\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 $$\label{intro:SL1} -(p(x)y')'+q(x)y=\lambda r(x) y,\quad x\in[c,d]\,\, a.e.$$ and the boundary conditions $$\label{intro:SL2} y(c)\cos\alpha=(py')(c)\sin\alpha,\quad y(d)\cos\beta=(py')(d)\sin\beta.$$ 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 $$\label{intro:asy} \lim_{n\to\infty} n^{-2}\lambda_n=\frac{\pi^2}{L^2},\quad L:=\int_c^d \Big(\frac{r(x)}{p(x)}\Big)^{1/2}\,dx.$$ 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 $$\label{intro:A1} u'=s(x)v,\quad v'=(q(x)-\lambda r(x))u,$$ and the boundary conditions \eqref{intro:SL2} as $$\label{intro:A2} u(c)\cos\alpha=v(c)\sin\alpha,\quad u(d)\cos\beta=v(d)\sin\beta.$$ 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 $$\label{mi:eq2} y_i(t)=y_i(a)+\sum_{j=1}^n \int_{[a,t)} y_j(s)d\omega_{ij}(s),\quad t\in[a,b],\, i=1,\dots,n .$$ 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 $$\label{mi:jump} y_i(t+)-y_i(t)=\sum_{j=1}^n \omega_{ij}(\{t\})y_j(t),\quad t\in[a,b) .$$ \end{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_00 \quad\text{whenever a\le a'0.} 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 $$\label{tmi:claim} W(H(x))z(x)=W(H(x))z(h(H(x))\quad\text{for all x\in[0,d].}$$ 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 $h(t)0$ 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 $$\label{SL:char} \Delta(\lambda):=\cos\beta\, u(d,\lambda)-\sin\beta\, v(d,\lambda)=0 .$$ 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 $$\label{SL:preq} \frac{d\theta}{dx}=s(x)\cos^2\theta+(\lambda r(x)-q(x))\sin^2\theta .$$ 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 $$\label{SL:beta} \theta(d,\lambda)=\beta+n\pi.$$ 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 $$\label{SL:deriv} (u(d)^2+v(d)^2)\frac{d}{d\lambda} \theta(d,\lambda)=\int_c^d r(x)u(x)^2\,dx,$$ 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 $c\le x_10$ 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 $$\label{MSL:A2} \cos\alpha\, U(a)=\sin\alpha\, V(a),\quad \cos\beta\,U(b+)=\sin\beta\,V(b+),$$ 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 $$\label{MSL:atomcond} S\cap (Q\cup R)=\emptyset.$$ 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 $a\le a'0$, $L^-<0$. 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 $$\label{MSL:min1} n^-=\sum_{J\in\mathcal J} n_J.$$ 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 $$\label{MSL:asyev} \lim_{n\to\infty} n^{-2}\lambda_n=\big(\frac{\pi}{GM(\sigma,\rho)}\big)^2.$$ 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. \begin{thebibliography}{99} \bibitem{A} F. V. Atkinson; \emph{Discrete and Continuous Boundary Problems}, Academic Press, New York 1964. \bibitem{BB} P. Binding and P. J. Browne; \emph{Eigencurves for two-parameter selfadjoint ordinary differential equations of even order}, J.\ Differential Equations 79 (1989), 289--303. \bibitem{BVasy} P. Binding and H. Volkmer; \emph{Existence and asymptotics of eigenvalues of indefinite systems of Sturm-Liouville and Dirac type}, J.\ Differential Equations 172 (2001), 116--133. \bibitem{BVasy2} P. Binding and H. 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