\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 48, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/48\hfil Eigenvalue problems]
{Eigenvalue problems of Atkinson, Feller and Krein, 
 and their mutual relationship}
\author[H. Volkmer\hfil EJDE-2005/48\hfilneg]
{Hans Volkmer}  

\address{Department of Mathematical Sciences\\
University of Wisconsin--Milwaukee\\
P. O. Box 413\\
Milwaukee, WI 53201 USA}
\email{volkmer@csd.uwm.edu}

\date{}
\thanks{Submitted September 13, 2004. Published April 28, 2005.}
\subjclass[2000]{34B24, 34B09}
\keywords{Sturm-Liouville problem; vibrating string; \hfill\break\indent
Krein-Feller eigenvalue problem; Atkinson eigenvalue problem}

\begin{abstract}
 It is shown that every regular Krein-Feller eigenvalue problem
 can be transformed to a semidefinite Sturm-Liouville problem
 introduced by Atkinson. This makes it possible to transfer results
 between the corresponding theories. In particular, Pr\"ufer angle
 methods become available for Krein-Feller problems.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{rem}[thm]{Remark}

\section{Introduction}

The classical regular Sturm-Liouville
eigenvalue problem consists of the quasi-differential equation
\begin{equation}\label{intro:SL1}
-(p(x)y')'+q(x)y=\lambda r(x) y,\quad x\in[c,d]\,\, a.e.
\end{equation}
and the boundary conditions
\begin{equation}\label{intro:SL2}
y(c)\cos\alpha=(py')(c)\sin\alpha,\quad y(d)\cos\beta=(py')(d)\sin\beta.
\end{equation}
It is assumed that $1/p$, $q$, $r$ are real-valued Lebesgue integrable
functions on $[c,d]$,
$p(x)>0$, $r(x)>0$ a.e.\ and $\alpha,\beta$ are real.
If \eqref{intro:SL1}, \eqref{intro:SL2} admit a nontrivial solution
$y$ for some $\lambda$, then $\lambda$ is called an eigenvalue and $y$ is a corresponding
eigenfunction. If the eigenfunction $y$ has exactly $n$
zeros in $(c,d)$ then $\lambda$ is called an eigenvalue with oscillation count $n$.
It is well known that, for every nonnegative integer $n$, there is a unique eigenvalue
$\lambda_n$ with oscillation count $n$, and these eigenvalues form an increasing sequence
$\lambda_0<\lambda_1<\lambda_2<\dots$ converging to infinity.
More precisely, the eigenvalues grow according to
\begin{equation}\label{intro:asy}
 \lim_{n\to\infty} n^{-2}\lambda_n=\frac{\pi^2}{L^2},\quad
 L:=\int_c^d \Big(\frac{r(x)}{p(x)}\Big)^{1/2}\,dx.
\end{equation}
These results can be proved conveniently with the help of the Pr\"ufer angle.

In this paper we will consider generalizations of the classical Sturm-Liouville
eigenvalue problem introduced by Atkinson, Feller and Krein, and
explore their relationship.

Setting $s=1/p$, $u=y$, $v=pu'$, Atkinson \cite[Chapter 8]{A} writes equation
\eqref{intro:SL1} as a Carath\'eodory system
\begin{equation}\label{intro:A1}
u'=s(x)v,\quad v'=(q(x)-\lambda r(x))u,
\end{equation}
and the boundary conditions \eqref{intro:SL2} as
\begin{equation}\label{intro:A2}
u(c)\cos\alpha=v(c)\sin\alpha,\quad u(d)\cos\beta=v(d)\sin\beta.
\end{equation}
He assumes that $s,q,r$ are real-valued integrable functions with $r(x)\ge 0$, $s(x)\ge 0$
for all $x\in[c,d]$.
Under some additional assumptions Atkinson \cite[Theorems 8.4.5, 8.4.6]{A}
proved existence and uniqueness
of eigenvalues with prescribed oscillation count provided the latter is
appropriately defined. We explain this and related results
on Atkinson's problem in Section \ref{SL} in more detail. Here we just mention that
the asymptotic formula \eqref{intro:asy} still holds \cite{BVasy}.

Krein \cite{GK,KK,K1,K2} introduced the eigenvalue problem for the vibrating string
with finite mass.
If we set $p=1$, $q=0$ in the classical Sturm-Liouville problem then we obtain
the eigenvalue problem of a vibrating string. The part of the string lying over the
interval $[c,x]$ has mass $\int_c^x r(t)\,dt$.
In Krein's eigenvalue problem the mass of the string occupying
$[c,x]$ is given by $\omega([c,x])$, where $\omega$ is a measure on the
Borel subsets of $[c,d]$.
Equations \eqref{intro:A1} have now to be interpreted as integral equations
\[ u(x)-u(c)=\int_{[c,x)} v(t)\,dt,\quad v(x)-v(c)=-\lambda \int_{[c,x)} u(t)\,
d\omega(t) .\]
These equations can be combined into one equation
\[ u(x)-u(c)-xu'(c)=-\lambda\int_{[c,x)} (x-t) u(t)\,d\omega(t);\]
see \cite[page 24]{A}. For Krein's problem we also have
a theorem on the existence of eigenvalues with prescribed oscillation count, and there
is also an asymptotic result similar to \eqref{intro:asy}.
Feller \cite{F1, F2} gave an alternative treatment of the vibrating string problem
based on generalized second derivatives.
Krein-Feller problems will be considered in Section \ref{MSL}.

Comparing the various eigenvalues problems, we note that Atkinson's problem
contains the classical Sturm-Liouville problem as a special case.
However, the relationship between
problems of Krein-Feller and Atkinson is less obvious. Atkinson \cite[page 202]{A}
points out that the vibrating string with beads (a string whose mass
is concentrated at finitely many points) can be
treated within his framework.

We show in this paper that every regular Krein-Feller eigenvalue problem can be transformed
to an equivalent problem of Atkinson's type. The construction of
the transformation depends on the Radon-Nikodym theorem.
Therefore, Atkinson's problem contains  all the other problems as special cases.
This makes it possible to create a unified theory where all results
are first proved in Atkinson's setting and then are specialized as needed.
In Section \ref{MSL} we apply this idea to obtain existence of eigenvalues
of Krein-Feller problems with prescribed oscillation counts and their asymptotics
from the corresponding results for Atkinson's problems.
We use these results as examples. In fact, it may be expected that
all results for Krein-Feller problems
can be obtained from corresponding ones for Atkinson's problem.
This should also apply to the
operator-theoretic aspect including expansion theorems as well as to
singular problems. For the theory of Krein-Feller operators see Fleige \cite{Fl}.

In Section \ref{MSL}, we introduce a measure Sturm-Liouville problem in which all
three functions $s,q,r$ are replaced by measures. These problems are
more general than those of Krein and Feller. We go on to show that under
a condition on the location of the atoms of these measures the measure
Sturm-Liouville problem is also a special case of Atkinson's problem.

In Sections \ref{mi} and \ref{tmi}, we deal with linear integral equations
that generalize \break
Carath\'eodory systems of differential equations and their
transformations.
The eigenvalue problems of Krein, Feller and our measure Sturm-Liouville
problem will involve such systems.

\section{Measure integral equations}\label{mi}

Let $\mathcal{B}$ denote the $\sigma$-algebra of Borel subsets of the interval $[a,b]$.
For each $i,j=1,\dots,n$, let $\omega_{ij}:\mathcal{B}\to\mathbb{C}$ be a complex-valued measure.
Consider the linear system
\begin{equation}\label{mi:eq2}
y_i(t)=y_i(a)+\sum_{j=1}^n \int_{[a,t)} y_j(s)d\omega_{ij}(s),\quad t\in[a,b],\, i=1,\dots,n .
\end{equation}
A solution is a bounded Borel measurable function $y:[a,b]\to\mathbb{C}^n$,
with $y(t)=(y_1(t),\dots,y_n(t))$,
that satisfies \eqref{mi:eq2}.
Note that since $y$ is bounded and Borel measurable, the integrals on the right-hand side of
\eqref{mi:eq2} are well-defined.
Atkinson \cite[Section 11.8]{A} considers such systems but he uses Stieltjes
integrals; see also \cite{N}. We prefer to work with integrals over measure spaces.

\begin{lemma}\label{mi:l1}
Let $y$ be a  solution of \eqref{mi:eq2}.
Then $y$ is of bounded
variation so that the one-sided limits $y(t+)$, $t\in[a,b)$ and $y(t-)$, $t\in(a,b]$ exist.
Moreover, $y$ is left-continuous at every $t\in(a,b]$, and
\begin{equation}\label{mi:jump}
 y_i(t+)-y_i(t)=\sum_{j=1}^n \omega_{ij}(\{t\})y_j(t),\quad t\in[a,b) .
\end{equation}
\end{lemma}

\begin{proof}
If $f$ is integrable with respect to a measure $\omega:\mathcal{B}\to\mathbb{C}$ then
\[ F(t):=\int_{[a,t)} f(s)\,d\omega(s)\]
defines a function $F:[a,b]\to\mathbb{C}$. For any partition
\[a=t_0<t_1<t_2<\dots<t_m=b,\]
we have
\[ \sum_{k=1}^m|F(t_k)-F(t_{k-1})|\le \int_{[a,b)} |f(t)|\,d|\omega|(t),\]
where $|\omega|$ denotes the total variation of $\omega$.
This shows that $F$ is of bounded variation.
Moreover, $F$ is left-continuous and
\[ F(t+)-F(t)=f(t)\omega(\{t\}) .\]
The statements of the lemma follow.
\end{proof}

By definition, atoms at $t=b$ do not enter the linear system \eqref{mi:eq2}.
However, we may use equation \eqref{mi:jump} to define $y(b+)$. Then $y(b+)$ depends on atoms at
$t=b$.
Note that if the measures $\omega_{ij}$, $j=1,\dots,n$,
have no atoms at $t$ then $y_i$ will be continuous at $t$.

Choose a measure $\omega:\mathcal B\to[0,\infty)$ such that each $\omega_{ij}$
is absolutely continuous with respect to $\omega$, for instance,
\[
\omega=\sum_{i=1}^n\sum_{j=1}^n |\omega_{ij}| .
\]
By the Radon-Nikodym theorem, there exist functions $w_{ij}$ (Radon-Nikodym derivatives),
integrable with respect to $\omega$, such that
\[ \omega_{ij}(E)=\int_E w_{ij}(t)\,d\omega(t),\quad  E\in\mathcal{B}.\]
In particular, for every bounded Borel measurable function $f:[a,b]\to\mathbb{C}$,
\[ \int_E f(t)\,d\omega_{ij}(t)=\int_E f(t)w_{ij}(t)\,d\omega(t) .\]
Therefore, we may write \eqref{mi:eq2} in the form
\begin{equation}\label{mi:eq3}
y(t)=y(a)+\int_{[a,t)} W(s)y(s)\,d\omega(s),\quad t\in[a,b],
\end{equation}
where $W(s)$ is the $n\times n$ matrix with entries $w_{ij}(s)$.
If $\omega$ is the Lebesgue measure then \eqref{mi:eq3} is
a standard Carath\'eodory system of linear differential equations.

The following theorem is well known \cite[Section 11.8]{A}.
We prove it for the sake of completeness.

\begin{thm}\label{mi:thm1}
For every $c\in\mathbb{C}^n$, there is a unique solution $y:[a,b]\to\mathbb{C}^n$
of \eqref{mi:eq2} with initial value $y(a)=c$.
\end{thm}

\begin{proof}
We take the system in the form \eqref{mi:eq3}.
Consider the space $Y$ of all bounded Borel measurable functions $y:[a,b]\to\mathbb{C}^n$
equipped with the sup norm
\[ \|y\|:=\sup_{t\in[a,b]} |y(t)|. \]
We may use here any norm $|\cdot|$ in $\mathbb{C}^n$.
Then $Y$ is a Banach space.
We define the operator $T:Y\to Y$ by
\[ (Ty)(t)=c+\int_{[a,t)} W(s)y(s)\,d\omega(s) .\]
Clearly, $T$ is well-defined.
Let $y,z\in Y$. We estimate
\[ |(Ty)(t)-(Tz)(t)|\le \int_{[a,t)} |W(s)||y(s)-z(s)|\,d\omega(s). \]
Hence
\[ \|Ty-Tz\|\le  L\|y-z\|,\]
where
\[ L:=\int_{[a,b)} |W(s)|\,d\omega(s) .\]
If $L<1$, application of Banach's fixed point theorem completes the proof.
If $L\ge 1$, we choose a partition $a=t_0<t_1<\dots<t_m=b$ such that
\[
\int_{(t_{k-1},t_k)} |W(s)|\,d\omega(s)<\frac12\quad
\text{for every $k=1,2,\dots,m$.}
\]
We set $y(a)=c$ and then define $y(a+)$ by \eqref{mi:jump}.
Then we remove the atom at $t=a$ if necessary and
apply Banach's fixed point theorem to $T$ with $y(a+)$ in place of $c$
and $[a,t_1]$ in place of $[a,b]$.
This way we get a solution on $[a,t_1]$.
After finitely many steps we obtain the desired solution of \eqref{mi:eq2}.
\end{proof}

\begin{rem}\label{mi:rem}
{\rm It should be noted that equation \eqref{mi:eq2} is solved
from left to right. In general, we cannot solve initial value
problems at points different from the left end point $a$. For
example, take $n=1$, $[a,b]=[0,2]$, and $\omega_{11}(E)=-1$ if
$1\in E$ and $\omega_{11}(E)=0$ if $1\not\in E$. The solution with
$y(0)=c$ is given by $y(t)=c$ for $0\le t\le 1$ and $y(t)=0$ for
$1<t\le 2$. There is no solution with $y(2)\ne 0$. }
\end{rem}

\section{Transformation of measure integral equations}\label{tmi}

Let $\omega:\mathcal{B}\to[0,\infty)$ be any nonnegative measure such that
\begin{equation}\label{tmi:ass}
\omega((a',b'))>0 \quad\text{whenever $a\le a'<b'\le b$.}
\end{equation}
Consider its distribution function $h:[a,b]\to[0,\infty)$ defined by
\begin{equation}\label{tmi:defh}
 h(t):=\omega([a,t)).
\end{equation}
This function is strictly increasing and left-continuous.
For all $e\in[a,b]$
\[ h(e+)-h(e)=\omega(\{e\}) .\]
For $e=b$ we take this is as the definition of $h(b+)$.
Let
\begin{equation}\label{tmi:defd}
 d:=h(b+)=\omega([a,b]).
\end{equation}
The function $h:[a,b]\to[0,d]$ is one-to-one but usually not onto. We
define a function $H:[0,d]\to[a,b]$ that is left-inverse to $h$ by
\[ H(x):=\max\{t\in[a,b] : h(t)\le x\} .\]
Hence $t=H(x)$ is the unique $t\in[a,b]$ such that $h(t)\le x\le h(t+)$.
The function $H$ is nondecreasing and continuous.
We use $H$ to transform integrals as follows; compare with
\cite[pages 194--195]{S}.

\begin{lemma}\label{tmi:l1}
Let $f:[a,b]\to\mathbb{C}$ be integrable with respect to $\omega$. Then
$f(H(x))$ is Lebesgue integrable and
\begin{equation}\label{tmi:eq1}
\int_{[a,t)} f(s)\,d\omega(s)=\int_0^{h(t)} f(H(x))\,dx\quad\text{for $t\in[a,b]$}.
\end{equation}
\end{lemma}

\begin{proof}
Equation \eqref{tmi:eq1} holds for the characteristic function $f=\chi_K$
with $K=(a',b')$ since
\[
 \{x\in[0,d]: H(x)\in(a',b')\}=(h(a'+),h(b')).
\]
In a similar way we see that \eqref{tmi:eq1} holds for characteristic functions
of any subinterval of $[a,b]$.
If we set $f=\chi_B$, $B\in\mathcal B$, then both sides of \eqref{tmi:eq1} define finite
measures on $\mathcal B$. Since these measures agree on intervals they also agree
on all Borel sets.
It follows that \eqref{tmi:eq1} holds for all nonnegative Borel measurable
simple functions.
Since every nonnegative Borel measurable functions is the pointwise
limit of a nondecreasing sequence of nonnegative simple functions, the monotone convergence
theorem shows that \eqref{tmi:eq1} holds for all nonnegative Borel measurable functions.
The statement of the lemma follows.
\end{proof}

Consider the linear system \eqref{mi:eq2} in the form \eqref{mi:eq3}.
Since we may add the Lebesgue measure to $\omega$, we may assume
that \eqref{tmi:ass} holds without loss of generality.
We define $d$, $h$ and $H$ relative to $\omega$.

We associate with \eqref{mi:eq3} the linear system
\begin{equation}\label{tmi:adi}
z'(x)=W(H(x))z(x),\quad x\in[0,d]\,a.e .
\end{equation}
By Lemma \ref{tmi:l1}, the entries of $W(H(x))$ are Lebesgue integrable on $[0,d]$.
Hence \eqref{tmi:adi} is a linear system in the sense of Carath\'eodory.
Solutions of equations \eqref{mi:eq3} and \eqref{tmi:adi} are connected as follows.

\begin{thm}\label{tmi:t1}
Assume that
\begin{equation}\label{tmi:cond}
W(t)^2=0\quad \text{whenever $\omega(\{t\})>0$.}
\end{equation}
Let $y:[a,b]\to\mathbb{C}^n$ be the solution of \eqref{mi:eq3} with $y(a)=c$,
and let $z:[0,d]\to\mathbb{C}^n$ be the solution of \eqref{tmi:adi} with $z(0)=c$.
Then
\[ y(t)=z(h(t)),\, y(t+)=z(h(t+)) \quad\text{for all $t\in[a,b]$} .\]
\end{thm}

\begin{proof}
We claim that
\begin{equation}\label{tmi:claim}
W(H(x))z(x)=W(H(x))z(h(H(x))\quad\text{for all $x\in[0,d]$.}
\end{equation}
If $x$ lies in the range of $h$ then $x=h(H(x))$ and \eqref{tmi:claim} holds.
If $x$ does not lie in the range of $h$, then
$h(t)<x\le h(t+)$ and $\omega(\{t\})>0$
where $t:=H(x)$.
On the interval $[h(t),x]$ system \eqref{tmi:adi} reads $z'=W(t)z$ with
constant coefficient $W(t)$.
Thus
\[ z(x)=e^{(x-h(t))W(t)}z(h(t)).\]
Therefore,
\[ W(t)z(x)=W(t)e^{(x-h(t))W(t)}z(h(t)) .\]
Using assumption \eqref{tmi:cond} and the power series expansion of the exponential
function of matrices, we see that
\[ W(t)e^{(x-h(t))W(t)}=W(t) .\]
This completes the proof of \eqref{tmi:claim}.

We have
\[ z(x)=c+\int_0^x W(H(r))z(r)\,dr\quad \text{for all $x\in[0,d]$.}\]
Setting $x=h(t)$ we obtain
\[ z(h(t))=c+\int_0^{h(t)} W(H(r))z(r)\,dr\quad\text{for all $t\in[a,b]$.} \]
By \eqref{tmi:claim}, we can write this as
\[ z(h(t))=c+\int_0^{h(t)} W(H(r))z(h(H(r)))\,dr\quad\text{for all $t\in[a,b]$.} \]
Transforming the integral on the right-hand side using Lemma \ref{tmi:l1},
we find
\[ z(h(t))=c+\int_{[a,t)} W(s)z(h(s))\,d\omega(s)\quad\text{for all $t\in[a,b]$.} \]
This implies $z(h(t))=y(t)$.
\end{proof}

If the measure $\omega$ has no atoms then condition
\eqref{tmi:cond} is satisfied. Thus all systems \eqref{mi:eq2}
which involve only measures without atoms can be transformed to
Carath\'eodory systems. However, the equation mentioned in
Remark \ref{mi:rem} cannot be
transformed to a Carath\'eodory equation.

\section{Atkinson's eigenvalue problem}\label{SL}

We consider system \eqref{intro:A1} subject to the boundary conditions
\eqref{intro:A2}.
We assume that $s,q,r:[c,d]\to\mathbb{R}$ are integrable functions, and $\alpha\in[0,\pi)$,
$\beta\in(0,\pi]$.
Note that we do not assume definiteness conditions unless stated explicitly.
The complex number $\lambda$ is called an {\it eigenvalue} if there exists a nontrivial
absolutely continuous solution $(u,v)$ of \eqref{intro:A1} satisfying
\eqref{intro:A2}.

Let $(u(x,\lambda),v(x,\lambda))$ be the unique solution of \eqref{intro:A1} with
initial values $u(c,\lambda)=\sin\alpha$, $v(c,\lambda)=\cos\alpha$.
A complex number
$\lambda$ is an eigenvalue if and only if
\begin{equation}\label{SL:char}
 \Delta(\lambda):=\cos\beta\, u(d,\lambda)-\sin\beta\, v(d,\lambda)=0 .
\end{equation}
Since $\Delta$ is an entire function, the set of eigenvalues is discrete or equals
the entire complex plane. In the latter case we call the eigenvalue problem {\it degenerate}.

The Pr\"ufer angle for \eqref{intro:A1}, \eqref{intro:A2}
is the absolutely continuous function $\theta(x)=\theta(x,\lambda)$
defined by
\[ \theta(x,\lambda)=\arg(v(x,\lambda)+i u(x,\lambda)),\quad \theta(c,\lambda)=\alpha .\]
It satisfies the first order differential equation
\begin{equation}\label{SL:preq}
 \frac{d\theta}{dx}=s(x)\cos^2\theta+(\lambda r(x)-q(x))\sin^2\theta .
\end{equation}
The real number $\lambda$ is an eigenvalue of \eqref{intro:A1}, \eqref{intro:A2} if and only if there
is an integer $n$ such that
\begin{equation}\label{SL:beta}
\theta(d,\lambda)=\beta+n\pi.
\end{equation}
Note that $\theta(d,\lambda)$ is an entire function of $\lambda$.

We are mainly interested in results on the eigenvalue problem
that permit $s$ to vanish on sets of positive measure.
Results of this nature can be carried over to measure Sturm-Liouville problems
in Section~\ref{MSL}.
The following asymptotic formula of the Pr\"ufer angle $\theta(d,\lambda)$ is
proved in \cite{BVasy}.

\begin{lemma}\label{SL:asy}
We have
\begin{gather}\label{SL:asy1}
\lim_{\lambda\to+\infty} \lambda^{-1/2}\theta(d,\lambda)
=\int_c^d\sqrt{r^+s^+}-
\int_c^d \sqrt{r^-s^-}=:L^+,\\
\lim_{\lambda\to-\infty} |\lambda|^{-1/2}\theta(d,\lambda)
=\int_c^d\sqrt{r^-s^+}-
\int_c^d \sqrt{r^+s^-}=:L^-,\label{SL:asy2}
\end{gather}
where $r^+(x):=\max(0,r(x))$, $r^-(x)=\max(0,-r(x))$.
\end{lemma}

From this lemma and the intermediate value theorem, we obtain existence of
real eigenvalues and their asymptotics without requiring any sign condition on $r$ or $s$.

\begin{thm}\label{SL:t1}
Assume $L^+>0$.
For every sufficiently large integer $n$, there exists a real positive eigenvalue $\lambda_n$
that satisfies $\theta(d,\lambda_n)=\beta+n\pi$.
For any choice of $\lambda_n$, we have
\[ \lim_{n\to\infty}n^{-2}\lambda_n=\pi^2 (L^+)^{-2} .\]
\end{thm}

There are similar theorems when $L^+<0$, $L^->0$, or $L^-<0$.
Under additional assumptions the statement of this theorem can be refined as follows.

\begin{lemma}\label{SL:l1}
The derivative of $\theta(d,\lambda)$ with respect to $\lambda$ is given by
\begin{equation}\label{SL:deriv}
 (u(d)^2+v(d)^2)\frac{d}{d\lambda} \theta(d,\lambda)=\int_c^d r(x)u(x)^2\,dx,
\end{equation}
where $u(x)=u(x,\lambda)$, $v(x)=v(x,\lambda)$.
\end{lemma}
For the proof of the above lemma, see \cite[Theorem 8.4.2]{A}.
We conclude the following uniqueness theorem.

\begin{thm}\label{SL:rp}
If the eigenvalue problem is not degenerate and $r\ge 0$,
then all eigenvalues are real and equation \eqref{SL:beta}
has at most one solution for every integer~$n$.
\end{thm}

\begin{proof}
If $\lambda_0$ is a nonreal eigenvalue then one shows as in \cite[Section 8.3]{A}
that $\int_c^d r(x)|u(x,\lambda_0)|^2\,dx=0$.
Since $r\ge 0$ we obtain that $r(x)u(x,\lambda_0)=0$ a.e.\ on $[c,d]$. This shows that
$(u(x,\lambda_0),v(x,\lambda_0))$ solves the system \eqref{intro:A1}, \eqref{intro:A2} for all
$\lambda$. Thus the problem is degenerate.
By Lemma \ref{SL:l1}, $\theta(d,\lambda)$ is increasing or constant. If $\theta(d,\lambda)$ is increasing
the second part of the statement is obvious. If $\theta(d,\lambda)$ is constant then either
the problem is degenerate or has no eigenvalues.
\end{proof}

\begin{lemma}\label{SL:l2}
If $s\ge 0$ and $\theta(x_0,\lambda)$ is an integer multiple of $\pi$, then
$\theta(x_1,\lambda)\le\theta(x_0,\lambda)\le\theta(x_2,\lambda)$ for
$c\le x_1<x_0<x_2\le d$.
\end{lemma}

The proof of the above lemma follows from an analysis of
equation \eqref{SL:preq}; see \cite[Theorem 8.4.1]{A}.


The zeros of $u(x)=u(x,\lambda)$ form a closed set that can be decomposed into (closed)
components.
Lemma \ref{SL:l2} yields the following theorem.

\begin{thm}\label{SL:sp}
Suppose $s\ge 0$.
If $\lambda_n$ is a solution of \eqref{SL:beta}
then the set of zeros of $u(x,\lambda_n)$ within $[c,d]$ has exactly
$n$ components if $\alpha\ne0$ and $\beta\ne\pi$, $n+1$ components if
either $\alpha=0$ or $\beta=\pi$, and $n+2$ components if $\alpha=0$ and $\beta=\pi$.
\end{thm}

Based on this theorem, we may call a solution $\lambda_n$ of \eqref{SL:beta}
an eigenvalue with {\it oscillation count} $n$ provided that $s\ge 0$.

In the rest of this section we will assume with Atkinson that $r\ge0$ and $s\ge 0$.
By Lemmas \ref{SL:l1} and \ref{SL:l2}, the function $\theta(d,\cdot)$
is nonnegative, and increasing or constant.
We write
\begin{equation}\label{SL:defell}
\ell^\pm:=\lim_{\lambda\to\pm\infty} \theta(d,\lambda) .
\end{equation}
If $\ell^-<\ell^+$ then the eigenvalue $\lambda_n$ with oscillation count $n$ exists
if and only if $\ell^-<\beta+n\pi<\ell^+$. Moreover, there are infinitely many
eigenvalues if and only if $\ell^+=+\infty$.
Formulas for $\ell^\pm$ are given in \cite{BVasy2}. They lead to the following results.

Let $\mathcal I$ be the collection of
closed intervals $I\subset[c,d]$ with positive length for which
$\int_I r=0$. Let $\mathcal J$ be the subcollection of $\mathcal
I$ consisting of those intervals $J\in\mathcal I$ with the
property that $I\in\mathcal I$, $J\subset I$ implies $I=J$
(maximal elements of $\mathcal I$). It is easy to show that the
intervals in $\mathcal J$ are mutually disjoint, and that every
interval in $\mathcal I$ is contained in an interval in~$\mathcal J$.
For $J=[c',d']\in \mathcal J$ let $(u_J,v_J)$ be the solution of \eqref{intro:A1}, that is,
\[ u'=s(x)v,\quad v'=q(x)u \]
with initial values
\begin{eqnarray*}
&u(c')=0, v(c')=1 & \text{if $c<c'<d'<d$,}\\
&u(c)=\sin\alpha, v(c)=\cos\alpha & \text{if $c=c'$,}\\
&u(d)=\sin\beta, v(d)=\cos\beta & \text{if $c<c'<d'=d$.}
\end{eqnarray*}
Let $\tilde n_J$ be the number of components of the set of zeros of $u_J$ within $J$.
Then we set $n_J:=\tilde n_J$ if $c=c'$, $\alpha\ne 0$ or if $c<c'<d'=d$, $\beta\ne \pi$.
In all other cases we define $n_J:=\tilde n_J-1$.

\begin{thm}\label{SL:min}
Suppose that $r,s\ge 0$, the problem is not degenerate and admits at least one eigenvalue.
Then the minimal oscillation count $n^-$ of eigenvalues
is given by
\begin{equation}\label{minus:limit}
 n^-=\sum_{J\in\mathcal J} n_J .
\end{equation}
\end{thm}

For the proof of the above theorem, see \cite[Section 3]{BVasy2}.


Under the additional assumption that $s(x)>0$ a.e.\
Theorem \ref{SL:min} is due to Everitt, Kwong and Zettl \cite{EKZ}
and, in an operator theoretic setting, to Binding and Browne \cite{BB}.
Atkinson \cite[Theorem 8.4.5]{A} gave the following sufficient condition for $n^-=0$.
It is a consequence of Theorem \ref{SL:min}

\begin{cor}\label{SL:mincor}
In addition to the assumptions of Theorem \ref{SL:min} suppose that
\[ \int_c^e r(x)\,dx>0,\,\int_e^b r(x)\,dx>0\quad\text{for all $e\in(c,d)$} \]
and
\[ \int_{c'}^{d'} r(x)\,dx=0 \text{ implies } \int_{c'}^{d'} q(x)\,dx =0 \text{ for all $[c',d']\subset [c,d]$}.
\]
Then $n^-=0$.
\end{cor}

The following theorem singles out the problems with finite spectrum.

\begin{thm}\label{SL:infinite}
Suppose $r,s\ge 0$.
The following statements are equivalent.
1. The eigenvalue problem is degenerate or has only finitely many eigenvalues;\\
2. $\ell^+$ is finite;\\
3. there is a partition
\[ c=\xi_0<\xi_1<\dots<\xi_m=d \]
such that, for all $j=1,2,\dots,m$,
\[ \int_{\xi_{j-1}}^{\xi_j} r(x)\,dx\times\int_{\xi_{j-1}}^{\xi_j} s(x)\,dx=0.\]
\end{thm}

For the proof of the above theorem, see \cite[Section 4]{BVasy2}.
We mention that, if $\ell^+$ is finite,
one can give a formula for the maximal oscillation count $n^+$;
see \cite[Section 4]{BVasy2}.

\section{Measure Sturm-Liouville problems}\label{MSL}

Let $\sigma,\chi,\rho:\mathcal{B}\to\mathbb{R}$ be (finite signed) measures, where again $\mathcal{B}$ denotes the
$\sigma$-algebra of Borel sets in $[a,b]$.
Consider the measure integral equations
\begin{gather}\label{MSL:IE1}
U(t)-U(a)=\int_{[a,t)} V(s)\,d\sigma(s),\quad t\in[a,b],\\
V(t)-V(a)=\int_{[a,t)}U(s)\,d\chi(s)-\lambda \int_{[a,t)} U(s)\,d\rho(s),\quad t\in[a,b] ,
\label{MSL:IE2}
\end{gather}
and the boundary conditions
\begin{equation}\label{MSL:A2}
\cos\alpha\, U(a)=\sin\alpha\, V(a),\quad \cos\beta\,U(b+)=\sin\beta\,V(b+),
\end{equation}
where $\alpha\in[0,\pi)$, $\beta\in(0,\pi]$.
In Section \ref{mi} we defined the notion of solution of
the system, and we also explained the meaning of $U(b+)$, $V(b+)$.
A complex number $\lambda$ is called an {\it eigenvalue} if there exists a nontrivial
solution $(U,V)$ of system \eqref{MSL:IE1}, \eqref{MSL:IE2}
satisfying the boundary conditions \eqref{MSL:A2}.

Note that the measure Sturm-Liouville problem includes
the one considered in Section \ref{SL} when we set
\[ \sigma(E)=\int_E s(x)\,dx,\quad \chi(E)
=\int_E q(x)\,dx,\quad \rho(E)=\int_E r(x)\,dx.
\]

Let $S$, $Q$, $R$ denote the set of atoms for $\sigma,\chi,\rho$, respectively.
We now transform the measure Sturm-Liouville problem
under the condition
\begin{equation}\label{MSL:atomcond}
S\cap (Q\cup R)=\emptyset.
\end{equation}
This condition permits the measures $\sigma,\chi,\rho$ to have atoms but
$\sigma$ and $\rho$ as well as $\sigma$ and $\chi$ may not have common atoms.
Choose a measure $\omega:\mathcal{B}\to[0,\infty)$ such that
each measure $\sigma,\chi,\rho$ is absolutely continuous with respect to $\omega$,
and $\omega((a',b'))>0$ for all $a\le a'<b'\le b$. For instance, we may take
\[ \omega=|\sigma|+|\chi|+|\rho|+\nu, \]
where $\nu$ denotes Lebesgue measure.
Let $f_\sigma$, $f_\chi$, $f_\rho$ denote the Radon-Nikodym derivatives
of $\sigma$, $\chi$, $\rho$ with respect to $\omega$.
Consider the distribution function $h(t)=\omega([a,t))$ of $\omega$,
and let $d:=h(b+):=\omega([a,b])$.
As in Section \ref{tmi} we define the function $H$ which is left-inverse to $h$.
Then we define Lebesgue integrable functions $s,q,r:[0,d]\to\mathbb{R}$ by
\[ s(x):=f_\sigma(H(x)),\, q(x):=f_\chi(H(x)),\, r(x):=f_\rho(H(x)) .\]
Employing these functions we call \eqref{intro:A1},
\eqref{intro:A2} (with $c=0$) the Sturm-Liouville problem associated with
the given measure Sturm-Liouville problem. The numbers $\alpha$, $\beta$ appearing
in \eqref{intro:A2} are the same as those in \eqref{MSL:A2}.

Let $(U(t,\lambda),V(t,\lambda))$ be the unique solution of \eqref{MSL:IE1}, \eqref{MSL:IE2}
with initial values $U(a,\lambda)=\sin\alpha$, $V(a,\lambda)=\cos\alpha$ according
to Theorem \ref{mi:thm1}.
Then $\lambda$ is an eigenvalue of the measure Sturm-Liouville problem if and only if
\begin{equation}\label{MSL:char}
 \cos\beta\, U(b+,\lambda)-\sin\beta\, V(b+,\lambda)=0 .
\end{equation}

\begin{thm}\label{MSL:t1}
If condition \eqref{MSL:atomcond} holds
then
\[ u(h(t),\lambda)=U(t,\lambda),\quad v(h(t),\lambda)=V(t,\lambda)\quad\text{for $t\in[a,b]$} \]
and
\[ u(d,\lambda)=U(b+,\lambda),\quad v(d,\lambda)=V(b+,\lambda) .\]
In particular, $\lambda$ is an eigenvalue of the measure Sturm-Liouville problem
if and only if $\lambda$ is an eigenvalue of the associated Sturm-Liouville problem.
\end{thm}
\begin{proof}
The first part of the statement
follows at once from Theorem \ref{tmi:t1} provided condition \eqref{tmi:cond} holds
for
\[
W(t)=\begin{pmatrix} 0 & f_\sigma(t)\\
f_\chi(t)-\lambda f_\rho(t) & 0\end{pmatrix}.
\]
If $\{t\}$ is an atom for $\omega$ then, by assumption \eqref{MSL:atomcond},
either $f_\sigma(t)=0$ or $f_\chi(t)=f_\rho(t)=0$. In both cases
$W(t)$ satisfies $W(t)^2=0$.

The second part of the statement follows by noting that the characteristic
equations
\eqref{SL:char} and \eqref{MSL:char} agree.
\end{proof}

We now express the numbers $L^+$ and $L^-$ from \eqref{SL:asy1}, \eqref{SL:asy2} involving $r$ and $s$
directly in terms of the given measures.
To this end let us introduce the geometric mean $GM(\omega_1,\omega_2)$ for two finite
nonnegative measures $\omega_1,\omega_2:\mathcal{B}\to [0,\infty)$.
We choose a nonnegative measure $\omega$ such that $\omega_1$ and $\omega_2$ are
absolutely continuous with respect to $\omega$.
Let $f_j$ be the Radon-Nikodym derivative of $\omega_j$ with respect to $\omega$.
Then we define
\[ GM(\omega_1,\omega_2):=\int_{[a,b]} \sqrt{f_1f_2}\,d\omega.\]
Note that $\sqrt{f_1f_2}$ is integrable with respect to $\omega$.
It is easy to see that the definition of the geometric mean does not depend
on the choice of~$\omega$.
If we use the Lebesgue decomposition $\omega_2=\omega_3+\omega_4$
with $\omega_3$ absolutely continuous with respect to $\omega_1$ and $\omega_4$
singular with respect to
$\omega_1$, and let $g$ denote
the Radon-Nikodym derivative of $\omega_3$ with respect to
$\omega_1$, then we may write
\[ GM(\omega_1,\omega_2)=\int_{[a,b]} \sqrt{g}\,d\omega_1 .\]
This shows that the geometric mean depends only on the absolutely continuous component of $\omega_2$ with
respect to $\omega_1$ (or the absolutely continuous component of $\omega_1$ with respect to $\omega_2$.)

Using the geometric mean, we can write $L^+$, $L^-$ as follows
\begin{gather}\label{MSL:L+}
L^+=GM(\sigma^+,\rho^+)-GM(\sigma^-,\rho^-),\\
L^-=GM(\sigma^+,\rho^-)-GM(\sigma^-,\rho^+),
\end{gather}
where $\sigma^+$ and $\sigma^-$ are the positive and negative variations
of $\sigma$ according
to the Jordan decomposition of $\sigma$.

Now Theorems \ref{SL:t1} and \ref{MSL:t1} yield the following theorem.

\begin{thm}\label{MSL:t2}
Assume \eqref{MSL:atomcond} holds, and the number $L^+$ in \eqref{MSL:L+}
is positive.
For sufficiently large $n$, there exists a real positive eigenvalue $\lambda_n$ of
the measure Sturm-Liouville problem that satisfies $\theta(d,\lambda_n)=\beta+n\pi$.
For any choice of $\lambda_n$, we have
\[ \lim_{n\to\infty}n^{-2}\lambda_n=\pi^2 (L^+)^{-2} .\]
\end{thm}

We have similar results when $L^+<0$, $L^->0$, $L^-<0$.
As in Section \ref{SL} we can say more under additional assumptions.

\begin{thm}\label{MSL:t3}
Assume the measure Sturm-Liouville problem is not degenerate,
satisfies \eqref{MSL:atomcond} and $\rho\ge 0$.
Then all of its eigenvalues are real and equation \eqref{SL:beta}
has at most one solution for every integer $n$.
\end{thm}

\begin{proof}
If $\rho\ge0$ then $r\ge0$, so the statement follows from Theorem \ref{SL:rp}
applied to the associated problem.
\end{proof}

We call $t\in[a,b]$ a {\it generalized zero}
of $U(t)=U(t,\lambda)$ if $U(t)U(t+)\le0$.
Generalized zeros form a closed set that can be decomposed into (closed)
components.

\begin{thm}\label{MSL:t4}
Suppose \eqref{MSL:atomcond}.
For $\lambda\in\mathbb{R}$, the collection of components of the set of zeros of the function $u(x,\lambda)$
within $[0,d]$ has the same cardinality as the collection of components of the
set of generalized zeros of $U(t,\lambda)$ within $[a,b]$.
If $\sigma\ge 0$ and $\lambda_n$ is a solution of \eqref{SL:beta} then the number of components of the
set of generalized zeros of $U(t,\lambda_n)$ is equal to $n$ if $\alpha\ne 0$ and $\beta\ne\pi$, equal to
$n+1$ if either $\alpha=0$ or $\beta=\pi$, and equal to $n+2$ if $\alpha=0$ and $\beta=\pi$.
\end{thm}

\begin{proof}
Let $X$ be the set of zeros of $u(x,\lambda)$, and let $T$ be the set of generalized zeros of $U(t,\lambda)$
for some real $\lambda$. Then $H$ defines a continuous map from $X$ onto $T$ which has the additional property that
the inverse image $H^{-1}(K)\subset X$ is connected whenever $K$ is connected.
It is an easy exercise in topology to show that the map $J\mapsto H(J)$ establishes a one-to-one correspondence
between the components of $X$ and $T$.
The second part of the statement now follows from Theorem \ref{SL:sp} applied to the associated problem.
\end{proof}

Based on this theorem we may call a solution $\lambda_n$ of \eqref{SL:beta}
an eigenvalue with {\it oscillation count} $n$ provided $\sigma\ge 0$.

As a simple example, consider $[a,b]=[0,1]$, $\rho$ Lebesgue measure,
$\chi=0$ and $\sigma\ge 0$ with
$\sigma(\{0\})=1$, $\sigma((0,1])=0$, $\alpha=\frac56\pi$, $\beta=\frac12\pi$.
Then
\begin{gather*}
  U(0,\lambda)=\frac12,\,\, U(t,\lambda)=\frac12(1-\sqrt3)\quad \text{for $t>0$,}\\
  V(t,\lambda)=-\frac12\sqrt{3}-\frac12\lambda(1-\sqrt{3}) t.
\end{gather*}
The problem has only one eigenvalue $\lambda_1=\sqrt3/(\sqrt3-1)$. According to
Theorem \ref{MSL:t4} its oscillation count is $1$.
When we set $\omega=\rho+\sigma$ then the associated Sturm-Liouville problem has $[0,d]=[0,2]$, $q=0$,
$r(x)=0$ for $x\in[0,1]$, $r(x)=1$ for $x\in(1,2]$ and $s=1-r$.
One may verify that $\theta(2,\lambda_1)=\beta+\pi$.
In this example, $U(t,\lambda_1)$ has no zero at $t=0$ but it has a
generalized zero there. This is the reason why
we avoid talking about ``interior'' generalized zeros in order to determine the oscillation count.

Suppose that $\rho\ge 0$. Let $\mathcal K$ be the
collection of all maximal subintervals $K$ of $[a,b]$ with $\rho(K)=0$.
The intervals in $K$ may be open, closed
or half-open.
We only consider intervals of positive length in $\mathcal K$ except possibly $\{a\}$
and $\{b\}$. We declare $\{a\}\in\mathcal K$ if $\rho(\{a\})=0$, $\rho((a,e))>0$ for all
$e\in(a,b)$ and $\sigma(\{a\})\ne 0$. Similarly, we define the meaning of $\{b\}\in\mathcal K$.
Let $\overline K=[a',b']$. We define $(U_K,V_K)$ as the solution of
\eqref{MSL:IE1}, \eqref{MSL:IE2} with $\lambda=0$ determined by the initial values
\begin{gather*}
U(a')=0,  V(a')=1 \quad\text{if $a,b\notin K$},\\
U(a)=\sin\alpha,  V(a)=\cos\alpha \quad\text{if $a\in K$},\\
U(b+)=\sin\beta, V(b+)=\cos\beta \quad \text{if $a\notin K$, $b\in K$}.
\end{gather*}
Let $\tilde m_K$ be the number of components of the set of generalized zeros of $U_K$ within
$\overline K$.
We define $m_K:=\tilde m_K$ if $a\in K$, $\alpha\ne0$ or if $a\not\in K$, $b\in K$, $\beta\ne\pi$.
In all other cases, let $m_K:=\tilde m_K-1$.

\begin{thm}\label{MSL:min}
Assume \eqref{MSL:atomcond} and $\sigma,\rho\ge 0$. Suppose the measure Sturm-Liouville problem is not
degenerate and admits at least one eigenvalue.
Then the minimal oscillation count $n^-$ of eigenvalues is
\[ n^-=\sum_{K\in\mathcal K} m_K .\]
\end{thm}

\begin{proof}
The minimal oscillation count of eigenvalues of the given measure Sturm-Liouville problem
agrees with that for the eigenvalues of the associated Sturm-Liouville problem.
Therefore, by Theorem \ref{SL:min}, we have
\begin{equation}\label{MSL:min1}
n^-=\sum_{J\in\mathcal J} n_J.
\end{equation}
Let $K$ be any subinterval of $[a,b]$, and let $J$
be the closure of the interval $\{x\in[0,d]: H(x)\in K\}$.
By Lemma \ref{tmi:l1}, we have $\rho(K)=\int_J r$.
If $K\in\mathcal K$ then $J\in\mathcal J$,
and if $J\in\mathcal J$ then
$K\in\mathcal K$ or $K=\{t_0\}$ is a singleton with $h(t_0)\ne h(t_0+)$.
Therefore, Theorem \ref{MSL:min} will follow from \eqref{MSL:min1} once we have shown
that $m_K=n_J$ for all $K\in\mathcal K$,
and $n_J=0$ if $K$ is a singleton different from
$\{a\}$ and $\{b\}$.
If $K$ is such a singleton and $J\in\mathcal J$ then $s,q,r$ are constant
on $J$ and either $r=q=0$ or $s=0$ by \eqref{MSL:atomcond}.
In both cases $u_J$ is affine linear
on $J$ and thus $n_J=0$.

Now consider an interval $K=(a',b')\in\mathcal K$ with corresponding
$J=[h(a'+),h(b')]$ in $
\mathcal J$.
Then $\rho(\{a'\})\ne 0$
and $\rho(\{b'\})\ne 0$. In particular, $c,d\notin J$,
and thus $(u_J,v_J)$ solves $u'=sv$, $v'=qu$ with initial conditions $u_J(h(a'+))=0$,
$v_J(h(a'+))=1$.
By \eqref{MSL:atomcond}, $s=0$ on $[h(a'),h(a'+)]$ and on $[h(b'),h(b'+)]$.
Hence $u_J$ is constant on these intervals. In particular, $u_J=0$ on $[h(a'),h(a'+)]$.
Therefore, by Theorem \ref{MSL:t1}, $U_K(t)$ is a constant multiple of $u_J(h(t))$.
By Theorem \ref{MSL:t4}, the number of components of the set of
generalized zeros of $U_K$ within
$\overline K$ agrees with the number of components of zeros of $u_J$ within $[h(a'),h(b'+)]$ and then also within $J$.
Therefore,
$m_K=n_J$. If $K$ is closed or half-open, we see in a similar way that also $m_K=n_J$.
\end{proof}

The example after Theorem \ref{MSL:t4} shows why we have to allow singletons $\{a\}$ or $\{b\}$ in
the collection $\mathcal K$.
Analogously to Corollary \ref{SL:mincor} we obtain the following sufficient condition for $n^-=0$.

\begin{cor}\label{MSL:mincor}
In addition to the assumption of Theorem \ref{MSL:min} suppose that
\begin{gather*}
 \sigma(\{a\})=\sigma(\{b\})=0,\\
 \rho([a,e))\rho((e,b])>0 \text{ for all $e\in(a,b)$},\\
 \text{$\rho(I)=0$ implies
$|\chi|(I)=0$ for all subintervals $I$ of $[a,b]$}.
\end{gather*}
Then $n^-=0$.
\end{cor}

\begin{thm}\label{MSL:infinite}
Suppose \eqref{MSL:atomcond} and $\rho,\sigma\ge 0$.
Then the measure Sturm-Liouville problem is degenerate or has only finitely many eigenvalues if and only
there is a partition
\[ a=\tau_0<\tau_1<\dots<\tau_m=b \]
such that
\[ \rho((\tau_{i-1},\tau_i))\sigma((\tau_{i-1},\tau_i))=0\quad\text{for all $i=1,2,\dots,m$.} \]
\end{thm}

\begin{proof}
Assume such a partition exists. Then we consider the partition of $[c,d]$ with
partition points $h(\tau_i)$, and, if $h$ is discontinuous at $\tau_i$, $h(\tau_i+)$.
Then, with $J=[h(\tau_{i-1}+),h(\tau_i)]$,
\[ \int_J r\times \int_J s=\rho((\tau_{i-1},\tau_i))\sigma((\tau_{i-1},\tau_i))=0.\]
This is also true for the intervals $J=[h(\tau_i),h(\tau_i+)]$ by virtue of our assumption
\eqref{MSL:atomcond}.
Therefore, by Theorem \ref{SL:infinite}, the associate Sturm-Liouville and thus also the
measure Sturm-Liouville problem is either degenerate or has only finitely many eigenvalues.
The proof of the converse statement is reduced to Theorem \ref{SL:infinite}
in a similar manner.
\end{proof}

Let us summarize our results for a given measure Sturm-Liouville problem with $\rho,\sigma\ge 0$.
First we check condition \eqref{MSL:atomcond}. If this condition fails we have given
no results and it appears that hitherto this type of problem has not been considered in
the literature.
So let us assume that \eqref{MSL:atomcond} holds. Next we verify if a partition of the type
described in Theorem \ref{MSL:infinite} exists.
If such a partition exists the problem is degenerate or has only finitely many eigenvalues.
Suppose that such a partition does not exist. Then the problem is not degenerate, its eigenvalues are real
and can be listed as an infinite sequence
\[ \lambda_{n^-}<\lambda_{n^-+1}<\lambda_{n^-+2}<\dots \]
converging to infinity. The eigenvalue $\lambda_n$ has oscillation count $n$ and $n^-$ is the
minimal oscillation count which can be determined from Theorem \ref{MSL:min}.
According to Theorem \ref{MSL:t2}, the eigenvalues satisfy the asymptotic formula
\begin{equation}\label{MSL:asyev}
 \lim_{n\to\infty} n^{-2}\lambda_n=\big(\frac{\pi}{GM(\sigma,\rho)}\big)^2.
\end{equation}
If $GM(\sigma,\rho)=0$ then the right hand side has to be interpreted as $+\infty$.
This case has been investigated in several papers; see
\cite{Fr1,Fr2,Fu,V}.

The vibrating string problem \cite{GK}, \cite{KK} is a special case of the
measure Sturm-Liouville problem.
We take $\sigma=\nu$ (Lebesgue measure), $\chi=0$ and $\rho(E)\ge 0$
is the mass of the string over $E\in\mathcal B$.
In this case, if $(U,V)$ is a solution of \eqref{MSL:IE1}, \eqref{MSL:IE2}
then $U$ is continuous and has left-hand derivatives
on $(a,b]$ and right-hand derivatives on $[a,b)$.
Moreover, $V(t)$ agrees with the left-hand derivative of $U$ at $t$ for all $t\in(a,b]$.
Since $\sigma$ has no atoms assumption \eqref{MSL:atomcond} holds.
Choosing $\omega:=\rho+\nu$
we transform
the vibrating string problem to Atkinson's  eigenvalue problem.
Unless the mass of the string is concentrated at finitely many points
(string with beads) we have infinitely many eigenvalues by Theorem \ref{MSL:infinite}.
Under the assumption that
\[ \rho([a,e))>0, \rho((e,b])>0\quad \text{for all $e\in(a,b)$} \]
we see from Corollary \ref{MSL:mincor} that the minimal oscillation count is $n^-=0$.
For the asymptotic formula \eqref{MSL:asyev} for the vibrating string see
\cite{K1}, \cite{K2}, \cite[(11.7)]{KK}.
Gantmacher and Krein \cite[Chapter 4]{GK} investigate the oscillations
of vibrating strings
by the method of oscillation kernels.
This method is very different from the method used
in this paper, namely, transformation to Atkinson's problem and use
of the Pr\"ufer angle.

If $\sigma$ is Lebesgue measure and $\rho\ge 0$
the asymptotic formula \eqref{MSL:asyev}
is also proved by McKean and Ray \cite{MR}.

\subsection*{Acknowledgement}
The author thanks Paul Binding for helpful discussions on the topic
of the paper, and the anonymous referee for pointing out some
additional references to the literature.

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\end{thebibliography}
\end{document}