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%%     date="OCT-1995",
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%%     journal="Am. J. Math. 118, 571-594 (1996)"
%%     doi="10.1353/ajm.1996.0024",
%%     copyright="Copyright (C) F.Gesztsy, B.Simon, &amp; G.Teschl"
%%     }
 
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\begin{document}

\title{Zeros of the Wronskian and Renormalized Oscillation Theory}


\author[F.\ Gesztesy]{Fritz Gesztesy}
\address{Department of Mathematics\\
University of Missouri\\ Columbia, MO 65211\\ USA}
\email{\mailto{gesztesyf@missouri.edu}}
\urladdr{\url{https://www.math.missouri.edu/people/gesztesy}}

\author[B.\ Simon]{Barry Simon}
\address{Division of Physics, Mathematics, and Astronomy\\
California Institute of Technology\\ Pasadena\\ CA 91125\\ USA}

\author[G. Teschl]{Gerald Teschl}
\address{Department of Mathematics\\
University of Missouri\\ Columbia, MO 65211\\ USA}
\curraddr{Faculty of Mathematics\\ University of Vienna\\
Oskar-Morgenstern-Platz 1\\ 1090 Wien\\ Austria\\ and International 
Erwin Schr\"odinger Institute for Mathematical Physics\\ 
Boltzmanngasse 9\\ 1090 Wien\\ Austria}
\email{\mailto{Gerald.Teschl@univie.ac.at}}
\urladdr{\url{http://www.mat.univie.ac.at/~gerald/}}


\date{October 12, 1995}

\thanks{This material is based upon work supported by the National Science Foundation under
Grant No.~DMS-9401491. The Government has certain rights in this material.}
\thanks {\it Amer. J. Math. {\bf118}, 571--594 (1996).}


\subjclass{Primary 34B24, 34C10; Secondary 34L15, 34L05}
\keywords{Oscillation theory, Sturm-Liouville operators, eigenvalues}

\begin{abstract}
For general Sturm-Liouville operators with separated boundary
conditions, we prove the following: If $E_{1,2}\in\mathbb{R}$
and if $u_{1,2}$ solve the differential equation $Hu_j=E_j u_j$, $j=1,2$
and respectively satisfy the boundary condition on the left/right, then
the dimension of the spectral projection $P_{(E_1, E_2)}(H)$ of
$H$ equals the number of zeros of the Wronskian of $u_1$ and $u_2$.
\end{abstract}

\maketitle

 
\section{Introduction}

 
For over a hundred and fifty years, oscillation theorems for
second-order differential equations have fascinated mathematicians.
Originating with Sturm's celebrated memoir \cite{20}, extended in a variety
of ways by B\^ocher \cite{2} and others, a large body of material has been
accumulated since then (thorough treatments can be found, e.g., in
\cite{4},\cite{13},\cite{18},\cite{19}, and the references therein). In this paper we'll
add a new wrinkle to oscillation theory by showing that zeros of
Wronskians can be used to count eigenvalues in situations where a
naive use of oscillation theory would give $\infty -\infty$.
 
To set the stage, we'll consider operators on $L^{2}((a,b); r\,dx)$
with $a&lt;b$ in $[-\infty,\infty]$ of the form
\[
(\tau u)(x)=r(x)^{-1}[-(p(x)u'(x))' + q(x)u(x)],
\]
where
\begin{equation}
r, p^{-1}, q \in L^1_\mathrm{loc}((a,b);dx) \text{ are real-valued
and $r,p &gt;0$ a.e.~on $(a,b)$}. \label{1.1}
\end{equation}
We'll use $\tau$ to describe the formal differentiation expression and
$H$ the operator given by $\tau$ with separated boundary conditions at
$a$ and/or $b$.
 
If $a$ (resp.~$b$) is finite and $q,p^{-1},r$ are in addition integrable
near $a$ (resp.~$b$), we'll say $a$ (resp.~$b$) is a {\it{regular}}
end point.  We'll say $\tau$ respectively $H$ is {\it{regular}} if
both $a$ and $b$ are regular. As is usual, (\cite{6}, Section XIII.2;
\cite{15}, Section~17; \cite{22}, Chapter 3), we consider the local domain
\begin{equation}
D_\mathrm{loc}=\{u \in AC_\mathrm{loc}((a,b)) \mid pu'\in
AC_\mathrm{loc}((a,b)),\ \tau u \in
L^{2}_\mathrm{loc}((a,b); r\,dx)\}, \label{1.2}
\end{equation}
where $AC_\mathrm{loc}((a,b))$ is the set of integrals
of $L^{1}_\mathrm{loc}((a,b); dx)$-functions (i.e., the set of locally
absolutely continuous functions) on $(a,b)$. General ODE
theory shows that for any $E\in\mathbb{C}$, $x_0\in (a,b)$, and $(\alpha,
\beta)\in\mathbb{C}^2$, there is a unique $u\in D_\mathrm{loc}$
such that $-(pu')'+qu - Eru=0$ for a.e.~$x \in (a,b)$ and $(u(x_0),
(pu')(x_0))=(\alpha,\beta)$.
 
The maximal and minimal operators are defined by taking
\[
D(T_\mathrm{max})=\{u\in L^{2}((a,b); r\,dx)\cap
D_\mathrm{loc}\mid \tau u\in L^{2}((a,b); r\,dx)\},
\]
with
\begin{equation}
T_\mathrm{max}u=\tau u. \label{1.3}
\end{equation}
$T_\mathrm{min}$ is the operator closure of
$T_\mathrm{max}\restriction D_\mathrm{loc}\cap\{u
\text{ has compact support in $(a,b)$}\}$. Then $T_\mathrm{min}$
is symmetric and $T^{*}_\mathrm{min}=T_\mathrm{max}$.
 
According to the Weyl theory of self-adjoint extensions (\cite{6}, Section
XIII.6; \cite{15}, Section 18; \cite{17}, Appendix to X.1; \cite{21}, Section 8.4;
\cite{22}, Chapters 4 and 5), the deficiency indices of
$T_\mathrm{min}$ are $(0,0)$ or $(1,1)$ or $(2,2)$ depending
on whether it is limit point at both, one or neither end point.
Moreover, the self-adjoint extensions can be described in terms of
Wronskians (\cite{6}, Section XIII.2; \cite{15}, Sections 17 and 18; \cite{21},
Section 8.4; \cite{22}, Chapter 3). Define
\begin{equation}
W(u_1,u_2)(x)=u_1(x)(pu_2')(x)-(pu_1')(x)u_2(x). \label{1.4}
\end{equation}
Then if $T_\mathrm{min}$ is limit point at both ends,
$T_\mathrm{min}=T_\mathrm{max}=H$. If $T_\mathrm{min}$
is limit point at $b$ but not at $a$, for $H$ any self-adjoint extension
of $T_\mathrm{min}$, if $\varphi_-$ is any function in $D(H)
\backslash D(T_\mathrm{min})$, then
\[
D(H)=\{u\in D(T_\mathrm{max})\mid W(u,\varphi_-)(x)\to 0
\text{ as $x\downarrow a$}\}.
\]
Finally, if $u_1$ is limit circle at both ends, the operators $H$ with
separated boundary conditions are those for which we can find
$\varphi_\pm \in D(H)$, $\varphi_+\equiv 0$ near $a$, $\varphi_-
\equiv 0$ near $b$, and $\varphi_\pm \in D(H)\backslash
D(T_\mathrm{min})$. In that case,
\[
D(H)=\{u\in D(T_\mathrm{max})\mid  W(u,\varphi_-)(x)\to 0
\text{ as $x\downarrow a$}, \,
W(u,\varphi_+)(x)\to 0 \text{ as $x\uparrow b$}\}.
\]
Of course, if $H$ is regular, we can just specify the boundary
conditions by taking values at $a,b$ since by regularity any $u\in
D(T_\mathrm{max})$ has $u,pu'$ continuous on $[a,b]$ (cf.~\eqref{A.4}).
It follows from this analysis that
 
\begin{proposition}\label{prop1.1} If $u_{1,2}\in D(H)$, then $W(u_1,u_2)(x)
\to 0$ as $x\to a$ or $b$.
\end{proposition}
 
We'll call such operators SL operators (for Sturm-Liouville, but SL
includes separated boundary conditions (if necessary)).
 
It will be convenient to write $\ell_-=a$, $\ell_+=b$.
 
Throughout this paper we will denote by $\psi_{\pm}(z,x) \in
D_\mathrm{loc}$ solutions of $\tau \psi = z \psi$ so that
$\psi_{\pm}(z,\,.\,)$ is $L^2(\cdot;r\,dx)$ at $\ell_{\pm}$ and $\psi_{\pm}
(z,\,.\,)$ satisfies the appropriate boundary condition at $\ell_\pm$
in the sense that for any $u \in D(H)$, $\lim\limits_{x\to\ell_{\pm}}
W(\psi_{\pm}(z),u)(x)=0$. If $\psi_{\pm}(z,\,.\,)$ exist, they are
unique up to constant multiples. In particular, $\psi_{\pm}(z,\,.\,)$
exist for $z$ not in the essential spectrum of $H$ and we can assume
them to be holomorphic with respect to $z$ in ${\mathbb{C}} \backslash
\text{spec}(H)$ and real for $z\in\mathbb{R}$. One can choose
\[
\psi_\pm(z,x) = ((H-z)^{-1} \chi_{(c,d)})(x) \quad
\text{for } x \smat{&gt; d}{&lt; c}, \quad a&lt;c&lt;d&lt;b
\]
and uniquely continue $\psi_\pm(z,x)$  for $x \smat{&lt; d}{&gt; c}$.
Here $(H-z)^{-1}$ denotes the resolvent of $H$ and $\chi_\Omega$ the
characteristic function of the set $\Omega  \subseteq \mathbb{R}$. Clearly
we can include a finite number of isolated eigenvalues in the domain
of holomorphy of $\psi_\pm$ by removing the corresponding poles. Moreover,
to simplify notations, all solutions $u$ of $\tau u = Eu$ are understood
to be not identically vanishing and solutions associated with real values
of the spectral parameter $E$ are assumed to be real-valued in this paper.
Thus if $E$ is real and in the resolvent set for $H$ or an isolated
eigenvalue, we are guaranteed there are solutions that obey the boundary
conditions at $a$ or $b$. It can happen if $E$ is in the essential
spectrum that such solutions do not exist or it may happen that they do.
In Theorems 1.3, 1.4 below, we'll explicitly assume such solutions exist
for the energies of interest. If these energies are not in the essential
spectrum, that is automatically fulfilled.
 
With these preliminaries out of the way, we can describe a theorem
Hartman proves in \cite{10} which gives an eigenvalue count in some cases
where oscillation theory would naively give $\infty -\infty$ (see
Weidmann \cite{22}, Chapter 14 for some results when $\tau$ is limit circle
at $b$). In fact, we have slightly generalized the theorem in order to
include, for instance, certain singular
cases like radial Schr\"odinger operators on
$(0,\infty)$ with potentials
singular near $0$ (we shall give a proof in Section~\ref{sec7}).
 
\begin{theorem}\label{thm1.2} Let $H$ be an SL operator on $(a,b)$ which
is non-oscillatory at $E_2$ near $a$ and limit point at $b$ and
suppose $E_1&lt; E_2$. Let $u_1$ (resp.~$u_2$) be $\psi_{-}
(E_1)$ (resp.~$\psi_-(E_2)$). Let $N(c)$, $c \in (a,b)$
denote the number of zeros of $u_1$ in $(a,c)$ minus the number of
zeros of $u_2$ in $(a,c)$. Let $P_\Omega(H)$ be the spectral projection
of $H$ corresponding to the Borel set $\Omega\subseteq\mathbb{R}$. Then,
if $\tau$ is oscillatory at $E_2$ near $b$,
\begin{equation}
\dim\mathrm{Ran}\, P_{(E_1, E_2)}(H) =
\varliminf\limits_{c\uparrow b}\, N(c), \tag{1.5a}\label{1.5a}
\end{equation}
and if $\tau$ is non-oscillatory at $E_2$ near $b$,
\begin{equation}
\dim\mathrm{Ran}\, P_{[E_1, E_2)}(H) =
\lim\limits_{c\uparrow b}\, N(c). \tag{1.5b}\label{1.5b}
\end{equation}
\addtocounter{equation}{1}
\end{theorem}
 
Theorem~\ref{thm1.2} is a bit more general than Hartman's result in \cite{10}
(see also \cite{9}, \cite{11}) since we assume $H$ to be non-oscillatory at
$E_2$ near $a$ while Hartman assumes $H$ to be regular at $a$. If
$\tau$ is oscillatory at $E_2$ near $b$ (i.e., $u_2$ has infinitely
many zeros near $b$), $N(c)$ is not constant for large $c$ but instead
varies between $N_0$ and $N_0+1$. This result leaves several questions
open: What happens if $H$ is limit circle at $b$ or in the case where
$H$ is not regular at either end (e.g., the important case of the
real line $(a,b)=(-\infty,\infty)$)? Moreover, it isn't clear when
$c$ is so large that $\varliminf\limits_{c\uparrow b}\,N(c)$ has been
reached. It would be better if we could actually count something
analogous to the zero count in ordinary oscillation theory. Our goal
in this paper is to prove such theorems.
 
The key is to look at zeros of the Wronskian. That zeros of the
Wronskian are related to oscillation theory is indicated by an old
paper of Leighton \cite{14}, who noted that if $u_j,pu_j' \in
AC_\mathrm{loc}((a,b))$, $j=1,2$ and $u_1$ and $u_2$ have a
non-vanishing Wronskian $W(u_1,u_2)$ in $(a,b)$, then their zeros must
intertwine each other. (In fact, $pu_1'$ must have opposite signs at
consecutive zeros of $u_1$, so by non-vanishing of $W$, $u_2$ must have
opposite signs at consecutive zeros of $u_1$ as well. Interchanging the
role of $u_1$ and $u_2$ yields strict interlacing of their zeros.)
Moreover, let $E_1&lt;E_2$ and $\tau u_j = E_j u_j$, $j=1,2$. If$x_0,x_1$
are two consecutive zeros of $u_1$, then the number of zeros of $u_2$
inside $(x_0,x_1)$ is equal to the number of zeros of the Wronskian
$W(u_1,u_2)$ plus one (cf.~Theorem~\ref{thm7.4}). Hence the Wronskian comes with
a built-in renormalization counting the additional zeros of $u_2$ in
comparison to $u_1$. In particular, this avoids taking limits of the
type \eqref{1.5a}.
 
We'll let $W_0(u_1,u_2)$ be the number of zeros of the Wronskian in the
open interval $(a,b)$ not counting multiplicities of zeros. Given
$E_1&lt;E_2$, we let $N_0(E_1, E_2)=\dim
\mathrm{Ran}\, P_{(E_1, E_2)}(H)$ be the dimension of the spectral
projection $P_{(E_1, E_2)}(H)$ of $H$.  Our main results are the
following two theorems:
 
\begin{theorem}\label{thm1.3}  Suppose $E_1 &lt; E_2$. Let $u_1=\psi_-(E_1)$ and
$u_2=\psi_+(E_2)$. Then
\[
W_0(u_1,u_2)=N_0(E_1, E_2).
\]
\end{theorem}
 
\begin{theorem}\label{thm1.4} Suppose $E_1 &lt; E_2$. Let $u_1=\psi_-(E_1)$ and
$u_2=\psi_-(E_2)$. Then either
\begin{equation}
W_0(u_1,u_2)=N_0(E_1, E_2) \label{1.6}
\end{equation}
or
\begin{equation}
W_0(u_1,u_2)=N_0(E_1, E_2) -1. \label{1.7}
\end{equation}
If either $N_0=0$ or $H$ is limit point at $b$, then \eqref{1.6} holds.
\end{theorem}
 
We'll see that if $b$ is a regular point and $E_2 &gt; e &gt; E_1$ with $e$
an eigenvalue and $|E_2-E_1|$ is small, then \eqref{1.7} holds rather than
\eqref{1.6}. We'll also see that if $u_{1,2}$ are arbitrary solutions of
$\tau u_j=E_j u_j$, $j=1,2$, then, in general, $|W_0-N_0|\leq 2$ (this
means that if one of the quantities is infinite, the other is as well)
and any of $0, \pm 1,\pm 2$ can occur for $W_0-N_0$. Especially, if either
$E_1$ or $E_2$ is in the interior of the essential spectrum of $H$
(or $\dim\mathrm{Ran}\, P_{(E_1, E_2)}(H)=\infty$), then $W_0
(u_1,u_2)=\infty$ for any $u_1$ and $u_2$ satisfying $\tau u_j=E_j
u_j$, $j=1,2$ (cf.~Theorem~\ref{thm7.3}).
 
Zeros of the Wronskians have two properties that are critical to these
results: First, zeros are precisely points where the Pr\"ufer angles
for $u_1$ and $u_2$ are equal $(\mod \pi)$. Second, if $\psi_- \in
D_\mathrm{loc}$ and $\psi_+ \in D_\mathrm{loc}$ satisfy
the boundary conditions at $a,b$, respectively, and $W(\psi_-,\psi_+)
(x_0)=0$ and if $(\psi_+(x_0), (p\psi_+')(x_0))\neq (0,0)$, then there
is a $\gamma$ such that
\[
\eta(x)= \begin{cases} \psi_-(x),  &amp; x\leq x_0 \\
\gamma\psi_+(x), &amp; x\geq x_0
\end{cases}
\]
satisfies $\eta\in D(H)$ and
\[
H \eta(x) =\begin{cases} (\tau\psi_-)(x), &amp; x\leq x_0 \\
\gamma (\tau\psi_+)(x), &amp; x\geq x_0.
\end{cases}
\]
We'll explore these properties further in Propositions \ref{prop3.1} and \ref{prop3.2}.
 
Section~\ref{sec2} provides a short proof of the ordinary oscillation theorem
in the regular case following the method in Courant-Hilbert (\cite{5},
page 454). Even though this result is well-known (see, e.g., \cite{1},
Theorem~8.4.5 and \cite{22}, Theorem 14.10 which describes the singular
case as well) we include it here since our overall strategy in this
paper is patterned after this proof: A variational argument will show
$N_0\geq W_0$ in Section~\ref{sec6} and a comparison-type argument in Sections
\ref{sec4} and \ref{sec5} will prove $N_0\leq W_0$. Explicitly, in Section~\ref{sec5} we'll show
 
\begin{theorem}\label{thm1.5} Let $E_1 &lt; E_2$. If $u_1=\psi_-(E_1)$ and
either $u_2=\psi_+(E_2)$ or $\tau u_2 = E_2 u_2$ and $H$ is limit point
at $b$, then
\[
W_0(u_1,u_2)\geq\dim\mathrm{Ran}\, P_{(E_1, E_2)}(H).
\]
\end{theorem}
 
In Section~\ref{sec6}, we'll prove that
 
\begin{theorem}\label{thm1.6} Let $E_1 &lt; E_2$. Let either $u_1=\psi_+(E_1)$ or
$u_1=\psi_-(E_1)$ and either $u_2=\psi_+(E_2)$ or $u_2=\psi_-(E_2)$.
Then
\begin{equation}
W_0(u_1,u_2)\leq\dim\mathrm{Ran}\, P_{(E_1, E_2)}(H). \label{1.8}
\end{equation}
\end{theorem}
 
\begin{remark} Of course, by reflecting about a point $c \in (a,b)$,
Theorems 1.3--1.5 hold for $u_1 = \psi_+(E_1)$ and $u_2 = \psi_-(E_2)$
(and either $N_0 = 0$ or $H$ is limit point at $a$ in the corresponding
analog of Theorem~\ref{thm1.4} yields \eqref{1.6} and similarly, $\tau u_2 = E_2 u_2$
and $H$ is limit point at $a$ yields the conclusion in the corresponding
analog of Theorem~\ref{thm1.5}).
\end{remark}
 
In Section~\ref{sec7}, we provide a number of comments, examples, and extensions
including:
 
\begin{theorem}\label{thm1.7} Let $E_{1,2} \in\mathbb{R}$, $E_1 \ne E_2$,
$\tau u_j = E_j u_j$, $j=1,2$, $\tau v_2=E_2 v_2$. Then $|W_0
(u_1,u_2)-W_0(u_1,v_2)|\leq 1$.
\end{theorem}
 
In addition, Theorem~\ref{thm7.5} addresses the problem of finite versus infinite
total number of eigenvalues in essential spectral gaps of $H$.
 
It is easy to see that Theorems \ref{thm1.5}, \ref{thm1.6}, and \ref{thm1.7} imply Theorems \ref{thm1.3}
and \ref{thm1.4}.
 
Some facts on quadratic forms are collected in the appendix.
 
Our interest in this subject originated in attempts to provide a general
construction of isospectral potentials for one-dimensional Schr\"odinger
operators (see \cite{8}) following previous work by Finkel, Isaacson, and
Trubowitz \cite{7} (see also \cite{3}) in the case of periodic potentials. In fact,
in the special case of periodic Schr\"odinger operators $H_p$, the
non-vanishing of $W(u_1, u_2)(x)$ for Floquet solutions $u_1 =
\psi_{\varepsilon_1}(E_1)$, $u_2 = \psi_{\varepsilon_2}(E_2)$, \;
$\varepsilon_{1,2} \in \{+,-\}$ of $H_p$, for $E_1$ and  $E_2$ in the
same spectral gap of $H_p$, is proven in \cite{7}.
 
 
\section{Oscillation Theory}
\label{sec2}
 
For background, we recall the following:
 
\begin{theorem}[{\cite{22}}, Theorem 14.10]\label{thm2.1} Let $H$ be an SL operator
which is bounded from below. If $e_1 &lt;\cdots &lt; e_n &lt; \cdots $ are its
eigenvalues below the essential spectrum and $\psi_1,\dots, \psi_n,
\dots$ its eigenfunctions, then $\psi_n$ has $n-1$ zeros in $(a,b)$.
All eigenvalues of $H$ are simple.
\end{theorem}
 
\begin{remark} (i) Those used to thinking of the Dirichlet boundary
condition case need to be warned that it is {\it{not}} in general true
that if $E$ is not an eigenvalue of $H$, then the number of zeros,
$Z$, of $\psi_\pm(E)$ is the number, $N(E)$, of eigenvalues less
than $E$. In general, all one can say is $N=Z$ or $N=Z+1$.
 
(ii) In the special case where $\tau$ is regular at $a$ and $b$, any
associated SL operator $H$ is well-known to be bounded from below with
compact resolvent (see, e.g., \cite{1}, Theorem~8.4.5; \cite{22}, Theorem 13.2).
Thus Theorem~\ref{thm2.1} applies to the regular case (to be used in our proof of
Proposition~\ref{prop4.1}).
\end{remark}
 
The first part of the proposition below is a simple integration by
parts and the second follows from the first.
 
\begin{proposition}\label{prop2.3} Let $E_1 \leq E_2$ and $\tau u_j = E_j u_j$,
$j=1,2$. Then for $a&lt;c&lt;d&lt;b$,
\[
W(u_1,u_2)(d)-W(u_1,u_2)(c)=(E_1-E_2) \int\limits^{d}_{c}
u_1(x)u_2(x) r(x)\, dx. \label{2.1}
\]
In particular, $W(u_1,u_2) \in AC_\mathrm{loc}((a,b))$ and
\[
\frac{dW(u_1,u_2)}{dx}\,(x)=
(E_1-E_2) r(x)u_1(x)u_2(x)\quad \text{a.e.} \label{2.2}
\]
If the problem is regular at $a$ (resp.~$b$), we can take $c$
(resp.~$d$) equal to $a$ (resp.~$b$). In the general
case we can take the limit $c \downarrow a$ (resp.~$d \uparrow
b$) in \eqref{2.1} if $u_1$ and $u_2$ are $L^2(\cdot; r\,dx)$ near $a$
(resp.~$b$).
\end{proposition}
 
\begin{corollary}\label{cor2.3} Let $E_1 &lt; E_2$ and $\tau u_j =E_j u_j$,
$j=1,2$. Suppose at each end of $[c,d]$, $a&lt;c&lt;d&lt;b$ either $W(u_1,u_2)=
0$ or $u_1=0$. If $\lim\limits_{x\downarrow a} W(u_1,u_2)(x)=0$
(resp.~$\lim\limits_{x\uparrow b}W(u_1,u_2)(x)=0$), we also
consider $c=a$ (resp.~$d=b$). Then $u_2$ must vanish at least
once in $(c,d)$.
\end{corollary}
 
\begin{proof} By decreasing $d$ to the first zero of $u_1$ in $(c,d]$
(and perhaps flipping signs), we can suppose $u_1&gt;0$ on $(c,d)$.
If $u_2$ has no zeros in $(c,d)$, we can suppose $u_2&gt;0$ on $(c,d)$ again
by perhaps flipping signs. At each end point, $W(u_1,u_2)$ vanishes or
else $u_1=0$, $u_2&gt;0$, and $u_1'(c)&gt;0$ (or $u_1'(d)&lt;0$). Thus,
$W(u_1,u_2)(c)\leq 0$, $W(u_1,u_2)(d)\geq 0$. Since the right side of
\eqref{2.1} is negative, this is inconsistent with \eqref{2.1}.
\end{proof}
 
\begin{proof}[Proof of Theorem \ref{thm2.1}] We first prove that $\psi_n$ has
at least $n-1$ zeros and then that if $\psi_n$ has $m$ zeros, then
$(-\infty, e_n]$ has at least $(m+1)$ eigenvalues. If $\psi_n$ has $m$
zeros at $x_1, x_2,\dots, x_m$ and we let $x_0=a$, $x_{m+1}=b$, then by
Corollary~\ref{cor2.3}, $\psi_{n+1}$ must have at least one zero in each of $(x_0,
x_1), (x_1, x_2),\dots, (x_m, x_{m+1})$, that is, $\psi_{n+1}$ has at
least $m+1$ zeros. It follows by induction that $\psi_n$ has at least
$n-1$ zeros.
 
On the other hand, if an eigenfunction $\psi_n$ has $m$ zeros, define
for $j=0,\dots, m$, $x_0=a,x_{m+1}=b$,
\[
\eta_j(x) = \begin{cases} \psi_n(x), &amp; x_j \leq x \leq x_{j+1} \\
0, &amp; \text{otherwise} \end{cases}, \quad 0 \leq j \leq m.
\]
Then $\eta_j$ is absolutely continuous with $p\eta'_j$ piecewise
continuous so $\eta_j$ is in the form domain of $H$ (see \eqref{A.6}) and
$\langle |H|^{1/2} \eta_j, \text{sgn}(H) |H|^{1/2} \eta_j \rangle_r =
e_n\,\| \eta_j \|_{2,r}$ (where $\langle \,\cdot\, ,\,\cdot\,\rangle_r$ and
$\| \cdot \|_{2,r}$ denote the scalar product and norm in $L^2((a,b);r\,dx$).
Thus if $\eta=\sum\limits^m_{j=0} c_j \eta_j$, then $\langle |H|^{1/2}
\eta, \text{sgn}(H)|H|^{1/2} \eta \rangle_r = e_n\,\| \eta \|_{2,r}$. It follows
by the spectral theorem that there are at least $m+1$ eigenvalues in
$(-\infty, e_n]$. Since $H$ has separated boundary conditions, its point
spectrum is simple.
\end{proof}
 
The second part of the proof of Theorem~\ref{thm2.1} also shows:
 
\begin{corollary}\label{cor2.4} Let $H$ be an SL operator bounded from below.
If $\psi_+(E,\,.\,)$ (resp.\ $\psi_-(E,\,.\,)$) has
$m$ zeros, then there are at least $m$ eigenvalues below $E$. In
particular, $E$ below the spectrum of $H$ implies that $\psi_\pm
(E,\,.\,)$ have no zeros.
\end{corollary}
 

 
\section{Zeros of the Wronskian}
\label{sec3}
 
Here we'll present the two aspects of zeros of the Wronskian which are
critical for the two halves of our proofs (i.e., for showing $N_0\geq
W_0$ and that $N_0\leq W_0$). First, the vanishing of the Wronskian lets
us patch solutions together:
 
\begin{proposition}\label{prop3.1}  Suppose that $\psi_{+,j}, \psi_-\in
D_\mathrm{loc}$ and that $\psi_{+,j}$ and $\tau\psi_{+,j}$,
$j=1,2$ are in $L^{2}((c,b); r\,dx)$ and that $\psi_-$ and $\tau\psi_-$ are
in $L^{2}((a,c);r\,dx)$ for all $c\in (a,b)$.  Suppose, in addition, that
$\psi_{+,j}$, $j=1,2$ satisfy the boundary condition defining $H$ at
$b$ (i.e., $W(u,\psi_{+,j})(c)\to 0$ as $c\uparrow b$ for all $u\in
D(H)$) and similarly, that $\psi_-$ satisfies the boundary condition
at $a$. Then
 
(i) If $W(\psi_{+,1}, \psi_{+,2})(c)=0$ and $(\psi_{+,2}(c),
(p\psi'_{+,2})(c)) \neq (0,0)$, then there exists a $\gamma$ such that
\[
\eta=\chi_{[c,b)}(\psi_{+,1}-\gamma\psi_{+,2})\in D(H)
\]
and
\begin{equation}
H\eta=\chi_{[c,b)}(\tau\psi_{+,1}-\gamma\tau\psi_{+,2}). \label{3.1}
\end{equation}
 
(ii) If $W(\psi_{+,1}, \psi_-)(c)=0$ and $(\psi_-(c),
(p\psi_-')(c))\neq (0,0)$, then there is a $\gamma$ such that
\[
\eta=\gamma\chi_{(a,c]}\psi_- + \chi_{(c,b)}\psi_{+,1}\in D(H)
\]
and
\begin{equation}
H\eta =\gamma\chi_{(a,c]}\tau\psi_- + \chi_{(c,b)}\tau\psi_{+,1}.
\label{3.2}
\end{equation}
\end{proposition}
 
\begin{proof} Clearly, $\eta$ and the right-hand-sides of \eqref{3.1}/\eqref{3.2} lie
in $L^2((a,b); r\,dx)$ and satisfy the boundary condition at $a$ and $b$, so it
suffices to prove that $\eta$ and $p\eta'$ are locally absolutely
continuous on $(a,b)$.
 
In case (i), if $\psi_{+,2}(c)\neq 0$, take $\gamma=-\psi_{+,1}(c)/
\psi_{+,2}(c)$ and otherwise (i.e., if $\psi_{+,2}(c) = 0$) take
$\gamma=- (p\psi'_{+,1})(c)/ (p\psi'_{+,2})(c)$. In either case,
$\eta$ and $p\eta'$ are continuous at $c$. Case (ii) is similar.
\end{proof}
 
The second aspect connects zeros of the Wronskian to Pr\"ufer
variables $\rho_u, \theta_u$ (for $u,pu'$ continuous) defined by
\[
u(x)=\rho_u(x)\sin(\theta_u(x)), \qquad (pu')(x)=\rho_u(x)
\cos(\theta_u(x)).
\]
If $(u(x), (pu')(x))$ is never $(0,0)$, then $\rho_u$ can be chosen
positive and $\theta_u$ is uniquely determined once a value of $\theta_u
(x_0)$ is chosen subject to the requirement $\theta_u$ continuous in $x$.
 
Notice that
\[
W(u_1,u_2)(x)= \rho_{u_1}(x)\rho_{u_2}(x)\sin(\theta_{u_1}(x)
- \theta_{u_2}(x)).
\]
Thus,
 
\begin{proposition}\label{prop3.2} Suppose $(u_j,pu_j')$, $j=1,2$ are never
$(0,0)$. Then $W(u_1,u_2)(x_0)$ is zero if and only if $\theta_{u_1}
(x_0)\equiv \theta_{u_2}(x_0)$ $(\mod \pi)$.
\end{proposition}
 
In linking Pr\"ufer variables to rotation numbers, an important role
is played by the observation that because of
\[
u(x) = \int\limits_{x_0}^x \frac{\rho_u(t) \cos(\theta_u(t))}{p(t)}\,
dt,
\]
$\theta_u(x_0)\equiv 0$ $(\mod \pi)$ implies $[\theta_u(x)-
\theta_u(x_0)]\big/ (x-x_0) &gt;0$ for $0 &lt;|x - x_0|$ sufficiently small
and hence for all $0 &lt;|x - x_0|$ if $(u,pu') \neq (0,0)$. (In fact,
suppose
$x_1 \ne x_0$ is the closest $x$ such that $\theta_u(x_1)=\theta_u
(x_0)$ then apply the local result at $x_1$ to obtain a contradiction.)
We summarize:
 
\begin{proposition}\label{prop3.3} If $(u,pu') \neq (0,0)$ then
$\theta_u(x_0)\equiv 0$ $(\mod \pi)$ implies
\[
[\theta_u(x)-\theta_u(x_0)]\big/ (x-x_0) &gt;0
\]
for $x \ne x_0$. In particular, if $\theta_u(c)\in [0,\pi)$ and $u$ has
$n$ zeros in $(c,d)$, then $\theta_u(d-\epsilon)\in (n\pi, (n+1)\pi)$
for sufficiently small $\epsilon &gt; 0$.
\end{proposition}
 
In exactly the same way, we have
 
\begin{proposition}\label{prop3.4} Let $E_1&lt;E_2$ and assume that $u_{1,2}$
solve $\tau u_j = E_j u_j$, $j=1,2$. Let $\Delta(x)=\theta_{u_2}(x)-
\theta_{u_1}(x)$. Then $\Delta(x_0)\equiv 0$ $(\mod \pi)$ implies
$(\Delta(x)-\Delta(x_0))/ (x-x_0)&gt;0$ for $0&lt;|x -x_0|$.
\end{proposition}
 
\begin{proof} If $\Delta(x_0)\equiv 0$ $(\mod 2\pi)$ and $\theta_{u_2}
(x_0)\not\equiv 0$ $(\mod \pi)$, then $\sin(\theta_{u_2}(x_0))
\sin(\theta_{u_1}(x_0))&gt;0$ so $u_1(x_0)u_2(x_0) &gt;0$ for $0&lt;|x-x_0|$
sufficiently small, and thus by \eqref{2.2}, $\frac{dW}{dx}(x_0)&gt;0$ for
a.e.~$x$ near $x_0$ and so $\Delta(x)$ is increasing. The same holds
for $\Delta(x_0)\equiv\pi$ $(\mod 2\pi)$ and $\theta_{u_2}(x_0)
\not\equiv 0$ $(\mod \pi)$.
 
If $\Delta(x_0)\equiv 0$ $(\mod 2\pi)$ and $\theta_{u_1}(x_0)\equiv
\theta_{u_2}(x_0)\equiv 0$ $(\mod \pi)$, then $(pu_1')(x_0) (pu_2')
(x_0)&gt;0$ and so since $u(x_0)=v(x_0)=0$, we see that it is still
true that $\frac{dW}{dx}(x)&gt;0$ a.e.~for $0&lt;|x-x_0|$ sufficiently
small.
\end{proof}
 
\begin{remark} (i) Suppose $r,p$ are continuous on $(a,b)$.
If $\theta_{u_1}(x_0)\equiv 0$ $(\mod \pi)$ then $\theta_{u_1}(x) -
\theta_{u_1}(x_0) = c_0(x-x_0) + o(x-x_0)$ with $c_0&gt;0$. If $\Delta
(x_0)\equiv 0$ $(\mod \pi)$ and $\theta_{u_1}(x_0)\not\equiv 0$
$(\mod \pi)$, then $\Delta(x)-\Delta (x_0)=c_1(x-x_0)+o(x-x_0)$ with
$c_{1}&gt;0$. If $\theta_{u_1} (x_0)\equiv 0\equiv\Delta(x_0)$ $(\mod \pi)$,
then $\Delta(x) - \Delta(x_0)=c_2(x-x_0)^{3}+o(x-x_0)^{3})$ with $c_2&gt;0$.
Either way, $\Delta$ increases through $x_0$. (In fact, $c_0=p(x_0)^{-1}$,
$c_{1}= (E_2-E_1)r(x_0)\sin^2 (\theta_{u_1}(x_0))$ and $c_{2}=\frac{1}{3}
r(x_0)p(x_0)^{-2}(E_2-E_1))$.
 
(ii) In other words, Propositions~\ref{prop3.3} and \ref{prop3.4} say that the integer parts
of $\theta_u/ \pi$ and $\Delta_{u,v}/ \pi$ are increasing with respect
to $x\in(a,b)$ (even though $\theta_u$ and $\Delta_{u,v}$ themselves
are not necessarily monotone in $x$).
 
(iii) Let $E \in [E_1,E_2]$ and assume $[E_1,E_2]$ to be outside the
essential spectrum of $H$. Then, for $x \in (a,b)$ fixed,
\begin{equation}
\frac{d\theta_{\psi_\pm}}{dE}\,(E,x) =
-\frac{\int\limits^{\ell_\pm}_x \psi_\pm(E,t)^2 \,dt}
{\rho_{\psi_\pm}(E,x)} \label{3.3}
\end{equation}
proves that $\mp\theta_{\psi_\pm}(E,x)$ is strictly increasing with
respect to $E$. In fact, from Proposition~\ref{prop2.3} one infers
\[
W(\psi_\pm(E), \psi_\pm(\tilde{E}))(x) = (\tilde{E}-E)
\int\limits^{\ell_\pm}_x \psi_\pm(E,t) \psi_\pm(\tilde{E},t)\, dt
\]
and using this to evaluate the limit $\lim\limits_{\tilde{E} \to E}
W(\psi_\pm(E), (\psi_\pm(E) - \psi_\pm(\tilde{E}))/(E-\tilde{E}))(x)$,
one obtains
\[
W(\psi_\pm(E), \frac{d\psi_\pm}{dE}\,(E))(x) =
\int\limits^{\ell_\pm}_x \psi_\pm(E,t)^2 \, dt.
\]
Inserting Pr\"ufer variables completes the proof of \eqref{3.3}.
\end{remark}
 

 
\section{The Hare and the Tortoise ($N_0 \leq W_0$ in the Regular Case)}
\label{sec4}
 
Our goal in this section is to prove Theorem~\ref{thm1.5} in the regular case
with opposite boundary conditions, that is,
 
\begin{proposition}\label{prop4.1} Let $H$ be a regular SL operator and suppose
$E_1&lt;E_2$. Then
\begin{equation}
W_0(\psi_-(E_1), \psi_+(E_2))\geq N_0(E_1, E_2).
\label{4.1}
\end{equation}
\end{proposition}
 
The proof will use Pr\"ufer angles. As a warm-up, let us prove equality
in the case that $H$ has $u(a)=u(b)=0$ boundary conditions and that
$E_{1,2}$ are not eigenvalues. Let $\theta_{\psi_\pm}(E,x)$ be the
Pr\"ufer angles for $\psi_\pm(E,x)$, normalized such that
$\theta_{\psi_\pm}(E,a)\in [0,\pi)$. Since $\psi_-(E_1)$ satisfies the
boundary condition at $a$, $\theta_{\psi_-}(E_1,a)=0$ and since $E_2$ is
not an eigenvalue, $\theta_{\psi_+}(E_2,a)&gt;0$. If there are $m$
eigenvalues below $E_1$ and $n_0+m$ below $E_2$, then, by standard
oscillation theory (essentially Proposition~\ref{prop3.3}), $\theta_{\psi_-}(E_1,b)
\in (m\pi,(m+1)\pi)$ and $\theta_{\psi_+}(E_2,b)=(n_0+m+1)\pi$. Let
$\Gamma_{\pm}(E,x)\equiv  \theta_{\psi_\pm}(E,x)$ $(\mod \pi)$, that is,
$\Gamma_{\pm}(E,x)\in [0,\pi)$ and $\Gamma_{\pm}-\theta_{\psi_\pm}\in
\mathbb{Z}\pi$.
 
Borrow a leaf from Aesop. Think of $\Gamma_-(E_1)$ as a tortoise and
$\Gamma_+(E_2)$ as a hare racing on a track of size $\pi$ with $0$
as the start and $\pi$ as the finish. Every time either runs through the
finish, it starts all over. Neither has to run only in the forward
direction (i.e., $\theta_{\psi_\pm}$ may not be monotone w.r.t.~$x$)
but they can't run in the wrong direction back through the start (i.e.,
Proposition~\ref{prop3.3} holds).
 
What makes $\Gamma_+(E_2)$ the hare to $\Gamma_-(E_1)$'s tortoise is
that $\Gamma_+(E_2)$ can only overtake $\Gamma_-(E_1)$, not the other
way around (i.e., Proposition~\ref{prop3.4} holds). Since $\Gamma_-(E_1,a)=0$ and
$\Gamma_+(E_2,a)&gt;0$, the hare starts out ahead of the tortoise. Since
$\Gamma_-(E_1,c)&lt;\pi$ but $\Gamma_+(E_1,c)\nearrow\pi$ as $c\nearrow b$,
the hare also ends up ahead (unlike in Aesop!).
 
Clearly, the number of times the hare crosses the finish line is the
sum of the number of times the tortoise does, plus the number of times
the hare ``laps,'' that is, passes the tortoise. Thus,
\[
n_0+m=m+W_0(\psi_-(E_1), \psi_+(E_2))
\]
so $W_0(\psi_-(E_1), \psi_+(E_2))=n_0$ in the Dirichlet case.
 
This picture also explains why it can happen that
\[
W_0(\psi_-(E_1), \psi_-(E_2))=n_0-1.
\]
For in this case, $\theta_{\psi_-}(E_1,a)=\theta_{\psi_-}(E_2,a)=0$.
The hare and tortoise start out together, so for $x=a+\epsilon$,
the hare is slightly ahead. If at $b$, $\Gamma_-(E_1,b)&gt;\Gamma_-(E_2,b)$,
then the tortoise \`a la Aesop wins the races; thus the hare has lapped
the tortoise one time too few, that is,
\[
n_0+m-1=m+W_0(\psi_-(E_1), \psi_-(E_2))
\]
and so
\begin{equation}
W_0=n_0-1. \label{4.2}
\end{equation}
 
Suppose $E_1&lt;e&lt;E_2$ with $e$ an eigenvalue. As $E_2\searrow e$,
$\Gamma_-(E_2,b)\searrow 0$ as $E_1\nearrow e$, $\Gamma_-(E_1,b)\nearrow
\pi$. Thus for $E_2-E_1$ sufficiently small, $\Gamma_-(E_2,b)&lt; \Gamma_-
(E_1,b)$ and \eqref{4.2} holds.
 
Now we turn to the proof of Proposition~\ref{prop4.1} in the general case
(assuming $H$ to be a regular SL operator for the rest of this section).
 
\begin{lemma}\label{lem4.2} Let $u_{1,2}$ be eigenfunctions of $H$ with
eigenvalues $E_1&lt;E_2$. Let $\ell$ be the number of eigenvalues of
$H$ in $(E_1, E_2)$. Then $W(u_1,u_2)(x)$ has exactly $\ell$ zeros in
$(a,b)$.
\end{lemma}
 
\begin{proof} Suppose $u_1$ is the $k$th eigenfunction. By Theorem~\ref{thm2.1},
$u_1$ has $k-1$ zeros and $u_2$ has $k+\ell$ zeros in $(a,b)$. Moreover,
$\Gamma_- (E_1,a)=\Gamma_+(E_2,a)$, $\Gamma_-(E_1,b)= \Gamma_+(E_2,b)$ so
$\Gamma_- (E_1,a+\epsilon)&lt;\Gamma_+(E_2,a+\epsilon)$, $\Gamma_+
(E_2,b-\epsilon)&lt;\Gamma_- (E_1,b-\epsilon)$. So the hare starts slightly
ahead and ends slightly behind and so it laps one less time than the
difference of the number of zeros, that is, $W_0(u_1,u_2)=(\ell+1)-1=
\ell$.
\end{proof}
 
\begin{lemma}\label{lem4.3} Let $E_1\leq E_2$ be eigenvalues of $H$ and suppose
$[E_1, E_2]$ has $\ell$ eigenvalues. Then for $\epsilon \geq 0$
sufficiently small, $W_0(\psi_-(E_1 - \epsilon), \psi_+(E_2+\epsilon))
=\ell$.
\end{lemma}
 
\begin{remark} Since $(E_1, E_2)$ has $\ell-2$ eigenvalues, Lemma~\ref{lem4.2}
says that $W(\psi_-(E_1), \psi_+(E_2))(x)$ has $\ell-2$ zeros in $(a,b)$
and clearly it has zeros at $a$ and $b$. Essentially, Lemma~\ref{lem4.3} says that
replacing $E_1$ by $E_1-\epsilon$ and $E_2$ by $E_2+\epsilon$ moves the
zeros at $a,b$ inside $(a,b)$ to give $\ell-2+2=\ell$ zeros.
\end{remark}
 
\begin{proof} Suppose first $E_1&lt;E_2$. Compare the tortoises associated
to $\psi_-(E_1-\epsilon)$ and $\psi_-(E_1)$. The first starts out at
$x=a$ in the same position as the second (i.e., $\Gamma_-(E_1-\epsilon,a)
= \Gamma_-(E_1,a)$), which means it must end slightly behind, that is,
$\Gamma_-(E_1-\epsilon,b) &lt; \Gamma_-(E_1,b)$. Similarly, since the faster
hare for energy $E_2+\epsilon$ ends up where the hare of energy $E_2$ does
(i.e, $\Gamma_+(E_2+\epsilon,b) = \Gamma_+(E_2,b)$), it must start out
slightly farther back, that is, $\Gamma_+(E_2+\epsilon,a) &lt; \Gamma_+
(E_2,a)$. Thus $W(\psi_-(E_1-\epsilon), \psi_+(E_2+\epsilon))(x)$ picks
up two zeros over the $\ell-2$ that $W(\psi_-(E_1), \psi_+(E_2))(x)$ has.
 
If $E_2=E_1\equiv E$, the $\Gamma_+$ for $\psi_+(E+\epsilon)$ starts out
slightly behind the one for $\psi_+(E)$ and ends up slightly ahead of
the $\Gamma_-$ for $\psi_-(E-\epsilon)$, and so there has to be one
crossing, that is, $W_0(\psi_-(E-\epsilon), \psi_+(E+\epsilon))=1$.
\end{proof}
 
\begin{lemma}\label{lem4.4} If $E_3&lt;E_4&lt;E$ and $u$ is any solution of
$\tau u=E u$, then
\begin{equation}
W_0(\psi_-(E_3), u)\geq W_0(\psi_-(E_4), u). \label{4.3}
\end{equation}
Similarly, if $E_3&gt;E_4&gt;E$ and $u$ is any solution of $\tau u=E u$,
then \eqref{4.3} holds.
\end{lemma}
 
\begin{proof} In the first case, think of $u$ as defining a hare and
$\psi_-(E_j)$, $j=3,4$ as defining tortoises. The $E_3$ and $E_4$
tortoises start out at the same place and the $E_3$ tortoise runs
``faster'' in that it is always ahead after the start. Clearly, the
hare will pass the slower tortoise at least as often as the faster one.
 
In the second case, there are two hares (defined by $\psi_-(E_j)$,
$j=3,4$), which start out at the same place, and one tortoise (defined
by $u$) and it is clear the faster hare (given by $\psi_-(E_3)$) has
to pass the tortoise at least as often as the slower one.
\end{proof}
 
\begin{lemma}\label{lem4.5} Lemma \ref{lem4.4} remains true if every $\psi_-$ is
replaced by a $\psi_+$.
\end{lemma}
 
\begin{proof} Reflect at some point $c \in (a,b)$ implying an
interchange of $\psi_+$ and $\psi_-$.
\end{proof}
 
\begin{proof}[Proof of Proposition \ref{prop4.1}] If $N_0=0$, there is nothing
to prove. If $N_0\geq 1$, let $\text{spec}(H)\cap(E_1, E_2)=
\{e_{m}\}_{m\in M}$ and let $e_s \leq e_\ell$ be the smallest and
largest of the $e_m$'s. Thus, $N_0$ is the number of eigenvalues in
$[e_s, e_\ell]$ and so
\[
N_0=W_0(\psi_-(e_s-\epsilon), \psi_+(e_\ell+\epsilon))
\]
by Lemma~\ref{lem4.3}. By Lemma~\ref{lem4.4},
\[
W_0(\psi_-(e_s-\epsilon), \psi_+(e_\ell+\epsilon))\leq
W_0(\psi_-(E_1), \psi_+(e_\ell+\epsilon))
\]
and then by Lemma~\ref{lem4.5}, this is no larger than $W_0(\psi_-(E_1),
\psi_+(E_2))$.
\end{proof}
 
 
\section{Strong Limits ($N_0 \leq W_0$ in the General Case)}
\label{sec5}
 
 
Using the approach of Weidmann (\cite{22}, Chapter 14) to control some limits,
we prove Theorem~\ref{thm1.5} in this section. Fix functions $u_1,u_2 \in
D_\mathrm{loc}$. Pick $c_{n}\downarrow a$, $d_{n}\uparrow b$.
Define $\tilde{H}_n$ on $L^{2}((c_{n}, d_{n}); r\,dx)$ by imposing
the following boundary conditions on $\eta\in D(\tilde{H}_{n})$
\begin{equation}
W(u_1,\eta)(c_{n})=0=W(u_2,\eta)(d_{n}). \label{5.1}
\end{equation}
On $L^{2}((a,b);r\,dx)= L^{2}((a,c_{n});r\,dx)\oplus L^{2}
((c_{n}, d_{n});r\,dx)\oplus L^{2}((d_{n}, b);r\,dx)$ take $H_{n}
=\alpha\mathbb{I}\oplus\tilde{H}_{n}\oplus\alpha\mathbb{I}$ with $\alpha$ a
fixed real constant. Then Weidmann proves:
 
\begin{lemma}\label{lem5.1} Suppose that either $H$ is limit point at
$a$ or that $u_1$ is an $\psi_-(E,x)$ for some $E$ and similarly, that
either $H$ is limit point at $b$ or $u_2$ is an $\psi_+(E',x)$ for
some $E'$. Then $H_n$ converges to $H$ in strong resolvent sense as
$n\to\infty$.
\end{lemma}
 
The idea of Weidmann's proof is that it suffices to find a core $D_0$ of
$H$ such that for every $\eta \in D_0$ there exists an $n_0 \in \mathbb{N}$
with $\eta\in D_0$ for $n \geq n_0$ and $H_{n}\eta\to H\eta$ as $n$
tends to infinity (see \cite{21}, Theorem~9.16(i)). If $H$ is limit point at
both ends, take $\eta\in D_0\equiv\{u\in D_\mathrm{loc}\mid
\text{supp}(u)\text{ compact in }(a,b)\}$. Otherwise, pick
$\tilde{u}_1,\tilde{u}_2\in D(H)$ with $\tilde{u}_2 =u_2$ near $b$ and
$\tilde{u}_2=0$ near $a$ and with $\tilde{u}_1=u_1$ near $a$ and
$\tilde{u}_1=0$ near $b$. Then pick $\eta\in D_0+\text{span}[\tilde{u}_1,
\tilde{u}_2]$ which one can show is a core for $H$ (\cite{22}, Chapter 14).
 
Secondly we note:
 
\begin{lemma}\label{lem5.2} Let $A_{n}\to A$ in strong resolvent sense as $n \to
\infty$. Then
\[
\dim\mathrm{Ran}\,P_{(E_1,E_2)}(A)\leq
\varliminf\limits_{n \to \infty} \dim\mathrm{Ran}\,
P_{(E_1,E_2)}(A_{n}). \label{5.2}
\]
\end{lemma}
 
\begin{proof}  Fix $m\leq\dim\mathrm{Ran}\,P_{(E_1,E_2)}(A)$ with
$m&lt;\infty$. We'll prove $m\leq$ RHS of \eqref{5.2}. Suppose
first that $(E_1,E_2)$ aren't eigenvalues of $A$. Then by Theorem
VIII.24 of \cite{16}, $P_{(E_1,E_2)}(A_{n})\to P_{(E_1,E_2)}(A)$ strongly
as $n \to\infty$. Picking orthonormal $\varphi_{1},\dots,\varphi_{m}$
in $\mathrm{Ran}\,P_{(E_1,E_2)}(A)$, we see that
\[
\varliminf\limits_{n \to \infty} \text{Tr}(P_{(E_1,E_2)}(A_{n}))
\geq \varliminf\limits_{n \to \infty} \sum_j \langle \varphi_j,
P_{(E_1,E_2)}(A_{n})\varphi_j \rangle_r = m
\]
as required.
 
If $E_{1,2}$ are arbitrary, we can always find a $\delta &gt; 0$ such that
$E_1+\delta, E_2-\delta$ are not eigenvalues of $A$ and such that
$\dim\mathrm{Ran}\, P_{(E_1+\delta, E_2-\delta)}(A)\geq m$. Thus,
\[
\varliminf\limits_{n \to \infty} \dim\mathrm{Ran}\,
P_{(E_1,E_2)}(A_{n})\geq \varliminf\limits_{n \to \infty}
\dim\mathrm{Ran}\, P_{(E_1+\delta, E_2-\delta)}(A_{n})\geq m.
\qed
\]
\end{proof}
 
\begin{proof}[Proof of Theorem \ref{thm1.5}] Let $c_{n}\downarrow a$, $d_{n}
\uparrow b$ and $H_n$ be as in Lemma~\ref{lem5.1} with $\alpha\notin [E_1, E_2]$.
Proposition~\ref{prop4.1} implies $W_0(u_1,u_2)\geq\dim\mathrm{Ran}\,
P_{(E_1, E_2)}(\tilde H_{n})= \dim\mathrm{Ran}\,P_{(E_1, E_2)}
(H_{n})$ since $\alpha\notin [E_1,E_2]$. Thus by Lemmas 5.1 and 5.2,
\[
W_0(u_1,u_2)\geq\dim\mathrm{Ran}\,P_{(E_1, E_2)}(H)
\]
as was to be proven.
\end{proof}
 

\section{A Variational Argument ($N_0 \geq W_0$)}
\label{sec6}
 
In this section, we'll prove Theorem~\ref{thm1.6}. Let $E_1&lt;E_2$. Suppose
first that $u_1=\psi_-(E_1)$ and $u_2=\psi_+(E_2)$. Let $x_1,\dots,
x_{m}$ be zeros of $W(u_1,u_2)(x)$. We'll prove that $\dim
P_{(E_1, E_2)}(H) \geq m$. If $W_0(u_1,u_2)=m$, this proves \eqref{1.8}. If
$W_0=\infty$, we can take $m$ arbitrarily large, and again \eqref{1.8} holds.
Define
\[
\eta_j(x) = \begin{cases} u_1(x), &amp; x\leq x_j \\
\gamma_j u_2(x), &amp; x\geq x_j \end{cases}, \quad 1 \le j \le m,
\]
where $\gamma_j$ is defined such that $\eta_j\in D(H)$ by Proposition
3.1. Let
\[
\tilde{\eta}_j(x) = \begin{cases} u_1(x), &amp; x\leq x_j \\
-\gamma_j u_2(x), &amp; x&gt;x_j \end{cases}, \quad 1 \le j \le m.
\]
If $E_2$ is an eigenvalue of $H$, we define in addition $\eta_0 = u_2
= -\tilde{\eta}_0$, $x_0=a$ and if $E_1$ is an eigenvalue of $H$,
$\eta_{m+1} = u_1 = \tilde{\eta}_{m+1}$, $x_{m+1}=b$.
 
\begin{lemma}\label{lem6.1} $\langle\eta_j, \eta_{k}\rangle_r =\langle
\tilde{\eta}_j, \tilde{\eta}_{k}\rangle_r$ for all $j,k$ where
$\langle \,\cdot\, ,\,\cdot\,\rangle_r$ is the $L^{2}((a,b);r\,dx)$
inner product.
\end{lemma}
 
\begin{proof}  Let $j&lt;k$. This just says that $\int\limits^{x_k}_{x_j}
u_1(x)u_2(x) r(x)\,dx=0$. But by \eqref{2.1}, this integral is $(E_1-E_2)^{-1}
[W(u_1,u_2)(x_{k})-W(u_1,u_2)(x_j)]=0$ since $W(u_1,u_2)(\,\cdot\,)$
vanishes at $x_\ell$ respectively in the limit (if $x_\ell = a,b$) by
Proposition~\ref{prop1.1}.
\end{proof}
 
Notice that by \eqref{3.2},
\begin{equation}
\biggl(H-\frac{E_2+E_1}{2}\biggr)\, \eta_j=
\biggl(\frac{E_1-E_2}{2}\biggr)\, \tilde{\eta}_j. \label{6.1}
\end{equation}
This result and Lemma~\ref{lem6.1} imply
 
\begin{lemma}\label{lem6.2} If $\eta$ is in the span of the $\eta_j$, then
\[
\left\|\biggl(H-\frac{E_2+E_1}{2}\biggr)\, \eta\right\|_{2,r} =
\frac{|E_2-E_1|}{2}\, \|\eta\|_{2,r}.
\]
\end{lemma}
 
Thus, $\dim\mathrm{Ran}\, P_{[E_1, E_2]}(H)\geq
\dim(\text{span}(\{\eta_j\}))$. But $u_1$ and $u_2$ are independent
on each interval (since their Wronskian is non-constant) and so the
$\eta_j$ are linearly independent. This proves Theorem~\ref{thm1.6} in the
$\psi_-(E_1), \psi_+(E_2)$ case.
 
The case $u_1=\psi_-(E_1)$, $u_2=\psi_-(E_2)$ is similar. We define
\[
\eta_j(x) = \begin{cases} u_1(x)-\gamma_j u_2(x), &amp; x\leq x_j \\
0, &amp; x\geq x_j \end{cases}, \quad 1 \le j \le m
\]
and
\[
\tilde{\eta}_j(x) = \begin{cases} u_1(x)+\gamma_j u_2(x), &amp; x\leq x_j \\
0, &amp; x&gt;x_j \end{cases}, \quad 1 \le j \le m.
\]
If $E_2$ is an eigenvalue of $H$, we define in addition $\eta_0 = u_2
= -\tilde{\eta}_0$, $x_0=b$ and if $E_1$ is an eigenvalue of $H$,
$\eta_{m+1} = u_1 = \tilde{\eta}_{m+1}$, $x_{m+1}=b$. Again, $\eta_j$'s
are linearly independent by considering their supports. To prove the
analog of Lemma~\ref{lem6.1}, we need
\[
\int\limits^{x_j}_{a}u_1(x)u_2(x)r(x)\, dx =0, \, 1\leq j \leq m.
\]
But by \eqref{2.1}, this integral is
\[
\lim\limits_{c\downarrow a}\, (E_1-E_2)^{-1}
[W(u_1,u_2)(x_j)-W(u_1,u_2)(c)].
\]
By hypothesis, $W(u_1,u_2)(x_j)=0$ and since $u_1$ and $u_2$ satisfy the
boundary condition at $a$, $W(u_1,u_2)(c)\to 0$ as $c \downarrow a$ by
Proposition~\ref{prop1.1}. The cases $u_1=\psi_+(E_1)$, $u_2=\psi_\pm(E_2)$ can be
obtained by reflection.
 

 
\section{Extensions, Comments, and Examples}
\label{sec7}
 
The following includes Theorem~\ref{thm1.7}:
 
\begin{theorem}\label{thm7.1} Let $E_1\neq E_2$. Let $\tau u_j=E_j u_j$,
$j=1,2$, $\tau v_2=E_2 v_2$ with $u_2$ linearly independent of $v_2$.
Then the zeros of $W(u_1,u_2)$ interlace the zeros of $W(u_1,v_2)$ and
vice versa (in the sense that there is exactly one zero of one
function in between two zeros of the other). In particular,
$|W_0(u_1,u_2)-W_0(u_1,v_2)| \leq 1$.
\end{theorem}
 
\begin{proof} We'll suppose $E_1&lt;E_2$. A similar argument works if
$E_2&lt;E_1$. In the language of Section~\ref{sec4}, $\Gamma_{u_1}$ represents
the tortoise and $\Gamma_{u_2},\Gamma_{v_2}$ are two hares. Since
$W(u_2,v_2)$ is a non-zero constant, one hare always stays ahead of
the other. It follows that if the hare $\Gamma_{u_1}$ crosses the tortoise
$\Gamma_{u_2}$ at $x_1$ and $x_2$, $x_1&lt;x_2$, the hare $\Gamma_{v_2}$
must cross it at some point in $(x_1, x_2)$.
\end{proof}
 
By applying this theorem twice, we conclude
 
\begin{theorem}\label{thm7.2} Let $E_1\neq E_2$. Let $u_1, u_2, v_1, v_2$ be
the linearly independent functions with $\tau u_j=E_j u_j$ and $\tau v_j
=E_j v_j$. Then
\[
|W_0(u_1, u_2)-W_0(v_1, v_2)|\leq 2.
\]
\end{theorem}
 
\begin{theorem}\label{thm7.3} If $\dim\mathrm{Ran} \, P_{(E_1,E_2)}(H)
=\infty$, then $W_0(u_1,u_2)=\infty$ for any $u_1$ and $u_2$ satisfying
$\tau u_j = E_j u_j$, $j=1,2$.
\end{theorem}
 
\begin{proof} Firstly, if $W_0(u_1,u_2)=\infty$ for one pair $u_{1,2}$
this is true for any pair by Theorem~\ref{thm7.2}. Secondly, pick $u_{1,2}$ such
that the operator $H_n$ of Lemma~\ref{lem5.1} converges to $H$ in strong resolvent
sense as $n \to \infty$. Hence by Theorem~\ref{thm1.3} (applied to $\tilde{H}_n$
defined before Lemma~\ref{lem5.1}) and Lemma~\ref{lem5.2} the number of zeros of the
Wronskian in $(c_n,d_n)$ must go to infinity as $n\to\infty$.
\end{proof}
 
\begin{example} Let us take $p=r=1$, $q=0$ with $[a,b]=[0,1]$
and Neumann boundary conditions $u'(0)=u'(1)=0$. Let $E_1= -k_1^2$,
$E_2=k_2^2$, and $u_1(x)= \psi_-(E_1,x)$, $u_2(x)=\psi_-(E_2,x)$. Then
$u_1(x)=\cosh(k_1 x)$, $u_2(x)=\cos(k_2 x)$, and
\[
W(u_1,u_2)(x) = -k_2 \cosh(k_1 x) \sin(k_2 x) -
k_1 \sinh(k_1 x)\cos (k_2 x)
\]
has no zero in $[0,1]$ if $0&lt;k_1$, $0&lt;k_2&lt;\frac{\pi}{2}$. Thus, while
$N_0=1$, $W_0=0$ so we see that $W_0=N_0-1$, that is, \eqref{1.7} in Theorem \ref{thm1.4}
can happen if the boundary conditions hold on the same side (note
that the problem is limit circle at $b=1$ as it must be, given Theorem \ref{thm1.4}).
This result is not surprising since $W(u_1,u_2)$ contains no
information about the boundary condition at $b$.
\end{example}
 
\begin{example} Again $p=r=1$, $q=0$. Take $[a,b]=[-1,1]$.
Consider the two sets of boundary conditions
\begin{align*}
&amp;\text{(B1)} &amp;&amp; u(\pm 1)=0, \\
&amp;\text{(B2)} &amp;&amp; u(\pm 1)=\pm u'(\pm 1),
\end{align*}
with corresponding operators $H_1$ and $H_2$. The lowest eigenvalue of
$H_1$ is $\frac1{4}\pi^2$. $H_2$ has $0$ as an eigenvalue with
eigenvector $\varphi(x)=x$. $H_2$ has the lowest eigenvalue at $\alpha$
where $\alpha$ satisfies $\sqrt{-\alpha}\,\tanh\sqrt{-\alpha}\,=1$
(i.e., $\alpha\sim -1.44$). Let $E_1=-2$, $E_2=0.5$, and $-u''_j= E_j
u_j$, $-v''_j= E_j v_j$, $j=1,2$, with $u_2(1)=v_1(-1)=0$ and $v_2(1)
-v'_2(1)=u_1(0)+u'_1(0)=0$. Since $H_2$ has two eigenvalues in $(E_1,
E_2)$ and $H_1$ has none, we see by Theorem~\ref{thm1.3} that $W_0(v_1,u_2)=0$,
$W_0(u_1,v_2)=2$, and thus any of $0, \pm 1, \pm 2$ can occur in
Theorem~\ref{thm7.2}.
\end{example}
 
\begin{theorem}\label{thm7.4} Let $E_1&lt;E_2$. Let $\tau u_j = E_j u_j$,
$j=1,2$. If $a &lt; x_0 &lt; x_1 &lt; b$ are zeros of $u_1$ or of $W(u_1, u_2)
(\,.\,)$, then the number of zeros of $u_2$ inside $(x_0,x_1)$ equals
the number of zeros of $W(u_1,u_2)(\,.\,)$ inside $(x_0,x_1)$ plus the
number of zeros of $u_1$ inside $(x_0,x_1)$ plus one.
\end{theorem}
 
\begin{proof} Let $\Gamma_{u_1}$ be the tortoise and $\Gamma_{u_2}$ the
hare. $\Gamma_{u_2}$ starts out ahead or equal and the number of times
$\Gamma_{u_2}$ laps (inside $(x_0,x_1))$ is equal to $W_0(u_1,u_2)$.
Since $\Gamma_{u_2}$ ends up slightly ahead (i.e., $\Gamma_{u_2}
(E_2,b-\epsilon) &gt;\Gamma_{u_1}(E_1,b-\epsilon)$), the number of zeros
of $u_2$ equals the number of laps plus the number of zeros of $u_1$
plus one.
\end{proof}
 
 
The following result is of special interest in connection with
the problem of whether the total number of eigenvalues of $H$ in
one of its essential spectral gaps is finite or infinite. In
particular, the energies $E_1, E_2$ in Theorem~\ref{thm7.5} below may lie
in the essential spectrum of $H$. For this purpose we consider an
auxiliary Dirichlet operator $H_{x_0}^D$, $x_0\in(a,b)$ associated
with $H$. $H_{x_0}^D$ is obtained by taking the direct sum of the
restrictions $H_{x_0,\pm}^D$ of $H$ to $(a,x_0)$, respectively
$(x_0,b)$, with a Dirichlet boundary condition at $x_0$. We
emphasize that the Dirichlet boundary conditions can be replaced
by boundary conditions of the type $\lim_{\epsilon\downarrow 0}
[u'(x_0\pm\epsilon)+ \beta u(x_0\pm\epsilon)] =0$, $\beta\in\mathbb{R}$.
 
\begin{theorem}\label{thm7.5}
Let $E_1&lt;E_2$. Let $\tau u_j = E_j u_j$, $\tau s_j = E_j s_j$,
and $s_j(E_j,x_0)=0$, $j=1,2$. Then we have
\begin{enumerate}
\item $\dim\mathrm{Ran}\, P_{(E_1,E_2)}(H)&lt;\infty
\quad \text {if and only if} \quad W_0(u_1,u_2)&lt;\infty$.
\item $\dim\mathrm{Ran}\, P_{(E_1,E_2)}(H)-1
\le \dim\mathrm{Ran}\, P_{(E_1,E_2)}(H^D_{x_0}) \le
\dim\mathrm{Ran}\, P_{(E_1,E_2)}(H)+2$.
\item $W_0(s_1,s_2)-1 \le \dim\mathrm{Ran}\,
 P_{(E_1,E_2)}(H^D_{x_0}) \le W_0(s_1,s_2)+1$.
\end{enumerate}
\end{theorem}
 
\begin{proof} Items (ii), (iii), and Theorem~\ref{thm7.2} imply (i). (ii) is
clear if the essential spectrum of $H$ and $(E_1,E_2)$ are not
disjoint. Otherwise, if the essential spectrum of $H$ and
$(E_1,E_2)$ are disjoint, a standard rank-one perturbation argument,
combined with the strict monotonicity of the Green's function
$G(E,x_0,x_0)$ of $H$ with respect to $E$ in essential spectral
gaps of $H$, applies.  For (iii) it suffices to prove
\[
W_{0,\pm}(s_1,s_2)\le \dim\mathrm{Ran}\,
P_{(E_1,E_2)}(H_{x_0,\pm}^D) \le
W_{0,\pm}(s_1,s_2)+1,
\]
where $W_{0,\pm}(s_1,s_2)$ abbreviates the number of zeros of
the Wronskian $W(s_1,s_2)$ inside $(x_0,b)$, respectively
$(a,x_0)$. But this is immediate from Theorems~1.5 and 1.6.
\end{proof}
 
Next we want to see how Theorem~\ref{thm1.2} (and hence Hartman's theorem
\cite{10}) follows from Theorem~\ref{thm1.4}. We start by assuming $\tau$ to be
oscillatory at $E_2$ near $b$. By Theorem~\ref{thm1.4}, $W_0(u_1,u_2)=N_0$
since $H$ in Theorem~\ref{thm1.2} is assumed to be limit point at $b$, so we
need only show that $W_0(u_1,u_2) = \varliminf\limits_{c\uparrow b}\,
N(c)$ in order to prove \eqref{1.5a}. Suppose first that $W_0(u_1,u_2)=
m&lt;\infty$. Since $\tau$ is non-oscillatory at $E_2$ near $a$ we can
pick $x_0$ such that $u_2$ and $W(u_1,u_2)$ have no zeros in $(a,x_0]$.
Hence we can assume without loss of generality that $\theta_{u_1}
(x_0)=\theta_0 \in (0,\pi)$ and $\theta_{u_2}(x_0) \in (\theta_0,\pi)$.
Let $x_m$ be the last zero of $W(u_1,u_2)(x)$ (set $m=0$ and skip
equation \eqref{7.1} if there are no zeros). At $x_m$,
\begin{equation}
\theta_{u_2}(x_{m})=\theta_{u_1}(x_{m})+m\pi \label{7.1}
\end{equation}
and then
\begin{equation}
\Gamma_{u_2}(x_{m}+\epsilon)&gt;\Gamma_{u_1}(x_{m}+\epsilon). \label{7.2}
\end{equation}
Let $N_{u_j}(x)$ be the number of zeros of $u_j$, $j=1,2$ in $(a,x)$.
By \eqref{7.1} and Proposition~\ref{prop3.3},
\[
N_{u_2}(x_{m})=N_{u_1}(x_{m})+m.
\]
As $x$ increases, \eqref{7.2} says that the next zero is of $u_2$ and then
since $W$ has no zeros, zeros of $u_1$ and $u_2$ must alternate. So
for $c&gt;x_{m}$, $N(c)\equiv N_{u_2}(c)-N_{u_1}(c)$ alternates between
$m$ and $m+1$ and since $\tau$ is assumed to be oscillatory at $E_2$
near $b$, we immediately get $\varliminf\limits_{c\uparrow b}N(c)=m$.
 
If $W_0(u_1,u_2)=\infty$, let $x_m$ be the $m$th zero. Then \eqref{7.1} and
\eqref{7.2} still hold so $N(x_{m})=m$. Since $u_2$ has zeros between any
pair of zeros of $u_1$, $N(x)\geq m$ for any $x\geq x_m$, so
$\varliminf\limits_{c\uparrow b}N(x)=\infty$, as required.
 
If $\tau$ is non-oscillatory at $E_2$ near $b$, we first assume
that $E_{1,2}$ are not eigenvalues. We need to show that the hare
ends up further along than the tortoise.  Without loss we assume
$u_{1,2}(x)&gt;0$ for $x$ near $b$ and claim in addition that $u_1 u_2$
is not $L_1$ near $b$. If $u_1&lt;u_2$ or $u_2&lt;u_1$ eventually near $b$,
we are done since $u_1 u_2 &gt; u_1^2$ or $u_1 u_2 &gt; u_2^2$ for $x$ near
$b$ and $u_j \notin L^2(\cdot; r\,dx)$ near $b$. In fact, by hypothesis, $u_j \in
L^2(\cdot; r\,dx)$ near $a$ and since $E_j$ are not eigenvalues and $\tau$ is limit
point at $b$, $u_j$ cannot be $L^2(\cdot; r\,dx)$ near $b$. Otherwise we can find
two points $x_0$ and $x_1$ close to $b$ such that $W(u_1,u_2)(x_0)
\ge 0$ and $W(u_1,u_2)(x_1) \le 0$, contradicting \eqref{2.1}. But $u_1 u_2$
not $L^1(\cdot; r\,dx)$ near $b$ together with \eqref{2.1} implies that $u_2'/u_2 &gt;
u_1'/u_1$ for $x$ near $b$ which, by monotonicity of $\cot(\,.\,)$,
yields that the hare ends up ahead.
 
It remains to treat the case where $E_{1,2}$ could be eigenvalues.
Choose $E' &lt; E''$ with $u(E')$ (resp.~$u(E'')$) equal to $\psi_{-}
(E')$ (resp.~$\psi_{-}(E'')$) the corresponding wave functions. Next,
choosing $E'$ below the spectrum of $H$ (implying that $u(E')$ has no
zeros by Corollary~\ref{cor2.4}) shows that the number of zeros of $u(E'')$
equals the number of eigenvalues below $E''$ (compare Corollary~\ref{cor2.4}),
that is, equals $\dim\mathrm{Ran}\, P_{(-\infty, E'')}(H)$
if $E''$ is not an eigenvalue. Theorem~\ref{thm2.1} then covers the case
where $E''$ is an eigenvalue. Applying this to $E''=E_1$ and
$E''=E_2$ proves \eqref{1.5b} since
\[
\dim\mathrm{Ran}\, P_{(-\infty, E'')}(H) -
\dim\mathrm{Ran}\, P_{(-\infty,E')}(H) =
\dim\mathrm{Ran}\, P_{[E', E'')}(H).
\]
 
\vskip 0.1in
 
Finally, we want to consider the relation to the density of states.
Given an SL operator $H$, let $H^D_{(L)}$ be the operator on $[-L,L]$
with Dirichlet boundary conditions. If the limit exists, we define
the integrated density of states (ids), $k(E)$, by the limit:
\[
k(E)=\lim\limits_{L\to\infty}\,(2L)^{-1}\dim\mathrm{Ran}\,
P_{(-\infty, E)} (H^D_{(L)}).
\]
 
\begin{theorem}\label{thm7.6} Suppose $H$ is such that the ids exists for all
$E$. Let $E_1&lt;E_2$ and suppose $\tau u=E_1u$, $\tau v= E_2v$. Let
$W_{(L)}$ be the number of zeros of $W(u_1,u_2)$ in $[-L,L]$. Then
\[
\lim\limits_{L\to\infty}\,(2L)^{-1}W_{(L)}=k(E_2)-k(E_1).
\]
\end{theorem}
 
\begin{proof} By Theorem~\ref{thm7.2} and Theorem~\ref{thm1.3}, $|W_{(L)}-
\dim\mathrm{Ran}\, P_{(E_1, E_2)}(H^D_{(L)})|\leq 2$, so the
result follows from
\[
\lim\limits_{L\to\infty}\,(2L)^{-1}\dim\mathrm{Ran}\,
P_{(E_1,E_2)}(H^D_{(L)}) =k(E_2)-k(E_1).
\]
\end{proof}
 
\subsection*{Acknowledgments} F.G.~is indebted to the Department of
Mathematics at Caltech for its hospitality and support during the
summer of 1994 and in February and March of 1995, where
some of this work was done.

 
 
\appendix
\section{Associated Quadratic Forms}

 
The purpose of this appendix is to clarify some form domain questions
which arise due to our general conditions on the local behavior on
$r,p$, and $q$ in \eqref{1.1}.
 
We'll consider regular SL operators and hence assume $(a,b)\subset
\mathbb{R}$ to be a finite interval with
\begin{equation}
r,p^{-1},q \in L^1((a,b); dx) \quad \text{real-valued and $r,p&gt;0$ a.e.~on }
(a,b). \label{A.1}
\end{equation}
Next, define in $L^2((a,b);r\,dx)$ the following linear operators
\begin{align*}
&amp; (H^0_{\alpha,\beta} u)(x) = -r(x)^{-1}(p(x)u'(x))', \\
&amp;D(H^0_{\alpha,\beta}) =
\{u \in L^2((a,b);r\,dx) \mid \!\!\!\begin{tabular}[t]{l}
$u,pu' \in AC([a,b]), r^{-1}(pu')' \in L^2((a,b);r\,dx)$, \\
$(pu')(a) + \alpha u(a) = (pu')(b) + \beta u(b) =0\}$, \end{tabular} \\
&amp;\qquad\qquad\qquad
\qquad\qquad\qquad \qquad\qquad\qquad \alpha,\beta \in
{\mathbb{R}} \cup \{ \infty \}
\end{align*}
(here $\alpha=\infty$ denotes a Dirichlet boundary condition $u(a)
=0$ and similarly at $b$),
\begin{align*}
&amp; S_{\alpha,\beta} u = s\, u, \quad (s\, u)(x) =
(p(x)/r(x))^{1/2}u'(x), \quad \alpha,\beta \in \{0,\infty\}, \\
&amp; D(S_{\alpha,\beta}) =
\{u \in L^2((a,b);r\,dx) \mid \!\!\!\begin{tabular}[t]{l}
$u \in AC([a,b]), s\, u \in L^2((a,b);r\,dx)$, \\
$u(a) = 0 \text{ if }
\alpha =\infty,\, u(b) = 0 \text{ if } \beta =\infty \}$,
\end{tabular}\\
&amp;S^+_{\alpha,\beta} u = s^+ u, \quad  (s^+ u)(x) =
-r(x)^{-1}[(p(x)r(x))^{1/2}u(x)]',
\quad \alpha,\beta \in \{0,\infty\},\\
&amp; D(S^+_{\alpha,\beta}) =
\{u \in L^2((a,b);r\,dx) \mid \!\!\!\begin{tabular}[t]{l}
$(pr)^{1/2}u \in AC([a,b]),s^+ u \in L^2((a,b);r\,dx)$, \\
$((pr)^{1/2}u)(a) = 0 \text{ if } \alpha = 0,
((pr)^{1/2}u)(b) = 0
\text{ if } \beta =0 \}$, \end{tabular}
\end{align*}
and the form
\[
R^0_{\alpha,\beta}(u,v) = \langle S_{\alpha,\beta}u,
S_{\alpha,\beta}v \rangle_r,
\quad D(R^0_{\alpha,\beta}) =  D(S_{\alpha,\beta}),
\quad \alpha,\beta \in
\{0,\infty\}
\]
($\langle\,.\, ,\,.\,\rangle_r$ the scalar product in $L^2((a,b);r\,dx)$).
 
\begin{lemma}\label{lemA.1} (i) $S_{\alpha,\beta} =
(S_{\alpha,\beta}^+)^*$ and $S_{\alpha,\beta}^+ = S^*_{\alpha,\beta}$
for all $\alpha,\beta \in \{0, \infty\}$.
 
(ii) $H^0_{\alpha,\beta} = S^*_{\alpha,\beta}
S_{\alpha,\beta}$, $\alpha,\beta \in \{0, \infty\}$.
\end{lemma}
 
\begin{proof} Define
\begin{align*}
&amp; K: L^2((a,b);r\,dx) \to D(S_{\infty,0}),
&amp;&amp; \widehat K: L^2((a,b);r\,dx) \to D(S^+_{0,\infty}),\\
&amp; \quad g \mapsto \int\limits_a^x
\frac{g(y)r(y)\,dy}{(p(y)r(y))^{1/2}},
&amp;&amp; \quad g \mapsto - (pr)(\,x\,)^{-1/2}
\int\limits_a^x g(y)r(y)\,dy .
\end{align*}
A straightforward calculation verifies $(K\,g)(a)=0$,
$sK\,g = g$ and $\big((pr)^{1/2} \widehat K\,g\big)(a)=0$, $s^+ \widehat K\,g = g$.
 
We only show $S_{\alpha,\beta}^* = S^+_{\alpha,\beta}$, the case
$(S^+_{\alpha,\beta})^* = S_{\alpha,\beta}$ being analogous. Moreover,
since $S_{\infty,\infty} \subseteq S_{\alpha,\beta}$ implies
$S_{\alpha,\beta}^* \subseteq S_{\infty,\infty}^*$ we only concentrate
on proving $S_{\infty,\infty}^* = S_{\infty,\infty}^+$, the rest
following from an additional integration by parts.
 
An integration by parts proves $S^+_{\infty,\infty}\subseteq
S^*_{\infty,\infty}$. Conversely, let $f \in D(S_{\infty,\infty}^*)$
and set $g = \widehat K S_{\infty,\infty}^* f$. Then
\[
\int\limits_a^b (\bar{f}-\bar{g}) (S_{\infty,\infty} h) r\, dx
= \int\limits_a^b (S_{\infty,\infty}^* \bar{f}-s^+ \bar{g}) hr \,
dx =0
\]
for all $h\in D(S_{\infty,\infty})$. Thus, $\mathrm{Ran}
(S_{\infty,\infty})$ is a subset of the kernel of the linear functional
$k\mapsto \langle f-g,k \rangle_r$. But $\mathrm{Ran} (S_{\infty,
\infty})= \{(pr)^{-1/2}\}^\perp$ (since $g\in \mathrm{Ran}
(S_{\infty,\infty})$ is equivalent to $(Kg)(b)=0$) and hence $f=g+c
(pr)^{-1/2} \in D(S^+_{\infty,\infty})$ for some constant $c$ proving
$S_{\infty,\infty}^* \subseteq S^+_{\infty,\infty}$ and hence (i).
 
By inspection, we obtain $D(S^+_{\alpha,\beta} S_{\alpha,\beta}) =
\{u\in D(S_{\alpha,\beta}) \mid S_{\alpha,\beta}u \in D(S^+_{\alpha,
\beta})\} = D(H^0_{\alpha,\beta})$ since $pu'\in AC([a,b])$ implies
$(p/r)^{1/2} u'= (pr)^{-1/2}(pu')\in L^2((a,b);r\,dx)$ and
$S^+_{\alpha,\beta} S_{\alpha,\beta}u = H^0_{\alpha,\beta}u.$ This
fact together with (i) proves (ii).
\end{proof}
 
Furthermore, we introduce the forms
\begin{align*}
Q_{q/r}(u,v) = \int\limits_a^b q(x)r(x)^{-1}\, \overline{u(x)}\,
v(x) r(x)\, dx, \\
D (Q_{q/r}) = \{u\in L^2((a,b);r\,dx)\mid (|q|/r)^{1/2} u \in
L^2((a,b);r\,dx)\},
\end{align*}
and 
\begin{align*}
q_{\alpha,\beta}(u,v)= \tilde{\beta}\, \overline{u(b)}\, v(b)
- \tilde{\alpha}\, \overline{u(a)}\, v(a), \quad D(q_{\alpha,\beta}) = AC([a,b]),\\
\tilde{\alpha} = \begin{cases} \alpha, &amp; \alpha\in\mathbb{R} \\
0, &amp; \alpha=\infty \end{cases},
\quad \tilde{\beta} = \begin{cases} \beta, &amp; \beta\in\mathbb{R} \\
0, &amp; \beta=\infty \end{cases},
\quad \alpha,\beta \in \mathbb{R} \cup \{\infty\}.
\end{align*}
Finally, we set
\begin{align*}
Q^0_{\alpha,\beta} = R^0_{\widehat{\alpha},\widehat{\beta}}
+ q_{\alpha,\beta}, \quad D (Q^0_{\alpha,\beta}) =
D(S_{\widehat{\alpha},\widehat{\beta}}), \\
\widehat{\alpha} = \begin{cases} 0, &amp; \alpha\in\mathbb{R} \\
\infty, &amp; \alpha=\infty \end{cases},
\quad \widehat{\beta} = \begin{cases} 0, &amp; \beta\in\mathbb{R} \\
\infty, &amp; \beta=\infty \end{cases},
\quad \alpha,\beta \in\mathbb{R} \cup \{\infty\} 
\end{align*}
and
\begin{equation}
Q_{\alpha,\beta} = Q^0_{\alpha,\beta} + Q_{q/r}, \quad
D (Q_{\alpha, \beta}) = D(S_{\widehat{\alpha}, \widehat{\beta}}),
\quad \alpha,\beta \in {\mathbb{R}} \cup \{\infty\}. \label{A.2}
\end{equation}
 
\begin{lemma}\label{lemA.2} (i) $q_{\alpha,\beta}$ is infinitesimally
form bounded with respect to $Q^0_{0,0}$.
 
(ii) $Q_{q/r}$ is relatively form compact with respect to
$Q^0_{\alpha,\beta}$, $\alpha,\beta \in\mathbb{R} \cup \{\infty\}$.
\end{lemma}

\begin{proof} (i) Since for arbitrary $c \in [a,b]$ and $u \in
D(S_{0,0})$,
\[
|u(c)|^2 = \biggl| u(x)^2 - 2 \int\limits_c^x u(t) u'(t)\,dt\biggr|
\leq |u(x)|^2 + 2 \int\limits_a^b |u(t) u'(t)|\,dt,
\]
one infers (after taking the supremum over all $c \in [a,b]$, multiplying by $r$, and integrating from $a$ to $b$) for 
any $\epsilon&gt;0$,
\begin{align}\nonumber
\| u \|_{L^\infty((a,b); dx)}^2 &amp;\leq \|r\|^{-1}_{L^1((a,b); dx)} \| u \|^2_{2,r} + 2 \int\limits_a^b
\frac{|u(t)|}{(\epsilon p(t)/2)^{1/2}}\, (\epsilon p(t)/2)^{1/2}
|u'(t)|\,dt \\
&amp; \leq \|r\|^{-1}_{L^1((a,b); dx)} \| u \|^2_{2,r} + \int\limits_a^b \biggl(\frac{2}
{\epsilon}\, \frac{|u(t)|^2}{p(t)} + \frac{\epsilon}{2}\, p(t)
|u'(t)|^2 \biggr)\, dt. \label{A.3}
\end{align}
Since $0&lt;p^{-1} \in L^1((a,b);dx)$, we can find a $\delta_1(\epsilon) &gt; 0$ such
that $\int\limits_{I_1(\epsilon)} p(t)^{-1}\, dt \leq
\frac{\epsilon}{8}$ with $I_1(\epsilon) = \{x\in (a,b)\mid
p(x)&lt; \delta_1(\epsilon)\}$. Thus,
\[
\int\limits_a^b \frac{|u(t)|^2}{p(t)}\, dt =
\int\limits_{I_1(\epsilon)} \frac{|u(t)|^2}{p(t)}\, dt +
\int\limits_{(a,b)\backslash I_1(\epsilon)}
\frac{|u(t)|^2}{p(t)}\, dt \leq \frac{\epsilon}{8}\,
 \| u \|_{L^\infty((a,b);dx)}^2 + \frac{1}{\delta_1(\epsilon)}\,  \int_a^b |u(t)|^2 dt.
\]
In addition, since $r &gt; 0$ a.e.\ on $(a,b)$, we can find a $\delta_2(\epsilon)&gt;0$ such
that $|I_2(\epsilon)| \leq \frac{\epsilon \delta_1(\epsilon)}{8}$ with $I_2(\epsilon) = \{x\in (a,b)\mid
r(x)&lt; \delta_2(\epsilon)\}$. Thus,
\[
\int\limits_a^b |u(t)|^2\, dt =
\int\limits_{I_2(\epsilon)}|u(t)|^2\, dt +
\int\limits_{(a,b)\backslash I_2(\epsilon)}
|u(t)|^2\, dt \leq \frac{\epsilon \delta_1(\epsilon)}{8}\,
 \| u \|_{L^\infty((a,b);dx)}^2 + \frac{1}{\delta_2(\epsilon)}\, \|u\|^2_{2,r}
\]
and one obtains from \eqref{A.3},
\begin{equation}
 \| u \|_{L^\infty((a,b);dx)}^2 \leq 2\big\{\|r\|^{-1}_{L^1((a,b); dx)} +
2[\epsilon\delta_1(\epsilon)\delta_2(\epsilon)]^{-1}\big\} \| u \|_{2,r}^2 +
\epsilon Q^0_{0,0}(u,u), \label{A.4}
\end{equation}
completing the proof of (i).
 
(ii) Let $G^0_{\alpha,\beta}(z,x,y)$ denote the Green's function of
$H^0_{\alpha,\beta}$, $\alpha,\beta \in\mathbb{R}\cup\{\infty\}$, that is,
\[
((H^0_{\alpha,\beta}-z)^{-1} u)(x) = \int\limits_a^b
G^0_{\alpha,\beta}(z,x,y) u(y)r(y)\, dy, \quad
z \in\mathbb{C} \backslash\mathbb{R}.
\]
Then $|q/r|^{1/2} (H^0_{\alpha,\beta}-z)^{-1} |q/r|^{1/2} \in
{\frak B}_2(L^2((a,b);r\,dx))$ (${\frak B}_2(\,.\,)$ the set of
Hilbert-Schmidt operators) since
\begin{align*}
\| |q/r|^{1/2} (H^0_{\alpha,\beta}-z)^{-1} |q/r|^{1/2} \|^2_2 &amp;=
\int\limits_a^b \!\! \int\limits_a^b \frac{|q(x)|}{r(x)}\,
|G^0_{\alpha,\beta}(z,x,y)|^2 \frac{|q(y)|}{r(y)}\,
r(x)r(y)\, dxdy \\
&amp;\leq M(z) \biggl[ \int\limits_a^b |q(x)|\, dx \biggr]^2
\end{align*}
using the fact that $|G^0_{\alpha,\beta}(z,\,.\,,\,.\,)|$ is bounded
on $(a,b) \times (a,b)$.
\end{proof}
 
Thus the forms $Q_{\alpha,\beta}$ in \eqref{A.2} are densely defined, closed,
and bounded from below (\cite{12}, Section VI.1). We denote by
$H_{\alpha,\beta}$ the uniquely associated self-adjoint operators
bounded from below guaranteed by the KLMN theorem (\cite{12},
Theorem VI.2.1; \cite{17}, Theorem X.17). The following theorem identifies
$H_{\alpha,\beta}$ as the usual regular SL operators (with separated
boundary conditions).
 
\begin{theorem}\label{thmA.3} $H_{\alpha,\beta}$ associated with
$Q_{\alpha,\beta}$ is given by
\begin{align}\nonumber
&amp;(H_{\alpha,\beta} u)(x) = r(x)^{-1}[-(p(x)u'(x))' +q(x) u(x)] ,\\ \nonumber
&amp;D(H_{\alpha,\beta}) =
\{u \in L^2((a,b);r\,dx)\mid \!\!\!\begin{tabular}[t]{l} $u,pu' \in AC([a,b]),
r^{-1}(-(pu')'+qu) \in L^2((a,b);r\,dx)$, \\
$(pu')(a) + \alpha u(a) = (pu')(b) + \beta u(b) =0\}$,
\end{tabular}\\
&amp;\qquad\qquad\qquad
\qquad\qquad\qquad \alpha,\beta \in\mathbb{R} \cup \{\infty\}. \label{A.5}
\end{align}
\end{theorem}
 
\begin{proof} It suffices to consider the Dirichlet case $\alpha
=\beta=\infty$, the other cases being similar. Denote by
$\widehat{H}_{\infty,\infty}$ the operator defined in \eqref{A.5} for
$\alpha=\beta=\infty$ and by $H_{\infty,\infty}$ the unique operator
associated with $Q_{\infty,\infty}$. Choose $u \in D
(Q_{\infty,\infty})$ and $v \in D(\widehat{H}_{\infty,\infty})$. Then
an integration by parts yields
\[
Q_{\infty,\infty}(u,v) = \langle u, \widehat{H}_{\infty,\infty}
v \rangle_r.
\]
Thus $\widehat{H}_{\infty,\infty} \subseteq H_{\infty,\infty}$ by
Corollary VI.2.4 of \cite{12} and hence $\widehat{H}_{\infty,\infty} =
H_{\infty,\infty}$ since $\widehat{H}_{\infty,\infty}$ is self-adjoint.
\qed
\end{proof}
 
\begin{remark} It follows from the above theorem, that for arbitrary
SL operators $H$ (not necessarily regular), elements 
$u \in L^2((a,b); r\,dx)$ which satisfy
\begin{equation}
u \in AC_\mathrm{loc}((a,b)), \quad (p/r)^{1/2}u' \in
L^2_\mathrm{loc}((a,b);r\,dx) \label{A.6}
\end{equation}
and which are in the domain of $H$ near $a$ and $b$, are in the form
domain of $H$. Moreover, let $u(x),v(x)$ be as in \eqref{A.6} and in $D(H)$
for $x \leq c$ and $x \geq d $, then
\begin{equation}
\begin{split}
Q_H(u,v) =&amp; \int\limits_{(a,b) \backslash (c,d)} \overline{u(x)}\,
(\tau v)(x)\, dx + \overline{u(d)}\,(pv')(d) - \overline{u(c)}\,
(pv')(c) \\
&amp; \qquad + \int\limits_c^d [\overline{u'(x)}\,(pv')(x) + q(x)
\overline{u(x)}\,v(x))]\,dx.
\end{split} \label{A.7}
\end{equation}
In fact, take $u$ as in \eqref{A.6} and in $D(H)$ for $x\leq c$ and $x\geq d$.
Consider the operator $\tilde{H}_{\alpha,\beta}$ associated with $\tau$
and boundary conditions induced by $u$ on the finite interval $(c,d)$
(cf.~\eqref{5.1}). Since $u\!\restriction_{(c,d)}\in D (\tilde{Q}_{\alpha,
\beta})$ ($\tilde{Q}_{\alpha,\beta}$ the form of $\tilde{H}_{\alpha,
\beta}$), we can pick a sequence $\tilde{u}_n$ in $D(\tilde{H}_{\alpha,
\beta})$ such that $\| \tilde{u}_n - u\!\restriction_{(c,d)} \|_{2,r} \to 0$
and $\langle(\tilde{u}_n - \tilde{u}_m),\tilde{H}_{\alpha,\beta}
(\tilde{u}_n -\tilde{u}_m) \rangle_r \to 0$ (implying $\| \tilde{u}_n -
u\restriction_{(c,d)} \|_{L^\infty((a,b);dx)} \to 0$ by \eqref{A.4} and Lemma~\ref{lemA.2}). Extend
$\tilde{u}_n$ to a function $u_n$ on $(a,b)$ by patching it with $u$
such that $u_n \in D(H)$ (which is possible since $u$ and $\tilde{u}_n$
satisfy the same boundary conditions at $c$ and $d$). By construction,
$u_n$ satisfies $\| u_n - u \|_{2,r} \to 0$ and $\langle (u_n - u_m),
H(u_n - u_m) \rangle_r \to 0$ and hence is in the form domain of $H$.
This proves \eqref{A.6} and an integration by parts then proves \eqref{A.7}.
\end{remark}
 

 
\begin{thebibliography}{99}
 
\bibitem{1} F.V.~Atkinson, {\em Discrete and Continuous Boundary
Problems}, Academic Press, New York, 1964.

\bibitem{2} M.~B\^ocher, {\em Le\c{c}ons de M\'ethodes de Sturm},
Gauthier-Villars, Paris, 1917.

\bibitem{3} M.~Buys and A.~Finkel, {\em The inverse periodic
problem for Hill's equation with a finite-gap potential}
J.~Diff.~ Eq. {\bf 55} (1984), 257--275.

\bibitem{4} W.A.~Coppel, {\em Disconjugacy}, Lecture Notes in
Mathematics {\bf 220}, Springer, Berlin, 1971.

\bibitem{5} R.~Courant and D.~Hilbert, {\em Methods of Mathematical
Physics, Volume I}, Wiley, New York, 1989.

\bibitem{6} N.~Dunford and J.T.~Schwartz, {\em Linear Operators},
Part II: Spectral Theory, Wiley, New York, 1988

\bibitem{7} A.~Finkel, E.~Isaacson, and E.~Trubowitz, {\em An
explicit solution of the inverse periodic problem for Hill's equation},
SIAM J.~Math.~Anal. {\bf 18} (1987), 46--53.

\bibitem{8} F.~Gesztesy, B.~Simon, and G.~Teschl, {\em Spectral
deformations of one-dimensional Schr\"odinger operators},  
J. d'Anal. Math. {\bf 70} (1996), 267--324.

\bibitem{9} P.~Hartman, {\em Differential equations with
non-oscillatory eigenfunctions},  Duke Math.~J. {\bf 15} (1948),
697--709.

\bibitem{10}same, {\em A characterization of the spectra of
one-dimensional wave equations},  Am.~J.~Math. {\bf 71} (1949),
915--920.

\bibitem{11} P.~Hartman and C.R.~Putnam, {\em The least cluster
point of the spectrum of boundary value problems},  Am.~J.~Math.
{\bf 70} (1948), 849--855.

\bibitem{12} T.~Kato, {\em Perturbation Theory for Linear Operators},
2nd ed., Springer, Berlin, 1980.

\bibitem{13} K.~Kreith, {\em Oscillation Theory}, Lecture Notes
in Mathematics {\bf 324}, Springer, Berlin, 1973.

\bibitem{14} W.~Leighton, {\em On self-adjoint differential
equations of second order},  J.~London Math.~Soc. {\bf 27}
(1952), 37--47.

\bibitem{15} M.A.~Naimark, {\em Linear Differential Operators,
Part II}, Ungar, New York, 1968.

\bibitem{16} M.~Reed and B.~Simon, {\em Methods of Mathematical
Physics.~I. Functional Analysis}, rev.~and enl.~ed.,
Academic Press, New York, 1980.

\bibitem{17} M.~Reed and B.~Simon, {\em Methods of Mathematical Physics.~II.
Fourier Analysis, Self-Adjointness}, Academic Press, New York, 1975.

\bibitem{18} W.T.~Reid, {\em Sturmian Theory for Ordinary
Differential Equations}, Springer, New York, 1980.

\bibitem{19} C.A.~Swanson, {\em Comparison and Oscillation Theory
of Linear Differential Equations}, Academic Press, New York, 1968.

\bibitem{20} J.C.F.~Sturm, {\em M\'emoire sur les \'equations
diff\'erentielles lin\'eaires du second ordre},  J.~Math.~Pures
Appl. {\bf 1} (1836), 106--186.

\bibitem{21} J.~Weidmann, {\em Linear Operators in Hilbert Spaces},
Springer, New York, 1980.

\bibitem{22}  J.~Weidmann, {\em Spectral Theory of Ordinary Differential
Operators}, Lecture Notes in Mathematics {\bf 1258}, Springer, Berlin, 1987.

\end{thebibliography}
 
\end{document}
 
 
 
Since $0&lt;p^{-1} \in L^1((a,b);dx)$, we can find a $\delta(\epsilon)$ such
that $\int\limits_{I_{\delta(\epsilon)}} p(t)^{-1}\, dt \leq
\frac{\epsilon}{4}$ with $I_{\delta(\epsilon)} = \{x\in (a,b)\mid
p(x)\leq \delta(\epsilon)\}$. Thus,
\[
\int\limits_a^b \frac{|u(t)|^2}{p(t)}\, dt =
\int\limits_{I_{\delta(\epsilon)}} \frac{|u(t)|^2}{p(t)}\, dt +
\int\limits_{(a,b)\backslash I_{\delta(\epsilon)}}
\frac{|u(t)|^2}{p(t)}\, dt \leq \frac{\epsilon}{4}\,
 \| u \|_{L^\infty ((a,b); dx)}^2 + \frac{1}{\delta(\epsilon)}\, \| u \|_{2,r}^2
\]
and one obtains from \eqref{A.3},
\begin{equation}
 \| u \|_{L^\infty ((a,b); dx)}^2 \leq 2[(b-a)^{-1} +
2(\epsilon\delta(\epsilon))^{-1}] \| u \|_2^2 +
\epsilon Q^0_{0,0}(u,u), \label{A.4}
\end{equation}
completing the proof of (i).
 
(ii) Let $G^0_{\alpha,\beta}(z,x,y)$ denote the Green's function of
$H^0_{\alpha,\beta}$, $\alpha,\beta \in\mathbb{R}\cup\{\infty\}$, that is,
\[
((H^0_{\alpha,\beta}-z)^{-1} u)(x) = \int\limits_a^b
G^0_{\alpha,\beta}(z,x,y) u(y)r(y)\, dy, \quad
z \in\mathbb{C} \backslash\mathbb{R}.
\]
Then $|q/r|^{1/2} (H^0_{\alpha,\beta}-z)^{-1} |q/r|^{1/2} \in
{\frak B}_2(L^2((a,b);r\,dx))$ (${\frak B}_2(\,.\,)$ the set of
Hilbert-Schmidt operators) since
\[
\align
\| |q/r|^{1/2} (H^0_{\alpha,\beta}-z)^{-1} |q/r|^{1/2} \|^2_2 &amp;=
\int\limits_a^b \!\! \int\limits_a^b \frac{|q(x)|}{r(x)}\,
|G^0_{\alpha,\beta}(z,x,y)|^2 \frac{|q(y)|}{r(y)}\,
r(x)r(y)\, dxdy \\
&amp;\leq M(z) \biggl[ \int\limits_a^b |q(x)|\, dx \biggr]^2
\endalign
\]
using the fact that $|G^0_{\alpha,\beta}(z,\,.\,,\,.\,)|$ is bounded
on $(a,b) \times (a,b)$.

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