\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 -10$, $-10$ 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 $c,d$, $-12$ and where $g$ is independent of $q$.
\item[(viii)] $p^{1/2}y'\in L^2(-1,1)$;
\item[(ix)] For any $-12$, 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
-\infty3,
\end{cases} \label{eq7.14}
\\
v(t)=\begin{cases}
-\frac{1}{2}\ln(\frac{t-1}{t+1}) & -13,
\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 $-10$ 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}
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\end{document}
\eqref{eq0eq.1
\eqref{eq0.1}