry')'=\xi y,\quad \xi=-\lambda. \label{eq0.5}
\end{equation}
Note that $r(t)>0$ for $t\in J_1\cup J_3$ so that
 \eqref{eq0.5} has the usual Sturm-Liouville form with
positive leading coefficient $r$.

Before proceeding to the details of the study of the Legendre
equation on each of the three intervals $J_i$, $i=1,2,3$ and on
the whole line $\mathbb{R}$ we make some general observations. (We
omit the study of the two-interval Legendre problems on any two of
the three intervals $J_1,J_2,J_3$ since this is similar to
the three-interval case. The two-interval theory could also be
applied to the two intervals $\mathbb{R}$ and $J_i$ for any
$i$.)

For $\lambda=\xi=0$ two linearly independent solutions are given by
\begin{equation}
u(t)=1,\quad v(t)=\frac{-1}{2}\ln(|\frac{1-t}{t+1}|) \label{eq0.6}
\end{equation}
Since these two functions $u,v$ play an important role below we make
some observations about them.

Observe that for all $t\in\mathbb{R}$, $t\neq\pm1$, we have
\begin{equation}
(pv')(t)=+1. \label{eq0.7}
\end{equation}
Thus the quasi derivative $(pv')$ can be continuously
extended so that it is well defined and continuous on the whole
real line $\mathbb{R}$ including the two singular points $-1$ and
$+1$. It is interesting to observe that $u$, $(pu')$ and (the
extended) $(pv')$ can be defined to be continuous on $\mathbb{R}$
and only $v$ blows up at the singular points $-1$ and $+1$.

These simple observations about solutions of \eqref{eq0.1} when
$\lambda=0$ extend in a natural way to solutions for all
$\lambda\in$ $\mathbb{C}$ and are given in the next theorem whose
proof may be of more interest than the theorem. It is based on a
`system regularization' of \eqref{eq0.1} using the above functions
$u$, $v$.

The standard system formulation of \eqref{eq0.1} has the form
\begin{equation}
Y'=(P-\lambda W)Y\quad\text{on }(-1,1), \label{eq0.8}
\end{equation}
where
\begin{equation}
Y=\begin{pmatrix}
y\\
py'
\end{pmatrix},
\quad P=\begin{pmatrix}

0 & 1/p\\
0 & 0
\end{pmatrix}
,\quad W=\begin{pmatrix}
0 & 0\\
1 & 0
\end{pmatrix}  \label{eq0.9}
\end{equation}
Let $u$ and $v$ be given by \eqref{eq0.6} and let
\begin{equation}
U=\begin{pmatrix}
u & v\\
pu' & pv'
\end{pmatrix}
  =\begin{pmatrix}
1 & v\\
0 & 1
\end{pmatrix} . \label{eq0.10}
\end{equation}
Note that $\det U(t)=1$, for $t\in J_2=(-1,1)$, and set
\begin{equation}
Z=U^{-1}Y. \label{eq0.11}
\end{equation}
Then
\begin{align*}
Z'
&  =(U^{-1})'Y+U^{-1}Y'=-U^{-1}U'
U^{-1}Y+(U^{-1})(P-\lambda V)Y\\
&  =-U^{-1}U'Z+(U^{-1})(P-\lambda W)UZ\\
&  =-U^{-1}(PU)Z+U^{-1}(DU)Z-\lambda(U^{-1}WU)Z=-\lambda(U^{-1}WU)Z.
\end{align*}
Letting $G=(U^{-1}WU)$ we may conclude that
\begin{equation}
Z'=-\lambda GZ.\; \label{eq0.13}
\end{equation}
Observe that
\begin{equation}
G=U^{-1}WU=\begin{pmatrix}
-v & -v^2\\
1 & v
\end{pmatrix} . \label{eq0.14}
\end{equation}

\begin{definition} \label{def1} \rm
We call \eqref{eq0.13} a `regularized' Legendre system.
\end{definition}

This definition is justified by the next theorem.

\begin{theorem}\label{T0.1}
Let $\lambda\in\mathbb{C}$ and let $G$ be given by
\eqref{eq0.14}.

\begin{enumerate}
\item Every component of $G$ is in $L^{1}(-1,1)$ and therefore \eqref{eq0.13}
is a \emph{regular} system.

\item For any $c_1,c_2\in\mathbb{C}$ \ the initial value problem
\begin{equation}
Z'=-\lambda GZ,\quad Z(-1)=\begin{pmatrix}
c_1\\
c_2
\end{pmatrix}  \label{eq0.15}
\end{equation}
has a unique solution $Z$ defined on the closed interval $[-1,1]$.

\item If $Y=\begin{pmatrix}
y(t,\lambda)\\
(py')(t,\lambda)
\end{pmatrix}$ is a solution of \eqref{eq0.8} and
$Z=U^{-1}Y=\begin{pmatrix}
z_1(t,\lambda)\\
z_2(t,\lambda)
\end{pmatrix}$, then $Z$ is a solution of \eqref{eq0.13} and for all
$t\in(-1,1)$ we have
\begin{gather}
y(t,\lambda)    =u z_1(t,\lambda)+v(t) z_2(t,\lambda)=z_1
(t,\lambda)+v(t) z_2(t,\lambda)\label{eq0.16}\\
(py')(t,\lambda)    =(pu') z_1(t,\lambda)+(pv')(t)
z_2(t,\lambda)=- z_2(t,\lambda) \label{eq0.17}
\end{gather}


\item For every solution $y(t,\lambda)$ of the singular scalar Legendre
equation \eqref{eq0.1} the quasi-derivative $(py')(t,\lambda)$ is
continuous on the compact interval $[-1,1]$. More specifically we
have
\begin{equation}
\lim_{t\to-1^{+}}(py')(t,\lambda)=-z_2(-1,\lambda),\quad
\lim_{t\to1^{-}}(py')(t,\lambda)=-z_2(1,\lambda).
\label{eq0.18}
\end{equation}
Thus the quasi-derivative is a continuous function on the closed
interval
$[-1,1]$ for every $\lambda\in\mathbb{C}$.

\item Let $y(t,\lambda)$ be given by \eqref{eq0.16}. If $z_2(1,\lambda
)\neq0$ then $y(t,\lambda)$ is unbounded at $1$;
 if $z_2(-1,\lambda)\neq0$ then $y(t,\lambda)$ is unbounded at $-1$.

\item Fix $t\in[-1,1]$, let $c_1,c_2\in\mathbb{C}$.
If $Z=\begin{pmatrix}
z_1(t,\lambda)\\
z_2(t,\lambda)
\end{pmatrix}$ is the solution of \eqref{eq0.13} determined by the
initial conditions
$z_1(-1,\lambda)=c_1,\;z_2(-1,\lambda)=c_2$, then
$z_i(t,\lambda)$ is an entire function of $\lambda$, $i=1,2$.
Similarly for the initial condition
$z_1(1,\lambda)=c_1,\;z_2(1,\lambda)=c_2$.

\item For each $\lambda\in\mathbb{C}$ there is a nontrivial solution
which is bounded in a (two sided) neighborhood of $1$; and there
is a (generally different) nontrivial solution which is bounded
in a (two sided) neighborhood of $-1$.

\item A nontrivial solution $y(t,\lambda)$ of the singular scalar
Legendre equation \eqref{eq0.1} is bounded at $1$ if and only if $ z_2
(1,\lambda)=0$; a nontrivial solution $y(t,\lambda)$ of the
singular scalar Legendre equation \eqref{eq0.1} is bounded at $-1$
if and only if $ z_2(-1,\lambda)=0$.
\end{enumerate}
\end{theorem}

\begin{proof}
Part (1) follows from \eqref{eq0.14}, (2) is a direct consequence
of (1) and the theory of regular systems, $Y=UZ$ implies
(3)$\Longrightarrow$(4) and (5); (6) follows from (2) and the
basic theory of regular systems. For (7) determine solutions
$y_1(t,\lambda)$, $y_{-1}(t,\lambda)$ by applying the Frobenius
method to obtain power series solutions of \eqref{eq0.1} in the
form: (see \cite{ALM}, page 5 with different notations)
\begin{gather}
y_1(t,\lambda)   =1+\sum_{n=1}^{\infty}a_{n}(\lambda)(t-1)^{n}
,\quad |t-1|<2;\\
y_{-1}(t,\lambda)   =1+\sum_{n=1}^{\infty}b_{n}(\lambda
)(t+1)^{n},\quad |t+1|<2; \label{eq0.19}
\end{gather}

Item (8) follows from \eqref{eq0.16} that if
$ z_2(1,\lambda)\neq0$, then $y(t,\lambda)$ is not bounded at
$1$. Suppose $ z_2(1,\lambda)=0$. If the corresponding
$y(t,\lambda)$ is not bounded at $1$ then there are two linearly
unbounded solutions at $1$ and hence all nontrivial solutions are
unbounded at $1$. This contradiction establishes (8) and completes
the proof of the theorem.
\end{proof}

\begin{remark} \label{rmk1} \rm
From Theorem \eqref{T0.1} we see that,\emph{ for every
}$\lambda\in$ $\mathbb{C}$, the equation \eqref{eq0.1} has a
solution $y_1$ which is bounded at $1$ and has a solution
$y_{-1}$ which is bounded at $-1$.

It is well known that for $\lambda_{n}=n(n+1):n\in\mathbb{N}_{0}
=\{0,1,2,\dots \}$ the Legendre polynomials $P_{n}$
(see \ref{P0} below) are
solutions on $(-1,1)$ and hence are bounded at $-1$ and at $+1$.
\end{remark}

For later reference we introduce the primary fundamental matrix of
the system \eqref{eq0.13}.

\begin{definition} \label{def2} \rm
Fix $\lambda\in\mathbb{C}$. Let $\Phi(\cdot,\cdot,\lambda)$ be the
primary fundamental matrix of \eqref{eq0.13}; i.e. for each
$s\in[-1,1]$, $\Phi(\cdot,s,\lambda)$ is the unique matrix
solution of the initial value problem:
\begin{equation}
\Phi(s,s,\lambda)=I \label{eq0.20}
\end{equation}
where $I$ is the $2\times2$ identity matrix. Since \eqref{eq0.13}
is regular, $\Phi(t,s,\lambda)$ is defined for all
$t,s\in[-1,1]$ and, for each fixed $t,s$,
$\Phi(t,s,\lambda)$ is an entire function of $\lambda$.
\end{definition}

We now consider two point boundary conditions for \eqref{eq0.13};
later we will relate these to singular boundary conditions for
\eqref{eq0.1}.
Let $A,B\in M_2(\mathbb{C})$, the set of $2\times2$ complex matrices,
and consider the boundary value problem
\begin{equation}
Z'=-\lambda GZ,\quad AZ(-1)+B Z(1)=0. \label{eq0.21}
\end{equation}


\begin{lemma}\label{L0.1}
A complex number $-\lambda$ is an eigenvalue of
\eqref{eq0.21} if and only if
\begin{equation}
\Delta(\lambda)=\det[A+B\Phi(1,-1,-\lambda)]=0. \label{eq0.22}
\end{equation}

Furthermore, a complex number $-\lambda$ is an eigenvalue of geometric
multiplicity two if and only if
\begin{equation}
A+B\Phi(1,-1,-\lambda)=0. \label{eq0.23}
\end{equation}
\end{lemma}

\begin{proof}
Note that a solution for the initial condition $Z(-1)=C$ is given by
\begin{equation}
Z(t)=\Phi(t,-1,-\lambda) C,\quad t\in[-1,1]. \label{eq0.24}
\end{equation}
The boundary value problem \eqref{eq0.21} has a nontrivial
solution for $Z$ if and only if the algebraic system
\begin{equation}
[ A+B\Phi(1,-1,-\lambda)] Z(-1)=0 \label{eq0.25}
\end{equation}
has a nontrivial solution for $Z(-1)$.

To prove the furthermore part, observe that two linearly
independent solutions of the algebraic system \eqref{eq0.25} for
$Z(-1)$ yield two linearly independent solutions $Z(t)$ of the
differential system and conversely.
\end{proof}

Given any $\lambda\in$ $\mathbb{R}$ and any solutions $y,z$ of
\eqref{eq0.1} the Lagrange form $[y,z](t)$ is defined by
\[
[ y,z](t)=y(t)(p\overline{z'})(t)-\overline{z}(t)(py^{\prime
})(t).
\]
So, in particular, we have
\begin{gather*}
[ u,v](t)    =+1,\quad [v,u](t)=-1,\quad [y,u](t)=-(py')(t),
\quad t\in \mathbb{R},\\
[y,v](t)    =y(t)-v(t)(py')(t),\quad t\in\mathbb{R},\;t\neq\pm1.
\end{gather*}


We will see below that, although $v$ blows up at $\pm1$, the
form $[y,v](t)$ is well defined at $-1$ and $+1$ since the limits
\[
\lim_{t\to-1}[y,v](t),\quad \lim_{t\to+1}[y,v](t)
\]
exist and are finite from both sides. This for any solution $y$ of
 \eqref{eq0.1} for any $\lambda\in$ $\mathbb{R}$. Note
that, since $v$ blows up at $1$, this means that $y$ must blow up
at $1$ except, possibly when $(py')(1)=0$. We will expand on this
observation below in the section on `Regular Legendre' equations.

Now we make the following additional observations:
For definitions of the technical terms used here, see \cite{zett05}.

\begin{proposition}\label{P0}
The following results are valid:

\begin{enumerate}
\item Both equations \eqref{eq0.1} and \eqref{eq0.5} are singular at
$-\infty$, $+\infty$ and at $-1$, $+1$, from both sides.

\item In the $L^2$ theory the endpoints $-\infty$ and $+\infty$ are in the
limit-point (LP) case, while $-1^{-}$, $-1^{+}$, $1^{-}$, $1^{+}$
are all in the limit-circle (LC) case. In particular both
solutions $u,v$ are in $L^2(-1,1)$. Here we use the notation
$-1^{-}$ \ to indicate that the equation is studied on an interval
which has $-1$ as its right endpoint. Similarly for $-1^{+}$,
$1^{-}$, $1^{+}$.

\item For every $\lambda\in\mathbb{R}$ the equation \eqref{eq0.1} has a
solution which is bounded at $-1$ and another solution which blows up
logarithmically at $-1$. Similarly for $+1$.

\item When $\lambda=0$, the constant function $u$ is a principal solution at
each of the endpoints $-1^{-}$, $-1^{+}$, $1^{-}$, $1^{+}$ but $u$
is a nonprincipal solution at both endpoints $-\infty$ and
$+\infty$. On the other hand, $v$ is a nonprincipal solution at
$-1^{-}$, $-1^{+}$, $1^{-}$, $1^{+}$  but is the principal
solution at $-\infty$ and $+\infty$. Recall that, at each
endpoint, the principal solution is unique up to constant
multiples but a nonprincipal solution is never unique since the
sum of a principal and a nonprincipal solution is nonprincipal.

\item On the interval $J_2=(-1,1)$ the equation \eqref{eq0.1} is
nonoscillatory at $-1^{-}$, $-1^{+}$, $1^{-}$, $1^{+}$ for
every real $\lambda$.

\item On the interval $J_3=(1,\infty)$ the equation \eqref{eq0.5} is
oscillatory at $\infty$ for every $\lambda>-1/4$ and nonoscillatory at
$\infty$ for every $\lambda<-1/4$.

\item On the interval $J_3=(1,\infty)$ the equation \eqref{eq0.1} is
nonoscillatory at $\infty$ for every $\lambda<1/4$ and oscillatory at $\infty$
\ for every $\lambda>1/4$.

\item On the interval $J_1=(-\infty,-1)$ the equation \eqref{eq0.1} is
nonoscillatory at $-\infty$ for every $\lambda<+1/4$ and oscillatory at
$-\infty$ for every $\lambda>+1/4$.

\item On the interval $J_1=(-\infty,-1)$ the equation \eqref{eq0.5} is
oscillatory at $-\infty$ for every $\lambda>-1/4$ and nonoscillatory at
$-\infty$ for every $\lambda<-1/4$.

\item The spectrum of the classical Sturm-Liouville problem (SLP) consisting
of equation \eqref{eq0.1} on $(-1,1)$ with the boundary condition
\[
(py')(-1)=0=(py')(+1)
\]
is discrete and is given by
\[
\sigma(S_{F})=\{n(n+1):n\in\mathbb{N}_{0}=\{0,1,2,\dots \}\}.
\]
Here $S_{F}$ denotes the classical Legendre operator; i.e., the
self-adjoint operator in the Hilbert space $L^2(-1,1)$ which
represents the Sturm-Liouville problem (SLP) \eqref{eq0.1},
\eqref{eq0.11}. The notation $S_{F}$ is used to indicate that this
is the celebrated Friedrichs extension. It's orthonormal
eigenfunctions are the Legendre polynomials $\{P_{n}
:n\in\mathbb{N}_{0}\}$ given by:
\[
P_{n}(t)=\sqrt{\frac{2n+1}{2}}
{\sum_{j=0}^{[n/2]}}
\frac{(-1)^{j}(2n-2j)!}{2^{n}j!(n-j)!(n-2j)!}t^{n-2j}\quad
(n\in\mathbb{N}_{0})
\]
where $[n/2]$ denotes the greatest integer $\leq n/2$.

 The special (ausgezeichnete) operator $S_{F}$ is one of an
uncountable number of self-adjoint realizations of the equation
\eqref{eq0.1} on $(-1,1)$ in the Hilbert space $H=L^2(-1,1)$.
The singular boundary conditions determining the other
self-adjoint realizations will be given explicitly below.

\item The essential spectrum of every self-adjoint realization of
equation \eqref{eq0.1} in the Hilbert space $L^2(1,\infty)$ and of
\eqref{eq0.1} in the Hilbert space $L^2(-\infty,-1)$ is given by
\[
\sigma_{e}=(-\infty,-1/4].
\]
For each interval every self-adjoint realization is bounded above and has at
most two eigenvalues. Each eigenvalue is $\geq-1/4$. The existence of $0,1$ or
$2$ eigenvalues and their location depends on the boundary condition. There is
no uniform bound for all self-adjoint realizations.

\item The essential spectrum of every self-adjoint realization of
equation \eqref{eq0.5} in the Hilbert space $L^2(1,\infty)$ and of
\eqref{eq0.5} in the Hilbert space $L^2(-\infty,-1)$ is given by
\[
\sigma_{e}=[1/4,\infty).
\]
For each interval every self-adjoint realization is bounded below and has at
most two eigenvalues. There is no uniform bound for all self-adjoint
realizations. Each eigenvalue is $\leq1/4$. The existence of $0,1$ or $2$
eigenvalues and their location depends on the boundary condition.
\end{enumerate}
\end{proposition}

\begin{proof}
Parts (1), (2), (4) are basic results in Sturm-Liouville theory
 \cite{zett05}.
The proof of (3) will be given below in the section on regular Legendre
equations. For these and other basic facts mentioned below the reader is
referred to the book \textquotedblleft Sturm-Liouville
Theory\textquotedblright\ \cite{zett05}. Part (10) is the well known
celebrated classical theory of the Legendre polynomials, see \cite{nize92} for
a characterization of the Friedrichs extension. In the other parts, the
statements about oscillation, nonoscillation and the essential spectrum
$\sigma_{e}$ follow from the well known general fact that, when the leading
coefficient is positive, the equation is oscillatory for all
$\lambda >\inf\sigma_{e}$ and nonoscillatory for all $\lambda<\inf\sigma_{e}$. Thus
$\inf\sigma_{e}$ is called the oscillation number of the equation.
It is well known that the oscillation number of equation
\eqref{eq0.5} on $(1,\infty)$ is $-1/4$. Since \eqref{eq0.5} is
nonoscillatory at $1^{+}$ for all $\lambda \in\mathbb{R}$
oscillation can occur only at $\infty$. The transformation
$t\to-1$ shows that the same results hold for
\eqref{eq0.5} on $(-\infty,-1)$. Since $\xi=-\lambda$ the above
mentioned results hold for the standard Legendre equation
\eqref{eq0.1} but with the sign reversed. To compute the essential
spectrum on $(1,\infty)$ we first note that the endpoint $1$ makes
no contribution to the essential spectrum since it is limit-circle
nonoscillatory. Note that
$\int_2^{\infty}1/\sqrt{r}=\infty$ and
\[
\lim_{t\to\infty}\frac{1}{4}(r''(t)-\frac{1}{4}
\frac{[r'(t)]^2}{r(t)})=\lim_{t\to\infty}\frac{1}{4}
(2-\frac{1}{4}\frac{4t^2}{t^2-1})=\frac{1}{4}.
\]


From this and Theorem XIII.7.66 in Dunford and Schwartz \cite{dusc63},
part (12) follows and part (11) follows from (12).
Parts (6)-(10) follow from the fact that the starting point of the
essential spectrum is the oscillation
point of the equation; that is, the equation is oscillatory for all
$\lambda$ above the starting point and nonoscillatory for all
$\lambda$ below. (Note that there is a sign change correction
needed in the statement of Theorem XIII.7.66 since $1-t^2$
is negative when $t>1$ and this theorem applies to a
positive leading coefficient.)
\end{proof}

\subsection*{Notation}
 $\mathbb{R}$ and $\mathbb{C}$ denote the
real and complex number fields respectively; $\mathbb{N}$ and
$\mathbb{N}_{0}$ denote the positive and non-negative integers
respectively; $L$ denotes Lebesgue integration; $AC_{\rm loc}(J)$ is
the set of complex valued functions which are Lebesgue integrable
on every compact subset of $J$; $(a,b)$ and $[\alpha ,\beta]$
represent open and compact intervals of $\mathbb{R}$,
respectively; other notations are introduced in the sections
below.

\section{Regular Legendre Equations}

In this section we construct \emph{regular} Sturm-Liouville
equations which are equivalent to the classical \emph{singular
equation} \eqref{eq0.1}. This construction is based on a
transformation used by Niessen and Zettl in \cite{nize92}. We
apply this construction to the Legendre problem on the interval
$(-1,1):$
\begin{equation}
My=-(py')=\lambda y\quad\text{on } J_2=(-1,1),\quad
p(t)=1-t^2,\quad -1<t<1.
\label{eq2.1}
\end{equation}
This transformation depends on a modification of the
function $v$ given by \eqref{eq0.6}. Note that $v$ changes sign in
$(-1,1)$ at $0$ and we need a function which is positive on the
entire interval $(-1,1)$ and is a nonprincipal solution at both
endpoints.

This modification consists of using a multiple of $v$ which
is positive near each endpoint and changing the function $v$
in the middle of $J_2$
\begin{equation}
v_{m}(t)=\begin{cases}
\frac{-1}{2}\ln\big(  \frac{1-t}{1+t}\big) , & 1/2\leq t<1\\
m(t), & -1/2\leq t\leq 1/2\\
\frac{1}{2}\ln\big(  \frac{1-t}{1+t}\big)  , & -1\leq t\leq -1/2
\end{cases}  \label{eq2.2}
\end{equation}
where the `middle function' $m$ is chosen so that the modified
function $v_{m}$ defined on $(-1,1)$ satisfies the following
properties:
\begin{enumerate}
\item $v_{m}(t)>0$, $-1<t<1$.

\item $v_{m},(pv_{m}')\in AC_{\rm loc}(-1,1)$, $v_{m},(pv_{m}')\in
L^2(-1,1)$.

\item $v_{m}$ is a nonprincipal solution at both endpoints.

\end{enumerate}
For later reference we note that
\begin{gather*}
(pv_{m}')(t)   =+1,\quad \frac{1}{2}\leq t<1,\\
(pv_{m}')(t)   =-1,\quad -1<t<\frac{-1}{2},\\
[ u,v_{m}](t)  =u(t)(pv_{m}')(t)-v(t)(pu'
)(t)=(pv_{m}')(t)=1,\quad \frac{1}{2}\leq t<1,\\
[ u,v_{m}](t)    =u(t)(pv_{m}')(t)-v(t)(pu'
)(t)=(pv_{m}')(t)=-1,\quad -1<t<-\frac{1}{2}. \label{eq2.3}
\end{gather*}


Niessen and Zettl \cite[Lemma 2.3 and Lemma 3.6]{nize92},
showed that such choices for $m$ are possible in general.
Although in the Legendre case studied
here an explicit such $m$ can be constructed we do not do so here
since our focus is on boundary conditions at the endpoints which
are independent of the choice of $m$.

\begin{definition} \label{def3} \rm
Let $M$ be given by \eqref{eq2.1}. Define
\begin{equation}
P=v_{m}^2p,\quad Q=v_{m}Mv_{m},\quad
W=v_{m}^2,\quad\text{on }J_2=(-1,1). \label{eq2.4}
\end{equation}
\end{definition}

Consider the equation
\begin{equation}
Nz=-(Pz')'+Qz=\lambda Wz,\quad\text{on }J_2=(-1,1). \label{eq2.5}
\end{equation}
In \eqref{eq2.4}, $P$ denotes a scalar function; this notation
should not be confused with $P$ defined in \eqref{eq0.9} where $P$
denotes a matrix.

\begin{lemma}\label{L2.1}
Equation \eqref{eq2.5} is regular with $P>0$ on
$J_2$, $W>0$ on $J_2$.
\end{lemma}

\begin{proof}
The positivity of $P$ and $W$ are clear. To prove that
\eqref{eq2.5} is regular\ on $(-1,1)$ we have to show that
\begin{equation}
\int_{-1}^{1}\frac{1}{P}<\infty,\quad
\int_{-1}^{1}Q<\infty, \quad
\int_{-1}^{1}W<\infty. \label{eq2.6}
\end{equation}
The third integral is finite since $v\in L^2(-1,1)$.

Since $v_{m}$ is a nonprincipal solution at both endpoints,
it follows from SL theory \cite{zett05} that
\[
\int_{-1}^{c}\frac{1}{pv_{m}^2}<\infty,\quad
\int_{d}^{1}\frac{1}{pv_{m}^2}<\infty,
\]
for some $c,d$, $-1<c<d<1$. By \eqref{eq2.3}, $1/v_{m}^2$ is
bounded on $[c,d]$ and therefore
\[
\int_{c}^{d}\frac{1}{p}<\infty
\]
so we can conclude that the first integral \eqref{eq2.6} is
finite. The middle integral is finite since $Mv_{m}$ is
identically zero near each endpoint and $v_{m},pv_{m}'\in
AC_{\rm loc}(-1,1)$.
\end{proof}

\begin{corollary}
Let $\lambda\in\mathbb{C}$. For every solution $z$ of
\eqref{eq2.5}, the limits
\begin{equation}
\begin{gathered}
z(-1)    =\lim_{t\to-1^{+}}z(t),\quad
z(1)=\lim_{t\to1^{-} }z(t),\\
(Pz')(-1)   =\lim_{t\to-1^{+}}(Pz')(t),\quad
(Pz')(1)=\lim_{t\to1^{-}}(Pz')(t)
\end{gathered} \label{eq2.7}
\end{equation}
exist and are finite.
\end{corollary}

\begin{proof}
This follows directly from SL theory \cite{zett05};
every solution and its
quasi-derivative have finite limits at each regular endpoint.
\end{proof}

We call equation \eqref{eq2.5} a `regularized Legendre equation'.
It depends on the function $v$ which depends on $m$. The key
property of $v$ is that it is a positive nonprincipal solution at
each endpoint. Note that $v_{m}$ in \eqref{eq2.2} is `patched
together' from two different nonprincipal solutions, one from each
endpoint, the `patching' function $m$ plays no significant role in
this paper.

Note that \eqref{eq2.5} is also defined on $(-1,1)$ but can be
considered on the compact interval $[-1,1]$ in contrast to the
singular Legendre equation \eqref{eq0.1}. A significant
consequence of this, as shown by \eqref{eq2.7}, is that, for each
$\lambda\in\mathbb{C}$, every solution $z$ of \eqref{eq2.5} and
its quasi-derivative $(Pz')$ can be continuously extended to the
endpoints $\pm1$. We use the notation $(Pz')$ to remind the reader
that the product $(Pz')$ has to be considered as one function when
evaluated at $\pm1$ since $P$ is not defined at $-1$ and at $1$.

\begin{remark} \label{rmk2} \rm
Note that we are using the theory of \emph{quasi-differential
}equations. The conditions \eqref{eq2.6} show that the equation
\eqref{eq2.5} is a regular quasi-differential equation. We take
full advantage of this fact in this paper.
\end{remark}

Let $S_{\rm min}(N)$ and $S_{\rm max}(N)$ denote the minimal and maximal
operators associated with \eqref{eq2.5}, and denote their domains
by $D_{\rm min}(N)$, $D_{\rm max}(N)$, respectively. Note that these are
operators in the weighted Hilbert space with weight function
$v_{m}^2$ which we denote by $L^2
(v_{m})=L^2(J_2,v_{m}^2)$. A self-adjoint realization $S(N)$
of \eqref{eq2.5} is an operator in $L^2(v_{m})$ which satisfies
\begin{equation}
S_{\rm min}(N)\subset S(N)=S^{\ast}(N)\subset S_{\rm max}(N). \label{eq2.8}
\end{equation}

Applying the theory of self-adjoint regular Sturm-Liouville
problems to the regularized Legendre equation \eqref{eq2.5} we
obtain the following result.

\begin{theorem}\label{T2.1}
If $A,B$ are $2\times2$ complex matrices satisfying the following
two conditions:
\begin{equation}
\operatorname{rank}(A:B)=2, \label{eq2.9}
\end{equation}
\begin{equation}
AEA^{\ast}=BEB^{\ast}, \label{eq2.10}
\end{equation}
then the set of all $z\in D_{\rm max}(N)$ satisfying
\begin{equation}
A\begin{pmatrix}
z(-1)\\
(Pz')(-1)
\end{pmatrix}
 +B\begin{pmatrix}
z(1)\\
(Pz')(1)
\end{pmatrix}
 =\begin{pmatrix}
0\\
0
\end{pmatrix}  \label{eq2.11}
\end{equation}
is a self-adjoint domain. Conversely, given any self-adjoint
realization of \eqref{eq2.5} in the space $L^2(v)$; i.e., any
operator $S(N)$ satisfying \eqref{eq2.8}, there exist $2\times2$
complex matrices $A,B$ satisfying \eqref{eq2.9} and \eqref{eq2.10}
such that the domain of $S(N)$ is the set of all $z\in
D_{\rm max}(N)$ satisfying \eqref{eq2.11}. Here $(A,B)$ is the
$2\times4$ matrix whose first two columns are the columns of $A$
and whose last two columns are those of $B$.
\end{theorem}

For a proof of the above theorem, see \cite{zett05}.
It is convenient to divide the self-adjoint boundary conditions
\eqref{eq2.11} into two disjoint mutually exclusive classes: the
separated conditions and the coupled ones. The former have the
well known canonical representation
\begin{equation}
\begin{gathered}
\cos(\alpha) z(-1)+\sin(\alpha)(Pz')(-1) =0,\quad 0\leq\alpha
<\pi,\\
\cos(\beta) z(1)+\sin(\beta)(Pz')(1)  =0,\quad 0<\beta\leq\pi.
\end{gathered}\label{eq2.12}
\end{equation}
The latter have the not so well known canonical representation
\begin{equation}
\begin{pmatrix}
z(1)\\
(Pz')(1)
\end{pmatrix}
  =e^{i\gamma}K
\begin{pmatrix}
z(-1)\\
(Pz')(-1)
\end{pmatrix}
,\quad -\pi<\gamma\leq\pi. \label{eq2.13}
\end{equation}

Examples of separated conditions are the well known Dirichlet
condition
\begin{equation}
z(-1)=0=z(1) \label{eq2.14}
\end{equation}
and the Neumann condition
\begin{equation}
(Pz')(-1)=0=(Pz')(1). \label{eq2.15}
\end{equation}
Examples of coupled conditions are the periodic conditions
\begin{gather}
z(-1)    =z(1)\\
(Pz')(-1)    =(Pz')(1) \label{eq2.16}
\end{gather}
and the semi-periodic (also called anti-periodic) conditions
\begin{equation}
\begin{gathered}
z(-1)   =-z(1)\\
(Pz')(-1)   =-(Pz')(1)
\end{gathered} \label{eq2.17}
\end{equation}
Note, however, that when $\gamma\neq0$ we have complex matrices
$A,B$ defining regular self-adjoint operators.

 Next we explore the relationship between solutions $y$ of the
singular equation \eqref{eq2.1} and solutions $z$ of the
regularized Legendre equation \eqref{eq0.1}.

\begin{lemma}\label{L2.2}
For any $\lambda\in\mathbb{C}$, the solutions
$y(\cdot,\lambda)$ of the singular equation \eqref{eq0.1} and the
solutions $z(\cdot,\lambda)$ of the regular equation \eqref{eq2.5}
are related by
\begin{equation}
\frac{y(t,\lambda)}{v_{m}(t)}=z(t,\lambda),\quad
-1<t<1,\;\lambda\in\mathbb{C}
\label{eq2.18}
\end{equation}
and the correspondence $y(\cdot,\lambda)$ $\to$ $z(\cdot,\lambda)$ is
$1-1$ onto. Note that there is the same $\lambda$ on both sides.
\end{lemma}

\begin{proof}
Fix $\lambda\in\mathbb{C}$ and simplify the notation for this proof
so that $v=v_{m}$ and let
\[
z=\frac{y}{v}\quad\text{on }(-1,1).
\]
Then $z'=\frac{vy'-yv'}{v^2}$ and
$((pv^2)z')'=(v(py')-y(pv'))'=v(py^{\prime })'+v'py'-y'pv'-y(pv'
)'=v(-\lambda y)+y(Mv)=-\lambda v^2\frac{y}{v}+\frac{y}
{v}vMv=-\lambda v^2z$ $+Qz$ and \eqref{eq2.5} follows. Reversing
the steps shows that the correspondence is $1-1$.
\end{proof}

\begin{remark}\label{R2.1}\rm
We comment on the relationship between the classical
singular Legendre equation \eqref{eq0.1} and its regularizations
\eqref{eq2.5}; this remark will be amplified below after we have
discussed the self-adjoint operators generated by the singular
Legendre equation \eqref{eq0.1}. In particular, we will see below
that the operator $S(N)$ determined by the Dirichlet condition
\eqref{eq2.14}, which we denote by $S_{F}(N)$, is a regular
representation of the celebrated classical singular Friedrichs
operator, denoted by $S_{F}$ below, whose eigenvalues are
$\{n(n+1):n\in \mathbb{N}_{0}\}$ and whose eigenfunctions are the
classical Legendre polynomials $P_{n}$ given above. Note that the
solutions $y(t,\lambda)$ and $z(t,\lambda)$ have exactly the same
zeros in the open interval $(-1,1)$ but not in the closed interval
$[-1,1]$ since $z$ may be zero at the endpoints and $y$ may not be
defined there.
\end{remark}

\begin{remark}\label{R2.2}\rm
Each solution $z$ and its quasi-derivative $(Pz')$ is
continuous on the compact interval $[-1,1]$. Note that $v(t)$ does
not depend on $\lambda$. Therefore the singularity of every
solution $y(t,\lambda)$ for all $\lambda\in\mathbb{C}$ is
contained in $v$, in other words, the nature of the singularities
of the solutions $y(t,\lambda)$ are invariant with respect to
$\lambda$. Although $v(t)$ does not exist for $t=-1$ and $t=1$ and
$y(t)$ also may not exist for $t=-1$ and $t=1$ the limits
\begin{equation}
\lim_{t\to-1^{+}}\frac{y(t,\lambda)}{v(t)}=z(-1,\lambda),\quad
\lim_{t\to1^{-}}\frac{y(t,\lambda)}{v(t)}=z(1,\lambda)
\label{eq2.19}
\end{equation}
exist for all solutions $y(t,\lambda)$ of the Legendre equation
\eqref{eq0.1}. If $z(1,\lambda)\neq0$, then $y(t,\lambda)$ blows
up logarithmically as $t\to1$; similarly at $-1$.
\end{remark}

\begin{remark} \label{R2.3}\rm
Applying the correspondence \eqref{eq2.18} to the
Legendre polynomials we obtain a factorization of these
polynomials:
\begin{equation}
P_{n}(t)=v(t) z_{n}(t),\quad -1<t<1,\;n\in\mathbb{N}_{0}. \label{eq2.20}
\end{equation}
Since $P_{n}$ is continuous at $-1$ and at $1$ and $v$ blows up at
these points it follows that $z_{n}(-1)=0=z_{n}(1)$,
$n\in\mathbb{N}_{0}$. Note that $z_{n}$ has exactly the same zeros
as $P_{n}$ in the open interval $(-1,1)$. However, also note that
this is not the case for the closed interval $[-1,1]$ since
$z_{n}(-1)=0=z_{n}(1)$ but $P_{n}(1)\neq0\neq P_{n}(-1)$ for each
$n\in\mathbb{N}_{0.}$
\end{remark}

\begin{remark} \label{R2.4}\rm
Below, following the characterization of the
self-adjoint Legendre realizations $S$ of the singular Legendre
equation\ \eqref{eq0.1} using singular SL theory, we will specify
a $1-1$ correspondence between the self-adjoint realizations
$S(N)$ of the regularized Legendre equation \eqref{eq2.5} and the
self-adjoint operators of the singular classical Legendre equation
\eqref{eq0.1}. In particular, we will see that the operator
$S_{D}(N)$ determined by the regular Dirichlet boundary
condition
\begin{equation}
z(-1)=0=z(1) \label{eq2.21}
\end{equation}
corresponds to the celebrated classical Friedrichs Legendre operator
$S_{F}$ determined by the singular boundary condition
\[
(py')(-1)=0=(py')(1)
\]
whose eigenvalues are $\{n(n+1),\;n\in\mathbb{N}_{0}\}$ and whose
eigenfunctions are the classical Legendre polynomials $P_{n}$
given by part (10) of Proposition \eqref{P0}. The Dirichlet
operator $S_{D}(N)$ has the same eigenvalues as $S_{F}$ but its
eigenfunctions are given by
\[
z_{n}=\frac{P_{n}}{v^2},\quad n\in\mathbb{N}_{0}.
\]
Note that each $z_{n}$ has exactly the same zeros in the open interval
$(-1,1)$ but not in the closed interval $[-1,1]$ since
$z_{n}(-1)=0=z_{n}(1)$.
Also note that $S_{F}$ is a self-adjoint operator in the space $L^2(-1,1)$
and $S_{D}(N)$ is a self-adjoint operator in the weighted Hilbert space
$L^2((-1,1),v^2)$.
\end{remark}

\section{Self-Adjoint Operators in $L^2(-1,1)$}

By a self-adjoint operator associated with equation \eqref{eq0.1}
in $H_2=L^2(-1,1)$ or a self-adjoint realization of equation
\eqref{eq0.1} in $H_2$ we mean a self-adjoint restriction of the
maximal operator $S_{\rm max}$ associated with \eqref{eq0.1}. This is
defined as follows:
\begin{gather}
D_{\rm max}=\{f:(-1,1)\to\mathbb{C}\mid f,\,
pf'\in AC_{\rm loc} (-1,1); f,\, pf'\in H_2\} \label{eq3.1}
\\
S_{\rm max}f=-(pf')',\quad f\in D_{\rm max} \label{eq3.2}
\end{gather}
We refer the reader to the classic texts of Akhiezer and Glazman
\cite{AG}, Dunford and Schwartz \cite{dusc63}, Naimark
\cite{Naimark}, and Titchmarsh \cite{Titchmarsh} for general, and
specific, information on the theory of self-adjoint extensions of
symmetric differential operators. We also refer to the excellent
account of \cite{Everitt} on the right-definite self-adjoint
theory of the Legendre expression \eqref{eq0.1}.

Note that all bounded continuous functions on $(-1,1)$ are in
$D_{\rm max}$; in particular all polynomials are in
$D_{\rm max}$. (More
precisely the restriction of every polynomial to $(-1,1)$ is in
$D_{\rm max})$. However $D_{\rm max}$ also contains functions which are
not bounded on $(-1,1)$, e.g. $f(t)=\ln(1-t)$.

\begin{lemma} \label{L3.1}
The operator $S_{\rm max}$ is densely defined in $H_2$ and
therefore has a unique adjoint in $H_2$ denoted by $S_{\rm min}:$
\begin{equation}
S_{\rm max}^{\ast}=S_{\rm min}. \label{eq3.3}
\end{equation}
Furthermore, the minimal operator $S_{\rm min}$ in $H_2$
is symmetric, closed,
densely defined and
\begin{equation}
S_{\rm min}^{\ast}=S_{\rm max} \label{eq3.4}
\end{equation}
Moreover, if $S$ is a self-adjoint extension of $S_{\rm min}$
then $S$ is also a
self-adjoint restriction of $S_{\rm max}$ and conversely.
Thus we have
\begin{equation}
S_{\rm min}\subset S=S^{\ast}\subset S_{\rm max}. \label{eq3.5}
\end{equation}
\end{lemma}

The above lemma is part of basic Sturm-Liouville theory;
see for example \cite{zett05}.

It is clear from \eqref{eq3.5} that each self-adjoint operator $S$
is determined by its domain $D(S)$. The operators $S$ satisfying
\eqref{eq3.5} are called self-adjoint realizations of
\eqref{eq0.1} in $H_2$ or on $(-1,1)$. We will also refer to
these as Legendre operators in $H_2$ or on $(-1,1)$.

Next we describe these self-adjoint domains. It is remarkable that
all self-adjoint Legendre operators can be described explicitly in
terms of two-point singular boundary conditions. For this the
functions $u$, $v$ given by \eqref{eq0.6} play an important role,
in a sense they form a basis for all self-adjoint boundary
conditions \cite{zett05}.
Let
\begin{equation}
My=-(py')'. \label{eq3.6}
\end{equation}
Of critical importance in the characterization of all self-adjoint
boundary conditions is the Lagrange sesquilinear form
$[\cdot,\cdot]$, now defined for all maximal domain functions,
\begin{equation}
[ f,g]=fp(\overline{g}')-gp(\overline{f}')\quad
(f,g\in D_{\rm max}), \label{eq3.7}
\end{equation}
and the associated Green's formula
\begin{equation}
\int_{a}^{b}\{\overline{g}Mf-\overline{f}
Mg\}=[f,g](b)-[f,g](a),\quad
f,g\in D_{\rm max},\;-1<a<b<1 \label{eq3.8}
\end{equation}
From this inequality it follows that the limits
\begin{equation}
\lim_{a\to-1^{+}}[f,g](t),\quad
\lim_{b\to+1^{-}}[f,g](t)
\label{eq3.9}
\end{equation}
exist and are finite.

 We can now give a characterization of all self-adjoint Legendre
operators in $L^2(-1,1)$.

\begin{theorem} \label{T3.1}
Let $u,v$ be given by \eqref{eq0.6}. Let $A,B$ be
$2\times2$ complex matrices satisfying the following two
conditions:
\begin{gather}
\operatorname{rank}(A:B)=2, \label{eq3.10}\\
AEA^{\ast}=BEB^{\ast},\quad E=\begin{pmatrix}
0 & -1\\
1 & 0
\end{pmatrix} . \label{eq3.11}
\end{gather}
Define $D(S)=\{y\in D_{\rm max}\}$ such that
\begin{equation}
A\begin{pmatrix}
(-py')(-1)\\
(ypv'-v(py'))(-1)
\end{pmatrix}
  +B\begin{pmatrix}
(-py')(1)\\
(ypv'-v(py'))(1)
\end{pmatrix}
=\begin{pmatrix}
0\\
0
\end{pmatrix}  . \label{eq3.12}
\end{equation}
Then $D(S)$ is a self-adjoint domain. Furthermore all self-adjoint
domains are generated this way. Here $(A:B)$ denotes the $2\times4$
matrix whose first two columns are those of $A$ and whose last two
 columns are the columns of $B$.
\end{theorem}

The proof of the above theorem is given in \cite[pages 183-185]{zett05}.

\begin{remark} \label{R3.1}\rm
We comment on some aspects of this remarkable characterization of
all self-adjoint Legendre operators in $L^2(-1,1)$.
\begin{enumerate}
\item Just as in the regular case, the singular self-adjoint boundary
conditions \eqref{eq3.12} are explicit since $v$ is explicitly
given near the endpoints by \eqref{eq0.6}.

\item Note that $[y,u]=-py'$ and $[y,v]=y(pv')-v(py')$. Hence $-py'$ and $(ypv'-v(py'))$ exist as finite limits at
$-1$ and at $1$ for all maximal domain functions $y$. In
particular, these limits exist and are finite for all solutions
$y$ of equation \eqref{eq0.1} for any $\lambda$. Thus a number
$\lambda$ is an eigenvalue of the singular boundary value problem
\eqref{eq0.1}, \eqref{eq3.12} if and only if the equation
\eqref{eq0.1} has a nontrivial solution $y$ satisfying
\eqref{eq3.12}. Note that the separate terms $y(pv')$ and $v(py')$
may not exist at $-1$ or at $+1$, they may blow up or oscillate
wildly at these points but the combination $[y,v]$ has a finite
limit at $-1$ and at $+1$ for any maximal domain functions $y,v$.

\item Choose
$A=\begin{pmatrix}
1 & 0\\
0 & 0
\end{pmatrix}$,
$B=\begin{pmatrix}
0 & 0\\
1 & 0
\end{pmatrix}$.
Then \eqref{eq3.11} holds and \eqref{eq3.12} reduces to
\begin{equation}
(py')(-1)=0=(py')(1) \label{eq3.13}
\end{equation}
This is the boundary condition which determines, among the
uncountable number of self-adjoint conditions, the special
(`ausgezeichnete') Friedrichs extension $S_{F}$. It is interesting
to observe that, even though \eqref{eq3.13} has the appearance of
a regular Neumann condition, in fact it is the singular analogue
of the regular Dirichlet condition. It is well known \cite{zett05}
that. in general, the Dirichlet boundary condition determines the
Friedrichs extension $S_{F}$ of regular SLP and that for singular
non-oscillatory limit-circle problems, in general, the Friedrichs
extension $S_{F}$ is determined by the conditions
\[
[ y,u_{a}](a)=0=[y,u_{b}](b)
\]
where $u_{a}$ is the principal solution at the left endpoint $a$
and $u_{b}$ is the principal solution at the right endpoint $b$.
Since the constant function $u=1$ is the principal solution
\emph{at both endpoints }$-1$ and $1$ in the Legendre case we
have $[y,u]=-(py')$ and \eqref{eq3.13} follows.

\item The condition \eqref{eq3.12} includes separated and coupled conditions.
Below we will give a canonical form for these two classes of
conditions which is analogous to the regular case. We will also
see below that \eqref{eq3.12} includes \emph{complex }boundary
conditions. These are coupled; it is known that all separated
self-adjoint conditions can be taken as real; i.e., each complex
separated condition \eqref{eq3.12} is equivalent to a real such
condition.

\item Since each endpoint is LCNO (limit-circle nonoscillatory) it is well
known that the spectrum $\sigma$ of every self-adjoint extension
$S$, $\sigma(S)$ is discrete, bounded below and unbounded above
with no finite cluster point. For $S_{F}$ we have the celebrated
result that
\[
\sigma(S_{F})=n(n+1),\;(n\in\mathbb{N}_{0})
\]
and the corresponding orthonormal eigenfunctions are
polynomials given by \eqref{eq0.13}. For other self-adjoint
Legendre operators $S$ the eigenvalues and eigenfunctions are not
known in closed form. However they can be computed numerically
with the FORTRAN code SLEIGN2, developed by Bailey, Everitt and
Zettl \cite{baez01}; this code, and a number of files related to
it, can be downloaded from\ www.math.niu.edu/SL2. It comes with a
user friendly interface.

\item It is known from general Sturm-Liouville theory that the eigenfunctions
of every self-adjoint Legendre realization $S$ are dense in
$L^2(-1,1)$. In particular the Legendre polynomials
\eqref{eq0.13} are dense in $L^2(-1,1)$.

\item If $S$ is generated by a separated boundary condition \eqref{eq3.12},
then the $n-th$ eigenfunction of $S$ has exactly $n$ zeros in the
open interval $(-1,1)$ for each $n\in\mathbb{N}_{0.}$ In
particular, this is true for the Legendre polynomials
\eqref{eq0.13}.

\item The self-adjoint boundary conditions \eqref{eq3.12} depend on the
function $v$ given by \eqref{eq0.6}. But note that only the values
of $v$ near the endpoints play a role in \eqref{eq3.12} and
therefore $v$ can be replaced by any function which is
asymptotically equivalent to it, in particular $v$ can be replaced
by any function which has the same values as $v$ in a neighborhood
of $-1$ and of $1$.
\end{enumerate}
\end{remark}

Now that we have determined all the self-adjoint singular Legendre operators
with Theorem \ref{T3.1}, we compare these with the self-adjoint operators
determined by the regularized Legendre equation which are given by Theorem
\ref{T2.1}. In making this comparison it is important to keep in mind that
these operators act in different Hilbert spaces: $L^2(-1,1)$ for the
singular classical case and $L^2(v^2)=$ $L^2((-1,1),v^2)$ for the
regularized case.

But first we show that the correspondence
\begin{equation}
\frac{y}{v}=z,\quad
y=v z \label{eq3.14}
\end{equation}
extends from solutions to functions in the domains of the operator
realizations of the classical Legendre equation and its
regularization. Since we now compare operator realizations of the
singular equation \eqref{eq0.1} and its regularization
\eqref{eq2.5} with each other we use the notation $S(M)$ for
operators associated with the former and $S(N)$ for those of the
latter. It is important to remember that the operators $S(M)$ are
operators in the Hilbert space $L^2(-1,1)$ and the operators
$S(N)$ are operators in the Hilbert space
$L(v^2)=L^2((-1,1),v^2)$.

We denote the Lagrange forms associated with equations
\eqref{eq0.1} and \eqref{eq2.5} by
\begin{equation}
[ y,f]_{M}=yp\overline{f'}-\overline{f}py',\quad
y,f\in D_{\rm max}(M), \label{eq3.15}
\end{equation}
and by
\begin{equation}
[ z,g]_{N}=zP\overline{g}'-\overline{g}Pz',\quad
z,g\in D_{\rm max}(N),\;P=v^2p, \label{eq3.16}
\end{equation}
respectively.

\subsection*{Notation}
We say that $D(N)$ is a self-adjoint domain for \eqref{eq2.5} if
the operator with this domain is a self-adjoint realization of
\eqref{eq2.5} in the Hilbert space $L(v^2)$. Similarly, $D(M)$
is a self-adjoint domain for \eqref{eq0.1} if the operator with
this domain is a self-adjoint realization of \eqref{eq2.1} in the
Hilbert space $L^2(-1,1)$.


\begin{theorem}\label{T3.2}
Let \eqref{eq0.1} and \eqref{eq2.5} hold; let $v$ be
given by \eqref{eq0.6}.
\begin{enumerate}
\item A function $z\in D_{\rm max}(N)$ if and only if
$v z\in D_{\rm max}(M)$.

\item $D(N)$ is a self-adjoint domain for \eqref{eq2.5}
if and only if $D(M)=\{y=vz:z\in D(M)\}$.

\item In particular we have a new characterization of the
Friedrichs domain
for \eqref{eq0.1}
\[
D(S_{F}(M))=\{vz:z\in D_{\rm max}(N):\;z(-1)=0=z(1)\}.
\]
\end{enumerate}
\end{theorem}

\begin{proof}
Assume $y,\,f\in D_{\rm max}(M)$.
Let $z=\frac{y}{v},\;g=\frac{f}{v}$. Then we
have
\begin{equation}
\begin{aligned}
[ z,g]_{N}
&  =[\frac{y}{v},\frac{f}{v}]_{N}=\frac{y}{v}P(\frac
{\overline{f}}{v})'-\frac{\overline{f}}{v}P(\frac{y}{v})'\\
&  =\frac{y}{v}pv^2\frac{v\overline{f'}-\overline{f}v'
}{v_2}-\frac{\overline{f}}{v}pv^2\frac{vy'-yv'}{v_2}\\
&  =yp\overline{f'}-\frac{y}{v}p\overline{f}v'-\overline
{f}py'+\frac{y}{v}p\overline{f}v'\\
&  =yp\overline{f'}-\overline{f}py'=[y,f]_{M}
\end{aligned}  \label{eq3.17}
\end{equation}
Part (2) follows from (1) and \eqref{eq3.17}. To prove (1). Assume
\[
y\in D_{\rm max}(M)=\{y\in L^2(J_2):py'\in AC_{\rm loc}(J_2
),\;My=(py')'\in L^2(J_2)\}.
\]
We must show that
\[
z\in D_{\rm max}(N)=\{z\in L^2(v^2),\;Pz'\in AC_{\rm loc}(J_2
),\;Nz=(Pz')'\in L^2(v^2)\}.
\]
Note that
\begin{gather*}
\int_{-1}^{1}|z^2|v^2=\int_{-1}^{1}|y^2|<\infty, \\
Pz'=v(py')-y(pv')=v(py')-y\in AC_{\rm loc} (J_2), \\
(Pz')'=v'py'+v(py')'
-y'(pv')-y(pv)'=vMy\in L^2(v^2).
\end{gather*}
The converse follows similarly by reversing the steps in this argument.
\end{proof}

\subsection{Eigenvalue Properties}

In this subsection we study the variation of the eigenvalues as
functions of the boundary conditions for the Legendre problem
consisting of the equation
\begin{equation}
My=-(py')'=\lambda y\quad\text{on }J_2=(-1,1),\;p(t)=1-t^2
,\;-1<t<1, \label{eq3.17.1}
\end{equation}
together with the boundary conditions
\begin{equation}
A\begin{pmatrix}
(-py')(-1)\\
(ypv'-v(py'))(-1)
\end{pmatrix}
  +B\begin{pmatrix}
(-py')(1)\\
(ypv'-v(py'))(1)
\end{pmatrix}
 =\begin{pmatrix}
0\\
0
\end{pmatrix}. \label{eq3.17.2}
\end{equation}
Here $v$ is given by \eqref{eq0.6} near the endpoints and the
matrices $A,B$ satisfy \eqref{eq3.10}, \eqref{eq3.11}.

 Since the homogeneous boundary conditions \eqref{eq3.17.2} are
invariant under multiplication by a nonsingular matrix, to study
the dependence of the eigenvalues on the boundary conditions it is
very useful to have a canonical represention of them. For such a
representation it is convenient to classify the boundary
conditions into two mutually exclusive classes: separated and
coupled. The separated conditions have the form \cite{zett05}
\begin{equation}
\begin{gathered}
\cos(\alpha)[y,u](-1)+\sin(\alpha)[y,v](-1)   =0,\quad
0\leq\alpha <\pi,\\
\cos(\beta)[y,u](1)+\sin(\beta)[y,v]\,(1)    =0,\quad
0<\beta\leq\pi.
\end{gathered} \label{eq3.17.3}
\end{equation}
The coupled conditions have the  canonical representation
\cite{zett05}
\begin{equation}
Y(1)=e^{i\gamma}K\,Y(-1),\; \label{eq3.17.4}
\end{equation}
where
\begin{equation}
Y=\begin{pmatrix}
[ y,u]\\
[ y,v]
\end{pmatrix}  ,\quad
-\pi<\gamma\leq\pi,\;K\in SL_2(\mathbb{R}); \label{eq3.17.5}
\end{equation}
i.e., $K=(k_{ij})$, $k_{ij}\in\mathbb{R}$, and $\det(K)=1$.

\begin{definition} \label{def4} \rm
The boundary conditions \eqref{eq3.17.3} are called
\emph{separated}\ and \eqref{eq3.17.4} are \emph{coupled}; if
$\gamma=0$ we say they are \emph{real coupled} and with
$\gamma\neq0$ they are \emph{complex coupled}.
\end{definition}

\begin{theorem} \label{thm5}
Let $S$ be a self-adjoint Legendre operator in $L^2(-1,1)$
according to Theorem \eqref{T3.1} and denote its spectrum by
$\sigma(S)$.
\begin{enumerate}
\item Then the boundary conditions determining $S$ are either given by
\eqref{eq3.17.3} or by \eqref{eq3.17.4} and each such boundary
condition determines a self-adjoint Legendre operator in
$L^2(-1,1)$.

\item The spectrum $\sigma(S)=\{\lambda_{n}:n\in N_{0}=\{0,1,2,\cdot\cdot
\cdot\}\}$ is real, discrete, and can be ordered to satisfy
\begin{equation}
-\infty<\lambda_{0}\leq\lambda_1\leq\lambda_2\leq\dots
\label{eq3.17.6}
\end{equation}
Here equality cannot hold for two consecutive terms.

\item If the boundary conditions are separated, then strict inequality holds
everywhere in \eqref{eq3.17.6} and if \ $u_{n}$ is an
eigenfunction of $\lambda_{n}$ then $u_{n}$ is unique up to
constant multiples and has exactly $n$ zeros in the open interval
$(-1,1)$ for each $n=0,1,2,3,\dots $

\item If the boundary conditions are coupled and real $(\gamma=0)$ and $u_{n}$
is a real eigenfunction of $\lambda_{n}$, then the number of zeros
of $u_{n}$ in the open interval $(-1,1)$ is $0$ or $1$ if $n=0$
and $n-1$ or $n$ or $n+1$, if $n\geq1$. (Note that, although there
may be eigenvalues of multiplicity $2$ and therefore some
ambiguity in the indexing of the eigenvalues, the eigenfunctions
$u_{n}$ are uniquely defined, up to constant multiples.)

\item If the boundary conditions are coupled and complex $(\gamma\neq0)$ then
all eigenvalues are simple and strict inequality holds in
\eqref{eq3.17.6}. If $u_{n}$ is an eigenfunction of $\lambda_{n}$,
then $u_{n}$ has no zero in the closed interval $[-1,1]$. The
number of zeros of both the real part $\operatorname{Re}(u_{n})$
and of the imaginary part $\operatorname{Im} (u_{n})$ in the
half-open interval $[-1,1)$ is $0$ or $1$ if $n=0$ and is $n-1$ or
$n$ or $n+1$ if $n\geq1$.

\item If the boundary condition is the classical condition
\begin{equation}
(py')(-1)=0=(py')(1), \label{eq3.17.7}
\end{equation}
then the eigenvalues are given by
\[
\lambda_{n}=n(n+1),\quad n\in\mathbb{N}_{0}=\{0,1,2,3,\dots \}
\]
and the normalized eigenfunctions are the classical Legendre
polynomials.

\item For any boundary conditions, separated, real coupled or
complex coupled, we have
\begin{equation}
\lambda_{n}\leq n(n+1),\quad n\in\mathbb{N}_{0}=\{0,1,2,3,\dots \}.
\label{eq3.17.8}
\end{equation}
In other words, the eigenvalues of the self-adjoint Legendre
operator determined by the classical boundary conditions
\eqref{eq3.17.7} maximize the eigenvalues of all other
self-adjoint Legendre operators.

\item For any self-adjoint boundary conditions, separated,
real coupled or complex coupled, we have
\begin{equation}
n(n+1)\leq\lambda_{n+2},\;n\in\mathbb{N}_{0}=\{0,1,2,3,\dots \}.
\label{eq3.17.9}
\end{equation}
In other words, the n-th eigenvalue of the self-adjoint Legendre
operator determined by the classical boundary conditions
\eqref{eq3.17.7} is a lower bound of $\lambda_{n+2}$ for all other
self-adjoint Legendre operators. These bounds are precise:

\item The range of $\lambda_{0}(S)=(-\infty,0]$ as $S$ varies over all
self-adjoint Legendre operators in $L^2(-1,1)$.

\item The range of $\lambda_1(S)=(-\infty,0]$ as $S$ varies over all
self-adjoint Legendre operators in $L^2(-1,1)$.

\item The range of $\lambda_{n}(S)=((n-2)(n-1),n\,(n+1)]$ as $S$ varies over
all self-adjoint Legendre operators in $L^2(-1,1)$.

\item The last three statements about the range of the eigenvalues are still
valid if the operators $S$ are restricted to those determined by real boundary
condition only.

\item Assume $S$ is any self-adjoint Legendre operator in $L^2(-1,1)$,
determined by separated, real coupled or complex coupled, boundary conditions
and let $\sigma(S)=\{\lambda_{n}:n\in N_{0}=\{0,1,2,\dots \}\}$
denote its spectrum. Then
\[
\frac{\lambda_{n}}{n^2}\to1,\quad\text{as }n\to\infty.
\]
\end{enumerate}
\end{theorem}

\begin{proof}
Part (7); i.e., \eqref{eq3.17.8}, is the well known classical
result about the Legendre equation and its polynomial solutions.
All the other parts follow from applying the known corresponding
results for regular problems, see \cite[Chapter 4]{zett05}, to the
above regularization of the singular Legendre equation.
\end{proof}

\subsection{Legendre Green's Function}

In this subsection we construct the Legendre Green's function. This seems to
be new even though, as mentioned in the Introduction, the Legendre equation
\begin{equation}
-(py')'=\lambda y,\;p(t)=1-t^2,\quad\text{on }J=(-1,1) \label{e0.1}
\end{equation}
is one of the simplest singular differential equations. Its
potential function $q$ is zero, its weight function $w$ is the
constant $1$, and its leading coefficient $p$ is a simple
quadratic. It is singular at both endpoints $-1$ and $+1$. These
singularities are due to the fact that $1/p$ is not Lebesgue
integrable in left and right neighborhoods of these points.

Our construction of the Legendre Green's functions is a five
step procedure:

\begin{enumerate}
\item Formulate the singular second order scalar
 equation \eqref{e0.1} as a
first order singular system.

\item `Regularize' this singular system by constructing regular
systems which
are equivalent to it.

\item Construct the Green's matrix for boundary value problems
of the regular system.

\item Construct the singular Green's matrix for the equivalent singular system
from the regular one.

\item Extract the upper right corner element from the singular
Green's matrix.
This is the Green's function for singular scalar boundary value
problems for equation \eqref{e0.1}.
\end{enumerate}

For the convenience of the reader we present these five steps here
even though some of them were given above.

For $\lambda=0$ recall the two linearly independent solutions
$u,v$ of \eqref{e0.1} given by
\begin{equation}
u(t)=1,\quad v(t)=\frac{-1}{2}\ln(|\frac{1-t}{t+1}|) \label{e0.6}
\end{equation}

The standard system formulation of \eqref{e0.1} has the form
\begin{equation}
Y'=(P-\lambda W)Y,\quad\text{on }(-1,1) \label{e0.8}
\end{equation}
where
\begin{equation}
Y=\begin{pmatrix}
y\\
py'
\end{pmatrix}  ,\quad
P=\begin{pmatrix}
0 & 1/p\\
0 & 0
\end{pmatrix} ,\quad
W=\begin{pmatrix}
0 & 0\\
1 & 0
\end{pmatrix}  \label{e0.9}
\end{equation}
Let
\begin{equation}
U=\begin{pmatrix}
u & v\\
pu' & pv'
\end{pmatrix}
  =\begin{pmatrix}
1 & v\\
0 & 1
\end{pmatrix}  . \label{e0.10}
\end{equation}
Note that $\det U(t)=1$, for $t\in J=(-1,1)$, and set
\begin{equation}
Z=U^{-1}Y. \label{e0.11}
\end{equation}
Then
\begin{align*}
Z'  &  =(U^{-1})'Y+U^{-1}Y'=-U^{-1}U'
U^{-1}Y+(U^{-1})(P-\lambda W)Y\\
&  =-U^{-1}U'Z+(U^{-1})(P-\lambda W)UZ\\
&  =-U^{-1}(PU)Z+U^{-1}(PU)Z-\lambda(U^{-1}WU)Z=-\lambda(U^{-1}WU)Z.
\end{align*}
Letting $G=(U^{-1}WU)$ we may conclude that
\begin{equation}
Z'=-\lambda GZ,\; \label{e0.13}
\end{equation}
where
\begin{equation}
G=U^{-1}W U=\begin{pmatrix}
-v & -v^2\\
1 & v
\end{pmatrix} , \label{e0.14}
\end{equation}
Note that \eqref{e0.14} is the regularized Legendre system of
Section 1.

The next theorem summarizes the properties of \eqref{e0.14} and
their relationship to \eqref{e0.1}.

\begin{theorem}\label{T0}
Let $\lambda\in\mathbb{C}$ and let $G$ be given by
\eqref{e0.14}.
\begin{enumerate}
\item Every component of $G$ is in $L^{1}(-1,1)$ and therefore \eqref{e0.13}
is a \emph{regular} system.

\item For any $c_1,c_2\in\mathbb{C}$  the initial value problem
\begin{equation}
Z'=-\lambda GZ,\;Z(-1)=\begin{pmatrix}
c_1\\
c_2
\end{pmatrix}  \label{e0.15}
\end{equation}
has a unique solution $Z$ defined on the closed interval $[-1,1]$.

\item If $Y=\begin{pmatrix}
y(t,\lambda)\\
(py')(t,\lambda)
\end{pmatrix}  $ is a solution of \eqref{e0.8} and
$Z=U^{-1}Y=\begin{pmatrix}
z_1(t,\lambda)\\
z_2(t,\lambda)
\end{pmatrix}$, then $Z$ is a solution of \eqref{e0.13} and for all
$t\in(-1,1)$ we have
\begin{gather}
y(t,\lambda)    =u z_1(t,\lambda)+v(t) z_2(t,\lambda)=z_1
(t,\lambda)+v(t) z_2(t,\lambda)\label{e0.16}\\
(py')(t,\lambda)    =(pu') z_1(t,\lambda)+(pv^{\prime
})(t) z_2(t,\lambda)= z_2(t,\lambda) \label{e0.17}
\end{gather}


\item For every solution $y(t,\lambda)$ of the singular scalar Legendre
equation \eqref{e0.1} the quasi-derivative $(py')(t,\lambda)$ is
continuous on the compact interval $[-1,1]$. More specifically we
have
\begin{equation}
\lim_{t\to-1^{+}}(py')(t,\lambda)
=z_2(-1,\lambda ),\quad
\lim_{t\to1^{-}}(py')(t,\lambda)=z_2(1,\lambda).
\label{e0.18}
\end{equation}
Thus the quasi-derivative is a continuous function on the closed
interval $[-1,1]$ for every $\lambda\in\mathbb{C}$.

\item Let $y(t,\lambda)$ be given by \eqref{e0.16}.
If $z_2(1,\lambda)\neq0$ then $y(t,\lambda)$ is unbounded at $1$;
 If $z_2(-1,\lambda)\neq0$ then
$y(t,\lambda)$ is unbounded at $-1$.

\item Fix $t\in[-1,1]$, let $c_1,c_2\in\mathbb{C}$.
If $Z=\begin{pmatrix}
z_1(t,\lambda)\\
z_2(t,\lambda)
\end{pmatrix}$ is the solution of \eqref{e0.13} determined by the
initial conditions
$z_1(-1,\lambda)=c_1,\;z_2(-1,\lambda)=c_2$, then
$z_i(t,\lambda)$ is an entire function of $\lambda$, $i=1,2$.
Similarly for the initial condition
$z_1(1,\lambda)=c_1$, $z_2(1,\lambda)=c_2$.

\item For each $\lambda\in\mathbb{C}$ there is a nontrivial solution
which is bounded in a (two sided) neighborhood of $1$; and there
is a (generally different) nontrivial solution which is bounded
in a (two sided) neighborhood of $-1$.

\item A nontrivial solution $y(t,\lambda)$ of the singular scalar
Legendre equation \eqref{e0.1} is bounded at $1$ if and only if
$ z_2(1,\lambda)=0$; A nontrivial solution $y(t,\lambda)$ of
the singular scalar Legendre equation \eqref{e0.1} is bounded at
$-1$ if and only if $ z_2(-1,\lambda)=0$.
\end{enumerate}
\end{theorem}

\begin{proof}
Part (1) follows from \eqref{e0.14}, (2) is a direct consequence
of (1) and the theory of regular systems, $Y=UZ$ implies
(3)$\Longrightarrow$(4) and (5); (6) follows from (2) and the
basic theory of regular systems. For (7) determine solutions
$y_1(t,\lambda)$, $y_{-1}(t,\lambda)$ by applying the Frobenius
method to obtain power series solutions of \eqref{eq0.1} in the
form: (see \cite{evlm}, page 5 with different notations)
\begin{gather}
y_1(t,\lambda)   =1+\sum_{n=1}^{\infty}a_{n}(\lambda)(t-1)^{n}
,\quad |t-1|<2;\\
y_{-1}(t,\lambda)   =1+\sum_{n=1}^{\infty}b_{n}(\lambda
)(t+1)^{n},\quad |t+1|<2; \label{e0.19}
\end{gather}
To prove (8) it follows from \eqref{e0.16} that if
$ z_2(1,\lambda)\neq0$, then $y(t,\lambda)$ is not bounded at
$1$. Suppose $ z_2(1,\lambda)=0$. If the corresponding
$y(t,\lambda)$ is not bounded at $1$ then there are two linearly
unbounded solutions at $1$ and hence all nontrivial solutions are
unbounded at $1$. This contradiction establishes (8) and completes
the proof of the theorem.
\end{proof}

\begin{remark} \label{rmk8} \rm
From Theorem \ref{T0} we see that, for every
$\lambda\in$ $\mathbb{C}$, the equation \eqref{e0.1} has a
solution $y_1$ which is bounded at $1$ and has a solution
$y_{-1}$ which is bounded at $-1$. It is well known
that for $\lambda_{n}=n(n+1):n\in\mathbb{N} _{0}=\{0,1,2,\dots \}$
the Legendre polynimials $P_{n}$ are solutions on $(-1,1)$ and
hence are bounded at $-1$ and at $+1$.
\end{remark}

We now consider two point boundary conditions for \eqref{e0.13};
later we will relate these to singular boundary conditions for
\eqref{e0.1}.

Let $A,B\in M_2(\mathbb{C})$, the set of $2\times2$
complex matrices, and
consider the boundary value problem
\begin{equation}
Z'=-\lambda GZ,\quad AZ(-1)+B Z(1)=0. \label{e0.21}
\end{equation}
Recall that $\Phi(t,s,-\lambda)$ is the primary fundamental
matrix of the system $Z'=-\lambda GZ$ constructed in Section 1.

\begin{lemma}\label{L3.1.1}
A complex number $-\lambda$ is an eigenvalue of
\eqref{e0.21} if and only if
\begin{equation}
\Delta(\lambda)=\det(A+B\Phi(1,-1,-\lambda)=0. \label{e0.22}
\end{equation}
Furthermore, a complex number $-\lambda$ is an eigenvalue of geometric
multiplicity two if and only if
\begin{equation}
A+B\Phi(1,-1,-\lambda)=0. \label{e0.23}
\end{equation}
\end{lemma}

\begin{proof}
Note that a solution for the initial condition $Z(-1)=C$ is given by
\begin{equation}
Z(t)=\Phi(t,-1,-\lambda) C,\;t\in[-1,1]. \label{e0.24}
\end{equation}
The boundary value problem \eqref{e0.21} has a nontrivial solution
for $Z$ if and only if the algebraic system
\begin{equation}
[ A+B \Phi(1,-1,-\lambda)] Z(-1)=0 \label{e0.25}
\end{equation}
has a nontrivial solution for $Z(-1)$.

To prove the furthermore part, observe that two linearly
independent solutions of the algebraic system \eqref{eq0.25} for
$Z(-1)$ yield two linearly independent solutions $Z(t)$ of the
differential system and conversely.
\end{proof}

Given any $\lambda\in$ $\mathbb{R}$ and any solutions $y,z$ of
\eqref{e0.1} the Lagrange form $[y,z](t)$ is defined by
\[
[ y,z](t)=y(t)(p\overline{z'})(t)-\overline{z}(t)(py^{\prime
})(t).
\]
So, in particular, we have
\begin{gather*}
[ u,v](t)   =+1,[v,u](t)=-1,[y,u](t)=-(py')(t),\;t\in
\mathbb{R},\\
[y,v](t)   =y(t)-v(t)(py')(t),\quad t\in\mathbb{R},\;t\neq\pm1.
\end{gather*}


We will see below that, although $v$ blows up at $\pm1$, the
form $[y,v](t)$ is well defined at $-1$ and $+1$ since the limits
\[
\lim_{t\to-1}[y,v](t),\quad \lim_{t\to+1}[y,v](t)
\]
exist and are finite from both sides. This for any solution $y$ of
 \eqref{e0.1} for any $\lambda\in$ $\mathbb{R}$. Note
that, since $v$ blows up at $1$, this means that $y$ must blow up
at $1$ except, possibly when $(py')(1)=0$.

We are now ready to construct the Green's function of the singular scalar
Legendre problem consisting of the equation
\begin{equation}
My=-(py')'=\lambda y+h\quad\text{on }J=(-1,1),\quad
p(t)=1-t^2,\;-1<t<1, \label{e3.18}
\end{equation}
together with two point boundary conditions
\begin{equation}
A\begin{pmatrix}
(-py')(-1)\\
(ypv'-v(py'))(-1)
\end{pmatrix}
+B\begin{pmatrix}
(-py')(1)\\
(ypv'-v(py'))(1)
\end{pmatrix}
 =\begin{pmatrix}
0\\
0
\end{pmatrix} , \label{e3.19}
\end{equation}
where $u,v$ are given by \eqref{e0.6} and $A,B$ are $2\times2$
complex matrices. This construction is based on the system
regularization discussed above and we will use the notation from
above. Consider the regular nonhomogeneous system
\begin{equation}
Z'=-\lambda GZ+F,\quad AZ(-1)+BZ(1)=0. \label{e3.20}
\end{equation}
where
\begin{equation}
F=\begin{pmatrix}
f_1\\
f_2
\end{pmatrix}  ,\quad
f_{j}\in L^{1}(J,\mathbb{C}),\;j=1,2. \label{e3.21}
\end{equation}


\begin{theorem} \label{thm7}
Let $-\lambda\in\mathbb{C}$ and let $\Delta(-\lambda)=[A+B\Phi
(1,-1,-\lambda)]$. Then the following statements are equivalent:

\begin{enumerate}
\item For $F=0$ on $J=(-1,1)$, the homogeneous problem \eqref{e3.20} has only
the trivial solution.

\item $\Delta(-\lambda)$ is nonsingular.

\item For every $F\in L^{1}(-1,1)$ the nonhomogeneous
problem \eqref{e3.20} has a unique solution $Z$ and this
solution is given by
\begin{equation}
Z(t,-\lambda)=\int_{-1}^{1}K(t,s,-\lambda)\,F(s)ds,\quad
-1\leq t\leq1,
\label{e3.22}
\end{equation}
where
\[
K(t,s,-\lambda)\,=\begin{cases}
\Phi(t,-1,-\lambda)\Delta^{-1}(-\lambda)(-B)\Phi(1,s,-\lambda),\\
\quad\text{if }-1\leq t<s\leq1,\\
\Phi(t,-1,-\lambda)\Delta^{-1}(-\lambda)(-B)\Phi(1,s,-\lambda
)+\phi(t,s-\lambda),\\
\quad\text{if }-1\leq s<t\leq1,\\
\Phi(t,-1,-\lambda)\Delta^{-1}(-\lambda)(-B)\Phi(1,s,-\lambda)+\frac{1}
{2}\phi(t,s-\lambda),\\
\quad\text{if }-1\leq s=t\leq1.
\end{cases}
\]
\end{enumerate}
\end{theorem}

The proof is a minor modification of the Neuberger construction
given in \cite{neub60}; see also \cite{zett05}.

From the regular Green's matrix we now construct the singular
Green's matrix and from it the singular scalar Legendre Green's
function.

\begin{definition} \label{def5} \rm
Let
\begin{equation}
L(t,s,\lambda)=U(t)  K(t,s,-\lambda) U^{-1}(s),\quad
-1\leq t,s\leq1. \label{e3.24}
\end{equation}
\end{definition}

The next theorem shows that $L_{12}$, the upper right component of
$L$, is the Green's function of the singular scalar Legendre
problem \eqref{e3.18}, \eqref{e3.19}.

\begin{theorem}
Assume that $[A+B\Phi(1,-1,-\lambda)]$ is nonsingular.
Then for every function $h$ satisfying
\begin{equation}
h,\,vh\in L^{1}(J,\mathbb{C}), \label{e3.25}
\end{equation}
the singular scalar Legendre problem \eqref{e3.18},
\eqref{e3.19} has a unique solution $y(\cdot,\lambda)$ given by
\begin{equation}
y(t,\lambda)=\int_{-1}^{1}L_{12}(t,s)\,h(s)ds,\;-1<t<1. \label{e3.26}
\end{equation}
\end{theorem}

\begin{proof}
Let
\begin{equation}
F=\begin{pmatrix}
f_1\\
f_2
\end{pmatrix}
 =U^{-1}H,\quad H=\begin{pmatrix}
0\\
-h
\end{pmatrix} . \label{e3.27}
\end{equation}
Then $f_{j}\in L^{1}(J_2,\mathbb{C})$, $j=1,2$. Since
$Y(t,\lambda )=U(t) Z(t,-\lambda)$, from \eqref{e3.22} we obtain
\begin{equation}
\begin{aligned}
Y(t,\lambda)
&  =U(t) Z(t,-\lambda)=U(t)\int_{-1}^{1}K(t,s,-\lambda)\,F(s)ds\\
&  =\int_{-1}^{1}U(t) K(t,s,-\lambda)U^{-1}(s)\,H(s)ds\\
& =\int_{-1}^{1}L(t,s,\lambda)H(s)ds,\quad
 -1<t<1.
\end{aligned} \label{e3.28}
\end{equation}
Therefore,
\begin{equation}
y(t,\lambda)=-\int_{-1}^{1}L_{12}(t,s,\lambda)h(s)ds,\,-1<t<1.
\label{e3.29}
\end{equation}
\end{proof}

An important property of the Friedrichs extension $S_{F}$ is  the
well known fact that it has the same lower bound as the minimal
operator $S_{\rm min}$. \ But this fact does not characterize the
Friedrichs extension of $S_{\rm min}$. Haertzen, Kong, Wu and
Zettl \cite{hkwz00} characterized all self-adjoint regular
Sturm-Liouville operators which preserve the lower bound of the
minimal operator, see also \cite[Proposition 4.8.1]{zett05}. The
next theorem, characterizes the Legendre Friedrichs extension
$S_{F}$ uniquely.

\begin{theorem}
[Everitt, Littlejohn and Mari\'{c}]\label{T3.3}
Suppose that $S$ $\neq S_{F}$ is a self-adjoint Legendre
operator in $L^2(-1,1)$. Then there exists $f\in D(S)$ such that
\[
pf''\notin L^2(-1,1)\quad\text{and}\quad
f'\notin L^2(-1,1).
\]
\end{theorem}
A proof can be found in \cite{evlm}.

\section{Maximal and Friedrichs Domains}

In this section we develop properties of the maximal and Friedrichs
domains including various characterizations of them.
Recall that the maximal domain
$D_{\rm max}$ is defined as follows. Let $H=L^2(-1,1)$ and
\[
D_{\rm max}=\{y\in H:(py')\in AC_{\rm loc}(-1,1),\;(py')'\in
H.
\]
The next lemma describes maximal domain functions and their
quasi-derivatives.

\begin{lemma}\label{L4.0}
Let $v$ be given by \eqref{eq0.6}. For every $y\in D_{\rm max}$
there exist two constants $c,d\in\mathbb{C}$ and a
function $g\in H$ such that
\begin{gather}
y(t)  =c+d\,v(t)+\int_{-1}^{t}
[v(t)-v(s)]\,g(s)ds,\quad -1<t<1.\label{eq4.1}\\
(py')(t)  =d+\int_{-1}^{t}g(s)\,ds,\quad
-1<t<1. \label{eq4.1a}
\end{gather}
Conversely, for every $c,d\in\mathbb{C}$ and $g\in H$, the
function $y$ defined by \eqref{eq4.1} is in $D_{\rm max}$.
\end{lemma}

\begin{proof}
Suppose $y\in D_{\rm max}$. Then $(py')'\in H$. Let $(py')'=g$.
Since $u,v$ are linearly independent solutions of
$(py')'=0$, \eqref{eq4.1} follows directly from the variation of
parameters formula. (The integrals exist since $v\in H$ and $v\in
L^{1}(-1,1).)$ Differentiating \eqref{eq4.1} yields, for almost
all $t\in(-1,1)$,
\[
y'(t)=d\,v'(t)+v'(t)\int_{-1}^{t}g(s)ds.
\]
Multiplying by $p(t)$, noting that $(pv')(t)=1$, yields
\eqref{eq4.1a} . To prove the converse statement note that $y$ is
in $H$ since each term of \eqref{eq4.1} is in $L^2(-1,1)$.
Clearly, $(py')\in AC_{\rm loc} (-1,1)$, and $(py')'=g\in H$.
\end{proof}

\begin{corollary}\label{C4.1}
The quasi-derivative $(py')$ of every maximal domain
function $y$ can be continuously extended to the compact
interval $[-1,1]$ and is therefore continuous and bounded on $[-1,1]$.
\end{corollary}

The proof of the above corollary follows directly from
\eqref{eq4.1a}.

\begin{lemma}\label{L4.1}
Let $v$ be given by \eqref{eq0.6}. For every $y\in D_{\rm max}$
we have:
\begin{enumerate}
\item Both limits
\begin{equation}
\lim_{t\to-1^{+}}\frac{y(t)}{v(t)},\quad\text{and}\quad
\lim_{t\to1^{-} }\frac{y(t)}{v(t)} \label{eq4.2}
\end{equation}
exist and are finite.

\item For any $c,d$, $-1<c<0<d<1$,
\begin{equation}
(\sqrt{p})v\big(  \frac{y}{v'}\big)  \in L^2(-1,c),\quad
(\sqrt{p})v\big(  \frac{y}{v'}\big)  \in L^2(d,1). \label{eq4.3}
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
In Section 2 we showed that $z=y/v_{m}$ is a solution of the
regular Legendre equation \eqref{eq2.5}. Therefore $z$ can be
continuously extended to both endpoints. Since $v_{m}$ agrees with
$v$ near both endpoints \eqref{eq4.2} follows.  Part (2) follows
from \cite[Theorem 4.2, Page 558]{nize92}.
\end{proof}

Recall the definition of the Friedrichs domain $D_{F}$:
\begin{equation}
D_{F}=\{y\in D_{\rm max}:(py')(-1)=0=(py')(1)\}. \label{eq4.4}
\end{equation}


The next theorem gives a number of equivalent characterizations of the
Friedrichs domain; see also \cite{evlm} and \cite{KKZ}.

\begin{theorem}\label{T4.1}
Let $v$ be given by \eqref{eq0.6}. For any $y\in
D_{\rm max}$, the following statements are equivalent:
\begin{itemize}
\item[(i)] In \eqref{eq4.1} of Lemma \eqref{L4.0} the constant $d=0$.

\item[(ii)] $y$ is bounded on $(-1,1)$.

\item[(iii)] The limits
\[
\frac{y(t)}{v(t)}\to0\quad\text{as $t\to-1^{+}$, and as $t\to+1$}
\]
exist and are finite.

\item[(iv)]
\[
\lim_{t\to-1^{+}}(py')(t)=0=\;\lim_{t\to1^{-}}(py')(t).
\]

\item[(v)] The limits
\[
\lim_{t\to-1^{+}}y(t),\quad \lim_{t\to1^{-}}y(t)
\]
exist and are finite.

\item[(vi)] $y\in AC[-1,1]$;

\item[(vii)] $y'\in L^2(-1,1)$. Furthermore, this result
is best possible in that there exists $g\in D(S_{F})$ such
that $g'\notin L^{q}(-1,1)$
for any $q>2$ and where $g$ is independent of $q$.

\item[(viii)] $p^{1/2}y'\in L^2(-1,1)$;

\item[(ix)] For any $-1<c<0<d<1$ we have
\[
\frac{y}{(\sqrt{p})\,v}\in L^2(-1,c),\quad
\frac{y}{(\sqrt{p})\,v}\in L^2(d,1),\quad -1<c<0<d<1.
\]


\item[(x)] $y,y'\in AC_{\rm loc}(-1,1)$ and $py''\in
L^2(-1,1)$. Furthermore, this result is best possible in the
sense that there exists $g\in D(S_{F})$ such that
$pg''\notin L^{q}(-1,1)$ for any $q>2$, and where $g$
is independent of $q$.
\end{itemize}
\end{theorem}

\begin{proof}
The equivalence of (i), (ii), (iii), (v) and (vi) is
clear from \eqref{eq4.1} of Lemma \eqref{L4.0} and the definition
of $v(t)$ in \eqref{eq0.6}. We now prove the equivalence of (ii)
and (iv) by using the method used to construct regular Legendre
equations above. In particular we use the `regularizing' function
$v_{m}$ and other notation from Section 2. Recall that $v_{m}$
agrees with $v$ near both endpoints and is positive on $(-1,1)$.
As in Section 2, $[\cdot,\cdot]_{M}$ and $[\cdot,\cdot]_{N}$
denote the Lagrange brackets of $\ M$ and $N$, respectively. Let
$z=y/v$ and $x=u/v$. Then
\begin{align*}
-(py')(1)
&  =[\frac{y}{v_{m}},\frac{u}{v_{m}}]_{M}(1)=[z,x]_{N}(1)\\
&  =\lim_{t\to1}z(t)\lim_{t\to1}(Px')(1)-\lim
_{t\to1}x(t)\lim_{t\to1}(Pz')(1)\\
&=\lim_{t\to 1}z(t)\lim_{t\to1}(Px')(1)=0.
\end{align*}
All these limits exist and are finite since $N$ is a regular
problem. Since $u$ is a principal solution and $v$ is a
nonprincipal solution it follows that
$\lim_{t\to 1}x(t)=0$. The proof for\ the endpoint $-1$ is
entirely similar. Thus we have shown that $(ii)$ implies $(iv)$.
The converse is obtained by reversing the steps. Thus we conclude
that $(i)$ through $(vi)$ are equivalent. Proofs of $(vii)$,
$(viii)$, $(ix)$ and $(x)$ can be found in \cite{ALM}.
\end{proof}

\section{Results on the Intervals $(-\infty,-1)$ and $(1,+\infty)$}

Here we expand on the observations of Proposition \ref{P0} regarding
the interval $(1,\infty)$. Similar remarks apply to $(-\infty,-1)$
as can be seen
from the change of variable $t\to-t$. Consider
\begin{equation}
My=-(py')'=\lambda y\quad\text{on }J_3=(1,\infty),\;p(t)=1-t^2.
\label{eq6.1}
\end{equation}
Note that $p(t)<0$ for $t>1$; so to conform to the standard notation
for Sturm-Liouville problems we study the equivalent equation
\begin{equation}
Ny=-(ry')'=\xi y\quad\text{on }J_3=(1,\infty),\quad
r(t)=t^2 -1>0,\quad \xi=-\lambda. \label{eq6.2}
\end{equation}
Recall from \eqref{eq0.6} that for $\lambda=\xi=0$ two linearly
independent solutions are given by
\begin{equation}
u(t)=1,\quad v(t)=\frac{1}{2}\ln(|\frac{t-1}{t+1}|) \label{eq6.3}
\end{equation}

Although we focus on the interval $(1,\infty)$ in this section we
make the following general observations: For all $t\in\mathbb{R}$,
$t\neq\pm1$, we have
\begin{equation}
(pv')(t)=-1, \label{eq6.4}
\end{equation}
so for any $\lambda\in\mathbb{R}$ and any solution $y$ of
\eqref{eq0.1}, we have the following Lagrange forms:
\begin{equation}
[ y,u]=-py',\quad [y,v]=-y-v(py'),\quad [u,v]=-1,\quad [v,u]=1.
\label{eq6.5}
\end{equation}
These play an important role in the theory of self-adjoint Legendre
operators and problems. Observe that, although $v$ blows up at $-1$
 and at $+1$ from both sides it turns out that these forms are
defined and finite at all points of $\mathbb{R}$ including $-1$
and $+1$ provided we define the appropriate one
sided limits:
\begin{equation}
[ y,u](1^{+})=\lim_{t\to1^{+}}[y,u](t),\quad
[y,u](1^{+})=\lim_{t\to-1^{-}}[y,u](t) \label{eq6.6}
\end{equation}
for all $y\in D_{\rm max}(J_3)$. Since $u\in L^2(1,2)$ and $v\in
L^2(1,2)$ it follows from general Sturm-Liouville theory that
$1$, the left endpoint of $J_3$ is limit-circle non-oscillatory
(LCNO). In particular, all solutions of equations \eqref{eq6.1},
\eqref{eq6.2} are in $L^2(1,2)$ for each $\lambda\in\mathbb{C}$.

In the mathematics and physics literature, when a singular
Sturm-Liouville problem is studied on a half line $(a,\infty)$, it
is generally assumed that the endpoint $a$ regular. Here the left
endpoint $a=1$ is singular. Therefore regular conditions such as
$y(a)=0$ or, more generally,
\[
A_1y(a)+A_2(py')(a)=0,\quad A_1,A_2\in\mathbb{R},\;
(A_1 ,A_2)\neq(0,0)
\]
do not make sense. Interestingly, as pointed out above, in the Legendre case
studied here, while the Dirichlet condition
\[
y(1)=0
\]
does not make sense, the Neumann condition
\begin{equation}
(py')(1)=0, \label{eq6.7}
\end{equation}
does in fact determine a self-adjoint Legendre operator in
$L^2(1,\infty)$ - the Friedrichs extension! So while
\eqref{eq6.7} has the appearance of a regular Neumann condition it
is in fact, in the Legendre case, the analogue of the Dirichlet
condition!

By a self-adjoint operator associated with equation \eqref{eq6.2}
in $H_3=L^2(1,\infty)$; i.e., a self-adjoint realization of
equation \eqref{eq6.2} in $H_3$ we mean a self-adjoint
restriction of the maximal operator $S_{\rm max}$ associated with
\eqref{eq6.2}. This is defined as follows:
\begin{gather}
D_{\rm max}=\{f:(-1,1)\to\mathbb{C}\mid f,\;pf'\in AC_{\rm loc}
(-1,1); f,\,pf'\in H_3\} \label{eq6.8}
\\
S_{\rm max}f=-(rf')',\quad f\in D_{\rm max} \label{eq6.9}
\end{gather}
Note that, in contrast to the $(-1,1)$ case, the Legendre
polynomials are not in $D_{\rm max}$; nor are solutions of
\eqref{eq6.2} in general. As in the case for $(-1,1)$ the
following basic lemma holds:

\begin{lemma}\label{L6.1}
The operator $S_{\rm max}$ is densely defined in $H_3$ and
therefore has a unique adjoint in $H_3$ denoted by $S_{\rm min}$:
\[
S_{\rm max}^{\ast}=S_{\rm min}.
\]
The minimal operator $S_{\rm min}$ in $H_3$ is symmetric, closed, densely
defined, and satisfies
\[
S_{\rm min}^{\ast}=S_{\rm max}.
\]
Its deficiency index $d=d(S_{\rm min})=1$. If $S$ is a self-adjoint
extension of $S_{\rm min}$, then $S$ is also a self-adjoint
restriction of $S_{\rm max}$ and conversely. Thus we have:
\[
S_{\rm min}\subset S=S^{\ast}\subset S_{\rm max}.
\]
\end{lemma}

The statements in the above lemma are well known facts
from Sturm-Liouville theory; for details, see \cite{zett05}.

It is clear from Lemma \eqref{L6.1} that each self-adjoint
operator $S$ is determined by its domain. Next we describe these
self-adjoint domains. For this the functions $u$, $v$ given by
\eqref{eq6.3} play an important role, in a sense they form a basis
for all self-adjoint boundary conditions \cite{zett05}.

The Legendre operator theory for the interval $(1,\infty)$ is
similar to the theory on $(-1,1)$ except for the fact that the
endpoint $\infty$ is in the limit-point case and therefore there
are no boundary conditions required or allowed at $\infty$.

Thus all self-adjoint Legendre operators in $H_3=L^2(1,\infty)$ are
generated by separated singular self-adjoint boundary conditions at $1$. These
have the form
\begin{equation}
A_1[y,u](1)+A_2[y,v](1)=0,\quad
A_1,A_2\in\mathbb{R},\quad (A_1,A_2)\neq(0,0). \label{eq6.10}
\end{equation}


\begin{theorem}\label{T6.1}Let
$A_1,A_2\in\mathbb{R}$, $(A_1,A_2)\neq(0,0)$ and define a
linear manifold $D(S)$ to consist of all $y\in D_{\rm max}$
satisfying \eqref{eq6.10}. Then the operator $S$ with domain$\
D(S)$ is self-adjoint in $L^2(1,\infty)$. Moreover, given any
operator $S$ satisfying $S_{\rm min}\subset S=S^{\ast}\subset
S_{\rm max}$ there exist $A_1,A_2\in\mathbb{R}$,
$(A_1,A_2)\neq(0,0)$ such that $D(S)$, the domain
of $S$, is given by \eqref{eq6.10}.
\end{theorem}

The proof of the above theorem is based on the next three lemmas.

\begin{lemma}\label{L6.2}
Suppose $S_{\rm min}\subset S=S^{\ast}\subset S_{\rm max}$.
Then there exists a function $g\in D(S)\subset D_{\rm max}$
 satisfying
\begin{enumerate}
\item $g$ is not in $D_{\rm min}$ and

\item
$[ g,g](1)=0$
such that $D(S)$ consists of all $y\in D_{\rm max}$ satisfying

\item
\begin{equation}
[ y,g](1)=0. \label{eq6.12}
\end{equation}
\end{enumerate}

Conversely, given $g\in D_{\rm max}$ which satisfies conditions
(1) and (2), the set $D(S)\subset D_{\rm max}$ consisting of all
$y$ satisfying (3) is a
self-adjoint extension of $S_{\rm min}$.
\end{lemma}

The proof of the above lemma follows from the GKN theory (see
\cite{AG} and \cite{Naimark}) applied to \eqref{eq6.2}. The next
lemma plays an important role and is called the `Bracket
Decomposition Lemma' in \cite{zett05}.

\begin{lemma}[Bracket Decomposition Lemma] \label{lem10}
For any $y,z\in D_{\rm max}$ we have
\begin{equation}
[ y,z](1)=[y,v](1)[\overline{z},u](1)-[y,u](1)[\overline{z},v](1).
\label{eq6.13}
\end{equation}
\end{lemma}

For a proof of the above lemma, see \cite[Pages 175-176]{zett05}.

\begin{lemma}[\cite{zett05}]\label{Lemma 5.5}
 For any $\alpha,\beta$ $\in\mathbb{C}$ there exists a function
$g\in D_{\rm max}(J_3)$ such that
\begin{equation}
[g,u](1^{+})=\alpha,[g,v](1^{+})=\beta. \label{eq6.14}
\end{equation}
\end{lemma}

Armed with these lemmas we can now proceed to the proof.

\begin{proof}[Proof of Theorem \ref{T6.1}]
Let $A_1,A_2\in\mathbb{R}$, $(A_1,A_2 )\neq(0,0)$. By Lemma
\eqref{Lemma 5.5} there exists a $g\in D_{\rm max}(J_3)$ such that
\begin{equation}
[ g,u](1^{+})=A_2,[g,v](1^{+})=-A_1. \label{eqn6.15}
\end{equation}
From \eqref{eq6.13} we get that for any $y\in D_{\rm max}$ we have
\begin{equation}
[ y,g](1)=[y,v](1)[g,u](1)-[y,u](1)[g,v](1)=A_1[y,u](1)+A_2
[y,v](1).\,
\end{equation}
Now consider the boundary condition
\begin{equation}
A_1[y,u](1)+A_2[y,v](1)=0. \label{eq6.17}
\end{equation}
If \eqref{eq6.17} holds for all $y\in D_{\rm max}$, then it follows
from Lemma 10.4.1, p175 of \cite{zett05} that $g\in D_{\rm min}$. But
this implies, also by Lemma 10.4.1, that $(A_1,A_2)\neq(0,0)$
which is a contradiction. From \eqref{eqn6.15} it follows that
\begin{align*}
[ g,g](1)  &  =[g,v](1)[g,u](1)-[g,u](1)[g,v](1)\\
&  =A_1[g,u](1)+A_2[g,v](1)=A_1A_2-A_2A_1=0.
\end{align*}
Therefore $g$ satisfies conditions (1) and (2) of Lemma \ref{L6.2} and
consequently
\begin{equation}
[ y,g](1)=A_1[y,u](1)+A_2[y,v](1)=0 \label{eq6.18}
\end{equation}
is a self-adjoint boundary condition.
To prove the converse, reverse the steps in this argument.
\end{proof}

It is clear from Theorem \eqref{T6.1} that there are an
uncountable number of self-adjoint Legendre operators in
$L^2(1,\infty)$. It is also clear that the Legendre polynomials
$P_{n}$ are not eigenfunctions of any such operator since they are
not in the maximal domain and therefore not in the domain of any
self-adjoint restriction $S$ of $D_{\rm max}$.

Next we study the spectrum of the self-adjoint Legendre operators in
$H_3=L^2(1,\infty)$.

\begin{theorem}\label{T6.2}
Let $S_{\rm min}\subset S=S^{\ast}\subset S_{\rm max}$ where
$S_{\rm min}$ and $S_{\rm max}$ are the minimal and maximal
operators in $L^2(1,\infty)$ associated with  \eqref{eq0.1}. Then
\begin{itemize}
\item $S$ has no discrete spectrum.

\item The essential spectrum $\sigma_{e}(S)$ is given by
$\sigma_{e}(S)=(-\infty,-\frac{1}{4}]$.
\end{itemize}
\end{theorem}

The proof of the above lemma is given in Proposition \ref{P0}. The
next theorem gives the version of Theorem \eqref{T6.2} for the
Legendre equation in the more commonly used form \eqref{eq0.5}.

\begin{theorem}\label{T6.3}
Let $S_{\rm min}\subset S=S^{\ast}\subset S_{\rm max}$ where
$S_{\rm min}$ and $S_{\rm max}$ are the minimal and maximal operators in
$L^2(1,\infty)$ associated with the equation \eqref{eq0.5}. Then
\begin{itemize}
\item $S$ has no discrete spectrum.

\item The essential spectrum $\sigma_{e}(S)$ is given by
$\sigma_{e}(S)=[\frac{1}{4},\infty)$.
\end{itemize}
\end{theorem}

The above theorem is obtained from the preceding theorem simply
by changing the sign.

\section{Legendre operators on the whole line}

In this section we study the Legendre equation \eqref{eq0.1} on
the whole real line $\mathbb{R}$ and note that, in addition to its
singular points at $-\infty$ and $+\infty$, it also has
singularities at the interior points $-1$ and $+1;$we refer to the
paper of Zettl \cite{zett} for further details in this setting.
Since we are studying the equation on both sides of these interior
singularities there are in effect interior singularities at
$-1^{-}$, $-1^{+}$ and at $+1^{-}$, $+1^{+}$. Our approach is
based on the direct sum method developed by Everitt and Zettl
\cite{evze86} for one interior singular point. The modifications
needed to apply this approach to two interior singularities, as we
do here, is straightforward. This method yields, in a certain
natural sense, all self-adjoint Legendre operators in the Hilbert
space $L^2(\mathbb{R})$ which we identify with the direct sum
\[
L^2(\mathbb{R})=L^2(-\infty,-1)\dotplus L^2(-1,1)\dotplus L^2
(1,\infty).
\]
One method for getting such operators is to simply take the direct sum of
three operators, one from each of the three separate spaces. However it is
interesting to note that not all self-adjoint operators in $L^2(\mathbb{R})$
are generated by such direct sums. This is what makes the three-interval
theory interesting: there are many other self-adjoint operators. These are
generated by interactions \emph{through }the interior singularities.

As above, let
\[
J_1=(-\infty,-1),\quad J_2=(-1,1),\quad
J_3=(1,\infty),\quad J_4=\mathbb{R} =(-\infty,\infty).
\]
Let $S_{\rm min}(J_i)$, $S_{\rm max}(J_i)$ denote the minimal and
maximal operators in $L^2(J_i)$, $i=1,2,3$ and denote their
domains by $D_{\min }(J_i)$, $D_{\rm max}(J_i)$, respectively.

\begin{definition} \label{def6} \rm
\label{D7.1}The minimal and maximal Legendre operators $S_{\rm min}$
and $S_{\rm max}$ in $L^2(\mathbb{R})$ and their domains
$D_{\rm min}$, $D_{\rm max}$ are defined as follows:
\begin{gather*}
D_{\rm min}   =D_{\rm min}(J_1)\dotplus D_{\rm min}(J_2)\dotplus D_{\rm min}
(J_3)\\
D_{\rm max}   =D_{\rm max}(J_1)\dotplus D_{\rm max}(J_2)\dotplus D_{\rm max}
(J_3)\\
S_{\rm min}   =S_{\rm min}(J_1)\dotplus S_{\rm min}(J_2)\dotplus S_{\rm min}
(J_3)\\
S_{\rm min}   =S_{\rm max}(J_1)\dotplus S_{\rm max}(J_2)\dotplus S_{\rm max}(J_3).
\end{gather*}
\end{definition}

\begin{lemma}\label{L7.1}
The minimal operator $S_{\rm min}$ is a closed, densely defined,
symmetric operator in $L^2(\mathbb{R})$ satisfying
\[
S_{\rm min}^{\ast}=S_{\rm max},\;S_{\rm max}^{\ast}=S.
\]
Its deficiency index, $d=d(S_{\rm min})=4$. Each self-adjoint
extension $S$ of $S_{\rm min}$ is a restriction of $S_{\rm max}$;
i.e., we have
\[
S_{\rm min}\subset S=S^{\ast}\subset S_{\rm max}.
\]
\end{lemma}

\begin{proof}
The adjoint properties follow from the corresponding properties of the
component operators and it follows that
\[
\operatorname{def}(S_{\rm min})
 =\operatorname{def}(S_{\rm min}(J_1))
+\operatorname{def}(S_{\rm min}(J_2))+\operatorname{def}(S_{\min
}(J_3))
  =1+2+1=4,
\]
since $-\infty$ and $+\infty$ are LP and $-1^{-}$, $-1^{+}$,
$+1^{-}$, $+1^{+}$ are all LC. For more details, see
\cite{evze86}.
\end{proof}

\begin{remark} \label{rmk9} \rm
Although the minimal and maximal operators $S_{\rm min}$, $S_{\rm max}$
are the direct sums of the corresponding operators on each of the
three intervals we will see below that there are many self-adjoint
extensions $S$ of $S_{\rm min}$ other than those which are simply
direct sums of operators from the three intervals.
\end{remark}

For $y,z\in D_{\rm max}$ , $y=(y_1,y_2,y_3)$,
$z=(z_1,z_2,z_3)$ we define the ``three interval''
 or ``whole line''  Lagrange sesquilinear from $[\cdot,\cdot]$
as follows:
\begin{equation} \label{eq7.1}
\begin{aligned}
[ y,z]  &  =[y_1,z_1]_1(-1^{-})-[y_1,z_1]_1(-\infty
)+[y_2,z_2]_2(+1^{-})-[y_2,z_2]_2(-1^{+})\\
&  +[y_3,z_3]_3(+\infty)-[y_3,z_3]_3(+1^{+})\\
&  =[y_1,z_1]_1(-1^{-})+[y_2,z_2]_2(+1^{-})-[y_2,z_2
]_2(-1^{+})-[y_3,z_3]_3(+1^{+}).
\end{aligned}
\end{equation}
Here $[y_i,z_i]_i$ denotes the Lagrange form on the interval
$J_i$, $i=1,2,3$. In the last step we noted that the Lagrange
forms evaluated at $-\infty$ and at $+\infty$ are zero because
these are LP endpoints. The fact that each of these one sided
limits exists and is finite follows from the one interval theory.

As noted above in \eqref{eq0.6} for $\lambda=0$ the Legendre
equation
\begin{equation}
My=-(py')'=\lambda y \label{eq7.2}
\end{equation}
has two linearly independent solutions
\[
u(t)=1,\quad v(t)=-\frac{1}{2}\ln(|\frac{t-1}{t+1}|).
\]
Observe that $u$ is defined on all of $\mathbb{R}$ but $v$ blows up
logarithmically at the two interior singular points from both sides.
Observe that
\begin{equation}
[ u,v](t)=u(t)(pv')(t)-v(t)(pu')(t)=1,\quad
-\infty<t<\infty\label{eq7.3}
\end{equation}
where we have taken appropriate one sided limits at $\pm1$ and
for all $y\in D$ we have
\begin{equation}
[ y,u]=-py',\quad [y,v]=y-v(py') \label{eq7.4}
\end{equation}
and again by taking appropriate one sided limits, if necessary,
$[y,u](t)$ is defined (finitely) for all $t\in\mathbb{R}$.
Similarly the vector
\begin{equation}
Y=\begin{pmatrix}
[ y,u]\\
[ y,v]
\end{pmatrix}
  =\begin{pmatrix}
-py'\\
y-v(py')
\end{pmatrix}  \label{eq7.5}
\end{equation}
is well defined. In particular,
\begin{equation}
Y(-1^{-}),\quad Y(-1^{+}),\quad Y(1^{-}),\quad Y(1^{+}) \label{eq7.6}
\end{equation}
are all well defined and finite. Note also that $Y(-\infty)$
and $Y(-\infty)$ are well defined and
\begin{equation}
Y(-\infty)=\begin{pmatrix}
0\\
0
\end{pmatrix}
 =Y(\infty). \label{eq7.7}
\end{equation}


\begin{remark} \label{rmk10} \rm
For any $y\in D_{\rm max}$ the one sided limits of  $py'$ and of
$y-v(py')$ exist and are finite at $-1$ and at $1$. Hence if
$py'$ has a nonzero finite limit then $y$ must blow up logarithmically.
\end{remark}

Now we can state the theorem giving the characterization of all
self-adjoint extensions $S$ of the minimal operator
$S_{\rm min}$; recall that these are all
operators $S$ satisfying
$S_{\rm min}\subset S=S^{\ast}\subset S_{\rm max}$.

\begin{theorem} \label{T7.1}
Suppose $A=(a_{ij})$, $B=(b_{ij})$, $C=(c_{ij}),\;D=(d_{ij})$,
 are $4\times2$ complex
matrices satisfying the following two conditions:
\begin{gather}
\operatorname{rank}(A,B,C,D)=4, \label{eq7.8} \\
AEA^{\ast}-BEB^{\ast}+CEC^{\ast}-DED^{\ast}=0,\quad
E=\begin{pmatrix}
0 & -1\\
1 & 0
\end{pmatrix}  . \label{eq7.9}
\end{gather}
Componentwise, conditions \eqref{eq7.9} are written for
$j,k=1,2,3,4$, as
\[
(a_{j1}\overline{a}_{k2}-a_{j2}\overline{a}_{k1})-(b_{j1}\overline{b}
_{k2}-b_{j2}\overline{b}_{k1})+(c_{j1}\overline{c}_{k2}-c_{j2}\overline
{c}_{k1})-(d_{j1}\overline{d}_{k2}-d_{j2}\overline{d}_{k1})=0.
\]

Define $D$ to be the set of all $y\in D_{\rm max}$ satisfying
\begin{equation}
AY(-1^{-})+BY(-1^{+})+CY(1^{-})+DY(1^{+})=0, \label{eq7.10}
\end{equation}
where
\[
Y=\begin{pmatrix}
-py'\\
y-v(py')
\end{pmatrix}  .
\]
Then $D$ is the domain of a self-adjoint extension $S$ of the
three interval minimal operator $S_{\rm min}$.

Conversely, given any self-adjoint operator $S$ satisfying
$S_{\rm min}\subset S=S^{\ast}\subset S_{\rm max}$ with
domain $D=D(S)$, there exist $2\times4$ complex matrices
$A=(a_{ij})$,
$B=(b_{ij}),\;C=(c_{ij}),\;D=(d_{ij})$, satisfying conditions
\eqref{eq7.8} and \eqref{eq7.9},
 such that $D(S)$ is given by \eqref{eq7.10}.
\end{theorem}

The proof of the above theorem is based on three lemmas which
we establish next. Also see Example
\cite[13.3.4, pp. 273-275]{zett05}.

\begin{remark} \label{rmk11} \rm
The boundary conditions are given by \eqref{eq7.10}; \eqref{eq7.8}
determines the \emph{number} of independent conditions and
\eqref{eq7.9} specifies the conditions on the boundary conditions
needed for self-adjointness.
\end{remark}

Using the three interval Lagrange form the next lemma gives an
extension of the GKN characterization for the whole line Legendre
 problem.

\begin{lemma}\label{L7.2}
Suppose $S_{\rm min}\subset S=S^{\ast}\subset S_{\rm max}$. Then there
exist $v_1,v_2,v_3,v_4\in D(S)\subset D_{\rm max}$
satisfying conditions
\begin{enumerate}
\item $v_1,v_2,v_3,v_4$ are linear independent modulo $D_{\rm min}$; i.e.,
no nontrivial linear combination is in $D_{\rm min}$;
\item $[ v_i,v_{j}]=0,\;i,j=1,2,3,4$, % \label{eq7.11}
such that $D(S)$ consists of all $y\in D_{\rm max}$ satisfying

\item $[ y,v_{j}]=0,\;j=1,2,3,4$. %\label{eq7.12}
\end{enumerate}
Conversely, given $v_1,v_2,v_3,v_4\in D_{\rm max}$ which satisfy
conditions (1) and (2) the set $D(S)\subset D_{\rm max}$
consisting of all $y$
satisfying (3) is a self-adjoint extension of $S_{\rm min}$.
\end{lemma}

The above lemma follows from \cite[Theorem 3.3]{evze86} extended
to three intervals and applied to the Legendre equation. The next
lemma is called the `Bracket Decomposition' Lemma in
\cite{zett05}. It applies to each of the intervals $J_i$,
$i=1,2,3$ but for simplicity of notation we omit the subscripts.

\begin{lemma}[Bracket Decomposition Lemma]
Let $J_i=(a,b)$, let $y,z,u,v\in
D_{\max }=D_{\rm max}(J_i)$, $J_i=(a,b)$ and assume that
$[v,u](c)=1$ for some $c$, $a\leq c\leq b$, then
\begin{equation}
[ y,z](c)=[y,v](c)[\overline{z},\overline{u}](c)-[y,\overline
{u}](c)[\overline{z},v](c). \label{eq7.13}
\end{equation}
\end{lemma}

For a proof of the above lemma, see \cite[Pages 175-176]{zett05}.
The next lemma extends \cite[Proposition 10.4.2, Pages
185-186]{zett05} from the one interval case to the three intervals
$J_i$, $i=1,2,3$.

For this lemma we extend the definitions of the functions $u,v$
given by \eqref{eq0.6} but we will continue to use the same
notation.
\begin{gather}
u(t)=\begin{cases}
1 & -1<t<1 ,\; -2<t<-1 ,\; 1<t<2,\\
0 & |t|>3,
\end{cases}  \label{eq7.14}
\\
v(t)=\begin{cases}
-\frac{1}{2}\ln(\frac{t-1}{t+1}) & -1<t<1 ,\; -2<t<-1 ,\; 1<t<2,\\
0 & |t|>3,
\end{cases} \label{eq7.15}
\end{gather}
and define both functions on the intervals $[-3,-2]$, $[2,3]$
so that they are
continuously differentiable on these intervals.

\begin{lemma}
Let $\alpha,\beta,\gamma,\delta$ $\in\mathbb{C}$.
\begin{itemize}
\item There exists a $g\in D_{\rm max}(J_2)$ which is not
in $D_{\rm min}(J_2)$ such that
\begin{equation}
[ g,u](-1^{+})=\alpha,[g,v](-1^{+})=\beta,[g,u](1^{+})=\gamma
,[g,v](1^{+})=\delta. \label{eq7.16}
\end{equation}


\item There exists a $g\in D_{\rm max}(J_1)$ which is not
in $D_{\rm min}(J_1)$ such that
\begin{equation}
[ g,u](-1^{-})=\alpha,[g,v](-1^{-})=\beta. \label{eq7.17}
\end{equation}

\item There exists a $g\in D_{\rm max}(J_3)$ which is not
in $D_{\rm min}(J_3)$ such that
\begin{equation}
[g,u](1^{+})=\gamma,[g,v](1^{+})=\delta. \label{eq7.18}
\end{equation}
\end{itemize}
\end{lemma}

\begin{proof}[Proof of Theorem \ref{T7.1}]
The method is the same as the method used in the proof of
Theorem \eqref{T6.1} but the computations are longer; it consists
in showing that each part of Theorem \eqref{T7.1} is equivalent to
the corresponding part of Lemma \eqref{L7.2}. For more details,
see \cite{zett05}.
\end{proof}


\begin{example} \label{exmp1} \rm
A Self-Adjoint Legendre Operator on the whole real line. The boundary
condition
\begin{equation}
(py')(-1^{-})=(py')(-1^{+})=(py')(1^{-})=(py^{\prime
})(1^{+})=0 \label{eq7.19}
\end{equation}
satisfies the conditions of Theorem \eqref{T7.1} and therefore
determines a self-adjoint operator $S_L$ in $L^2(\mathbb{R})$.
Let $S_1$ in $L^2(-\infty,-1)$ be determined by
$(py')(-1^{-})=0$ , $S_2=S_{F}$ in $(-1,1)$ by
$(py')(-1^{+})=(py')(1^{-})=0$, and $S_3$ by $(py')(1^{+})=0$.
then each $S_i$ is self-adjoint and the direct sum:
\begin{equation}
S=S_1\dotplus S_2\dotplus S_3 \label{eq7.20}
\end{equation}
is a self-adjoint operator in $L^2(-\infty,\infty)$. It is well
known that the essential spectrum of a direct sum of operators is
the union of the essential spectra of these operators. From this,
Proposition \eqref{P0}, and the fact that the spectrum of $S_2$
is discrete we have
\[
\sigma_{e}(S)=(-\infty,-1/4].
\]
\end{example}

Note that the Legendre polynomials satisfy all four conditions of
\eqref{eq7.19}. Therefore the triples
\begin{equation}
P_L=(0,P_{n},0)\;(n\in\mathbb{N}_{0}), \label{eq7.21}
\end{equation}
are eigenfunctions of $S_L$ with eigenvalues
\begin{equation}
\lambda_{n}=n(n+1)\;(n\in\mathbb{N}_{0}). \label{eq7.22}
\end{equation}
Thus we may conclude that
\begin{equation}
(-\infty,-1/4]\cup\{\lambda_{n}=n(n+1),\;n\in\mathbb{N}_{0}\}\subset\sigma(S).
\label{eq7.23}
\end{equation}
We conjecture that
\begin{equation}
(-\infty,-1/4]\cup\{\lambda_{n}=n(n+1):\;n\in\mathbb{N}_{0}\}=\sigma(S).
\label{eq7.24}
\end{equation}



\begin{example} \label{exmp2} \rm
By using equation \eqref{eq0.1} on the interval $(-1,1)$ and
equation \eqref{eq0.5} on the intervals $(-\infty,-1)$ and
$(1,\infty)$, in other words by using $p(t)=1-t^2$ for $-1<t<1$
and $p(t)=t^2-1$ for $-\infty<t<-1$ and for $1<t<\infty$ and
applying the three-interval theory as in Example (1) we obtain an
operator whose essential spectrum is $[1/4,\infty)$ and whose
discrete spectrum contains the classical Legendre eigenvalues:
\[
\{\lambda_{n}=n(n+1):\;n\in\mathbb{N}_{0}\}.
\]
Note that $\lambda_{0}=0$ is below the essential spectrum and all other
eigenvalues $\lambda_{n}$ for $n>0$ are embedded in the essential spectrum.
Each triple
\[
(0,P_{n},0)\;\text{when }n\in\mathbb{N}_{0}
\]
is an eigenfunction with eigenvalue $\lambda_{n}$ for $n\in\mathbb{N}_{0}$.
\end{example}

\subsection*{Conclusion}
In this paper we have studied spectral theory in Hilbert spaces of
square-integrable functions associated with the Legendre
expression \eqref{eq0.1}, this is known as the right-definite
theory. There is also a left-definite theory, stemming from the
work of Pleijel \cite{Pleijel1}, see also \cite{evlw},
\cite{Vonhoff}, \cite{zett05} and the references in these papers.
This takes place in the setting of Hilbert-Sobolev spaces. There
is a third approach, developed by Littlejohn and Wellman
\cite{LW}, and used in \cite{evlw} for \eqref{eq0.1} - also called
`left-definite' by these authors - which takes place in the
setting of an infinite number of Hilbert-Sobolev spaces. We plan
to write a sequel to this paper discussing these other two
approaches.

\begin{thebibliography}{00}


\bibitem{AG} N. I. Akhiezer and I. M. Glazman;
\emph{Theory of linear
operators in Hilbert space,} Volumes 1 and 2, Dover Publications Inc.,
New York, 1993.

\bibitem{ALM} J. Arves\'{u}, L. L. Littlejohn, and F. Marcell\'{a}n;
\emph{On the right-definite and left-definite spectral theory of the Legendre
polynomials, } J. Comput. Anal. and Appl., J. Comput. Anal. and Appl.,
4(4)(2002), 363-387.

\bibitem{baez01} P. B. Bailey, W. N. Everitt and A. Zettl;
\emph{The SLEIGN2 Sturm-Liouville code, }ACM TOMS, ACM Trans.
Math. Software 21, (2001), 143-192.

\bibitem{BEN} C. Bennewitz;
\emph{A generalisation of Niessen's limit-circle
criterion.} Proc. Roy. Soc. Edinburgh Sect. (A) 78, 1977/78, 81-90.

\bibitem{CHEL} R. S. Chisholm, W. N. Everitt, and L. L. Littlejohn;
 \emph{An integral operator inequality with applications, }
J. of Inequal. \& Appl., 3, 1999, 245-266.

\bibitem{dusc63} N. Dunford and J. T. Schwartz;
\emph{Linear Operators,  II}, Wiley, New York, 1963.

\bibitem{Everitt} W. N. Everitt;
 \emph{Legendre polynomials and singular
differential operators, }Lecture Notes in Mathematics, Volume 827,
Springer-Verlag, New York, 1980, 83-106.

\bibitem{evze86} W. N. Everitt and A. Zettl;
 \emph{Sturm-Liouville Differential
Operators in Direct Sum Spaces}, Rocky Mountain J. of Mathematics,
(1986), 497-516.

\bibitem{evlw} W. N. Everitt, L. L. Littlejohn and R. Wellman;
\emph{Legendre Polynomials, Legendre-Stirling Numbers,
And the Left-Definite Spectral
Analysis of the Legendre Expression}, J. Comput. Appl. Math., 148(2002), 213-238.

\bibitem{evlm} W. N. Everitt, L. L. Littlejohn and V. Mari\'{c};
\emph{On Properties of the Legendre Differential Expression},
Result. Math. 42(2002), 42-68.

\bibitem{hkwz00} K. Haertzen, Q. Kong, H. Wu and A. Zettl;
\emph{Geometric aspects of Sturm-Liouville Problems I},
 Structures on Spaces of Boundary
Conditions, Proc. Roy. Soc. Edinburgh, 130A, (2000), 561-589.

\bibitem{KKZ} H. G. Kaper, M. K. Kwong, and A. Zettl;
\emph{Characterizations of the Friedrich's extensions of singular
Sturm-Liouville expressions,} SIAM
J. Math. Anal., 17(4), 1986, 772-777.

\bibitem{LW} L. L. Littlejohn and R. Wellman;
 \emph{A general left-definite theory for certain self-adjoint
operators with applications to differential
equations, } J. Differential Equations, J. Differential Equations,
181(2), 2002, 280-339.

\bibitem{Naimark} M. A. Naimark;
\emph{Linear differential operators II,}
Frederick Ungar Publishing Co., New York, 1968.

\bibitem{nize92} H.-D. Niessen and A. Zettl;
\emph{Singular Sturm-Liouville Problems:
The Friedrichs Extension and Comparison of Eigenvalues, }
Proc. London Math. Soc. (3) 64 (1992), 545-578.

\bibitem{neub60} J. W. Neuberger;
\emph{Concerning Boundary Value Problems},
Pacific J. Math. 10 (1960), 1385-1392.

\bibitem{Pleijel1} \AA . Pleijel;
\emph{On Legendre's polynomials,}
Mathematical Studies 21, North Holland Publishing Co., 1975, 175-180.

\bibitem{Szego}G. Szeg\"{o};
 \emph{Orthogonal polynomials } (4th edition),
American Mathematical Society Colloquium Publications,
 vol. 23, Providence, Rhode Island, 1975.

\bibitem{Titchmarsh} E. C. Titchmarsh; \emph{Eigenfunction expansions
associated with second-order differential equations I, }
Clarendon Press, Oxford, 1962.

\bibitem{Vonhoff} R. Vonhoff;
\emph{A left-definite study of Legendre's
differential equation and of the fourth-order Legendre
type differential equation, } Result. Math. 37, 2000, 155-196.

\bibitem{zett05} Anton Zettl;
\emph{Sturm-Liouville Theory}, American
Mathematical Society, Surveys and Monographs 121, 2005.

\bibitem{zett} A. Zettl;
\emph{The Legendre equation on the whole real line},
in Differential Equations and Applications, Vol. I and II (Columbus, OH),
1988, 502-508, Ohio Univ. Press, Athens, OH, 1989.

\end{thebibliography}


\end{document}


\eqref{eq0eq.1

\eqref{eq0.1}