\input amstex \documentstyle{amsppt} \magnification=1200 \baselineskip=15 pt \TagsOnRight \redefine\mod{\text{\rom{mod}}\,} \def\smat#1#2{\hskip -3mm\spreadmatrixlines{-2mm}\matrix \\\endmatrix} \def\tsmat#1#2{\hskip -4mm\matrix &{\ssize#1}\\ \vspace{-2mm} &{\ssize#2} \endmatrix} \loadbold \topmatter \title Zeros of the Wronskian and Renormalized Oscillation Theory \endtitle \author F.~Gesztesy$^{1}$, B.~Simon$^{2}$, and G.~Teschl$^{1}$ \endauthor \leftheadtext{F.~Gesztesy, B.~Simon, and G.~Teschl} \date July 13, 1995 \enddate \thanks $^{1}$ Department of Mathematics, University of Missouri, Columbia, MO 65211. E-mail for F.G.: mathfg\@\linebreak mizzou1.missouri.edu; e-mail for G.T.: mathgr42\@mizzou1.missouri.edu \endthanks \thanks $^{2}$ Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA 91125. 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. \endthanks \thanks {\it{To appear in Am.~J.~Math.}} \endthanks \abstract For general Sturm-Liouville operators with separated boundary conditions, we prove the following: If $E_{1,2}\in\Bbb R$ 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 $u_2$. \endabstract \endtopmatter \document \flushpar{\bf{\S 1. Introduction}} \medpagebreak For over a hundred and fifty years, oscillation theorems for second-order differential equations have fascinated mathematicians. Originating with Sturm's celebrated memoir [20], extended in a variety of ways by B\^ocher [2] and others, a large body of material has been accumulated since then (thorough treatments can be found, e.g., in [4],[13],[18],[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 $a0$ a.e.~on $(a,b)$}. \tag 1.1 $$We'll use \tau to describe the formal differentiation expression and H the operator given by \tau with separated boundary conditions at a and/or b. If a (resp.~b) is finite and q,p^{-1},r are in addition integrable near a (resp.~b), we'll say a (resp.~b) is a {\it{regular}} end point. We'll say \tau respectively H is {\it{regular}} if both a and b are regular. As is usual, ([6], Section XIII.2; [15], Section 17; [22], Chapter 3), we consider the local domain$$ D_{\text{\rom{loc}}}=\{u \in AC_{\text{\rom{loc}}}((a,b)) \mid pu'\in AC_{\text{\rom{loc}}}((a,b)),\ \tau u \in L^{2}_{\text{\rom{loc}}}((a,b); r\,dx)\}, \tag 1.2 $$where AC_{\text{\rom{loc}}}((a,b)) is the set of integrals of L^{1}_{\text{\rom{loc}}} functions (i.e., the set of locally absolutely continuous functions) on (a,b). General {\eightpoint{ODE}} theory shows that for any E\in\Bbb C, x_0\in (a,b), and (\alpha, \beta)\in\Bbb C^2, there is a unique u\in D_{\text{\rom{loc}}} such that -(pu')'+qu - Eru=0 for a.e.~x \in (a,b) and (u(x_0), (pu')(x_0))=(\alpha,\beta). The maximal and minimal operators are defined by taking$$ D(T_{\text{\rom{max}}})=\{u\in L^{2}((a,b); r\,dx)\cap D_{\text{\rom{loc}}}\mid \tau u\in L^{2}((a,b); r\,dx)\}, $$with$$ T_{\text{\rom{max}}}u=\tau u. \tag 1.3 $$T_{\text{\rom{min}}} is the operator closure of T_{\text{\rom{max}}}\restriction D_{\text{\rom{loc}}}\cap\{u \text{ has compact support in (a,b)}\}. Then T_{\text{\rom{min}}} is symmetric and T^{*}_{\text{\rom{min}}}=T_{\text{\rom{max}}}. According to the Weyl theory of self-adjoint extensions ([6], Section XIII.6; [15], Section 18; [17], Appendix to X.1; [21], Section 8.4; [22], Chapters 4 and 5), the deficiency indices of T_{\text{\rom{min}}} are (0,0) or (1,1) or (2,2) depending on whether it is limit point at both, one or neither end point. Moreover, the self-adjoint extensions can be described in terms of Wronskians ([6], Section XIII.2; [15], Sections 17 and 18; [21], Section 8.4; [22], Chapter 3). Define$$ W(u_1,u_2)(x)=u_1(x)(pu_2')(x)-(pu_1')(x)u_2(x). \tag 1.4 $$Then if T_{\text{\rom{min}}} is limit point at both ends, T_{\text{\rom{min}}}=T_{\text{\rom{max}}}=H. If T_{\text{\rom{min}}} is limit point at b but not at a, for H any self-adjoint extension of T_{\text{\rom{min}}}, if \varphi_- is any function in D(H) \backslash D(T_{\text{\rom{min}}}), then$$ D(H)=\{u\in D(T_{\text{\rom{max}}})\mid W(u,\varphi_-)(x)\to 0 \text{ as $x\downarrow a$}\}. $$Finally, if u_1 is limit circle at both ends, the operators H with separated boundary conditions are those for which we can find \varphi_\pm \in D(H), \varphi_+\equiv 0 near a, \varphi_- \equiv 0 near b, and \varphi_\pm \in D(H)\backslash D(T_{\text{\rom{min}}}). In that case,$$ D(H)=\{u\in D(T_{\text{\rom{max}}})\mid W(u,\varphi_-)(x)\to 0 \text{ as $x\downarrow a$}, \, W(u,\varphi_+)(x)\to 0 \text{ as $x\uparrow b$}\}. $$Of course, if H is regular, we can just specify the boundary conditions by taking values at a,b since by regularity any u\in D(T_{\text{\rom{max}}}) has u,pu' continuous on [a,b] (cf.~(A.4)). It follows from this analysis that \proclaim{Proposition 1.1} If u_{1,2}\in D(H), then W(u_1,u_2)(x) \to 0 as x\to a or b. \endproclaim We'll call such operators SL operators (for Sturm-Liouville, but SL includes separated boundary conditions (if necessary)). It will be convenient to write \ell_-=a, \ell_+=b. Throughout this paper we will denote by \psi_{\pm}(z,x) \in D_{\text{\rom{loc}}} solutions of \tau \psi = z \psi so that \psi_{\pm}(z,\,.\,) is L^2 at \ell_{\pm} and \psi_{\pm} (z,\,.\,) satisfies the appropriate boundary condition at \ell_\pm in the sense that for any u \in D(H), \lim\limits_{x\to\ell_{\pm}} W(\psi_{\pm}(z),u)(x)=0. If \psi_{\pm}(z,\,.\,) exist, they are unique up to constant multiples. In particular, \psi_{\pm}(z,\,.\,) exist for z not in the essential spectrum of H and we can assume them to be holomorphic with respect to z in {\Bbb C} \backslash \text{spec}(H) and real for z\in\Bbb R. One can choose$$ \psi_\pm(z,x) = ((H-z)^{-1} \chi_{(c,d)})(x) \quad \text{for } x \smat{> d}{< c}, \quad a d}{< c}$. Here$(H-z)^{-1}$denotes the resolvent of$H$and$\chi_\Omega$the characteristic function of the set$\Omega \subseteq \Bbb 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$is in the essential spectrum that such solutions do not exist or it may happen that they do. In Theorems 1.3, 1.4 below, we'll explicitly assume such solutions exist for the energies of interest. If these energies are not in the essential spectrum, that is automatically fulfilled. With these preliminaries out of the way, we can describe a theorem Hartman proves in [10] which gives an eigenvalue count in some cases where oscillation theory would naively give$\infty -\infty$(see Weidmann [22], Chapter 14 for some results when$\tau$is limit circle at$b$). \proclaim{Theorem 1.2 (Hartman [10], see also [9],[11])} Let$H$be an SL operator on$(a,b)$which is regular at$a$and limit point at$b$and suppose$E_1< E_2$. Let$u_1$\rom(resp.~$u_2$\rom) be$\psi_{-} (E_1)$\rom(resp.~$\psi_-(E_2)$\rom). Let$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)$be the spectral projector of$H$corresponding to the Borel set$\Omega\subseteq\Bbb R$. Then, if$\tau$is oscillatory at$E_2$, $$\dim\text{\rom{Ran}}\, P_{(E_1, E_2)}(H) = \varliminf\limits_{c\uparrow b}\, N(c), \tag 1.5a$$ and if$\tau$is non-oscillatory at$E_2$, $$\dim\text{\rom{Ran}}\, P_{[E_1, E_2)}(H) = \lim\limits_{c\uparrow b}\, N(c). \tag 1.5b$$ \endproclaim If$\tau$is oscillatory at$E_2$(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$N_0+1$. Hartman's 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 [14], who noted that if$u_j,pu_j' \in AC_{\text{\rom{loc}}}((a,b))$,$j=1,2$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$E_1 e > E_1$with$e$an eigenvalue and$|E_2-E_1|$is small, then (1.7) holds rather than (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$\dim\text{\rom{Ran}}\, P_{(E_1, E_2)}(H)=\infty$), 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 7.3). 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$\psi_- \in D_{\text{\rom{loc}}}$and$\psi_+ \in D_{\text{\rom{loc}}}$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 $$\eta(x)= \cases \psi_-(x), & x\leq x_0 \\ \gamma\psi_+(x), & x\geq x_0 \endcases$$ satisfies$\eta\in D(H)$and $$H \eta(x) =\cases (\tau\psi_-)(x), & x\leq x_0 \\ \gamma (\tau\psi_+)(x), & x\geq x_0. \endcases$$ We'll explore these properties further in Propositions 3.1 and 3.2. Section 2 provides a short proof of the ordinary oscillation theorem in the regular case following the method in Courant-Hilbert ([5], page 454). Even though this result is well-known (see, e.g., [1], Theorem 8.4.5 and [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 6 and a comparison-type argument in Sections 4 and 5 will prove$N_0\leq W_0$. Explicitly, in Section 5 we'll show \proclaim{Theorem 1.5} Let$E_1 < E_2$. 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 $$W_0(u_1,u_2)\geq\dim\text{\rom{Ran}}\, P_{(E_1, E_2)}(H).$$ \endproclaim \flushpar In Section 6, we'll prove that \proclaim{Theorem 1.6} Let$E_1 < E_2$. 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 $$W_0(u_1,u_2)\leq\dim\text{\rom{Ran}}\, P_{(E_1, E_2)}(H). \tag 1.8$$ \endproclaim \remark{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 1.4 yields (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 1.5). \endremark In Section 7, we provide a number of comments, examples, and extensions including: \proclaim{Theorem 1.7} Let$E_{1,2} \in\Bbb R$,$E_1 \ne E_2$,$\tau u_j = E_j u_j$,$j=1,2$,$\tau v_2=E_2 v_2$. Then$|W_0 (u_1,u_2)-W_0(u_1,v_2)|\leq 1$. \endproclaim It is easy to see that Theorems 1.5, 1.6, and 1.7 imply Theorems 1.3 and 1.4. 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 [8]) following previous work by Finkel, Isaacson, and Trubowitz [7] (see also [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 [7]. \vskip 0.3in \flushpar {\bf{\S 2. Oscillation Theory}} \medpagebreak For background, we recall the following: \proclaim{Theorem 2.1 ([22], Theorem 14.10)} Let$H$be an SL operator which is bounded from below. If$e_1 <\cdots < e_n < \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$are simple. \endproclaim \remark{Remarks} (i) Those used to thinking of the Dirichlet boundary condition case need to be warned that it is {\it{not}} in general true that if$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$is well-known to be bounded from below with compact resolvent (see, e.g., [1], Theorem 8.4.5; [22], Theorem 13.2). Thus Theorem 2.1 applies to the regular case (to be used in our proof of Proposition 4.1). \endremark The first part of the proposition below is a simple integration by parts and the second follows from the first. \proclaim{Proposition 2.2} Let$E_1 \leq E_2$and$\tau u_j = E_j u_j$,$j=1,2$. Then for$a0$on$(c,d)$. If$u_2$has no zeros in$(c,d)$, we can suppose$u_2>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>0$, and$u_1'(c)>0$(or$u_1'(d)<0$). Thus,$W(u_1,u_2)(c)\geq 0$,$W(u_1,u_2)(d)\leq 0$. Since the right side of (2.1) is positive, this is inconsistent with (2.1). \qed \enddemo \demo{Proof of Theorem {\rom{2.1}}} We first prove that$\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$, then by Corollary 2.3,$\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$, $$\eta_j(x) = \cases \psi_n(x), & x_j \leq x \leq x_{j+1} \\ 0, & \text{otherwise} \endcases, \quad 0 \leq j \leq m.$$ Then$\eta_j$is absolutely continuous with$p\eta'_j$piecewise continuous so$\eta_j$is in the form domain of$H$(see (A.6)) and$\langle |H|^{1/2} \eta_j, \text{sgn}(H) |H|^{1/2} \eta_j \rangle = e_n\,\| \eta_j \|$(where$\langle \,\cdot\, ,\,\cdot\,\rangle$and$\| \cdot \|$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$\langle |H|^{1/2} \eta, \text{sgn}(H)|H|^{1/2} \eta \rangle = e_n\,\| \eta \|$. 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. \qed \enddemo The second part of the proof of Theorem 2.1 also shows: \proclaim{Corollary 2.4} Let$H$be an SL operator bounded from below. If$\psi_+(E,\,.\,)$\rom(resp.\linebreak$\psi_-(E,\,.\,)$\rom) 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. \endproclaim \vskip 0.3in \flushpar{\bf{\S 3. Zeros of the Wronskian}} \medpagebreak 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: \proclaim{Proposition 3.1} Suppose that$\psi_{+,j}, \psi_-\in D_{\text{\rom{loc}}}$and that$\psi_{+,j}$and$\tau\psi_{+,j}$,$j=1,2$are in$L^{2}((c,b))$and that$\psi_-$and$\tau\psi_-$are in$L^{2}((a,c))$for all$c\in (a,b)$. Suppose, in addition, that$\psi_{+,j}$,$j=1,2$satisfy the boundary condition defining$H$at$b$\rom(i.e.,$W(u,\psi_{+,j})(c)\to 0$as$c\uparrow b$for all$u\in D(H)$\rom) and similarly, that$\psi_-$satisfies the boundary condition at$a$. Then {\rom{(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 $$H\eta=\chi_{[c,b)}(\tau\psi_{+,1}-\gamma\tau\psi_{+,2}). \tag 3.1$$ {\rom{(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 $$H\eta =\gamma\chi_{(a,c]}\tau\psi_- + \chi_{(c,b)}\tau\psi_{+,1}. \tag 3.2$$ \endproclaim \demo{Proof} Clearly,$\eta$and the right-hand-sides of (3.1)/(3.2) lie in$L^2((a,b))$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. \qed \enddemo 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, \proclaim{Proposition 3.2} 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$\theta_{u_1} (x_0)\equiv \theta_{u_2}(x_0)(\mod \pi)$. \endproclaim 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) >0$for$0 <|x - x_0|$sufficiently small and hence for all$0 <|x - x_0|$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: \proclaim{Proposition 3.3} If$(u,pu') \neq (0,0)$then$\theta_u(x_0)\equiv 0(\mod \pi)$implies $$[\theta_u(x)-\theta_u(x_0)]\big/ (x-x_0) >0$$ 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$\epsilon > 0$. \endproclaim In exactly the same way, we have \proclaim{Proposition 3.4} Let$E_10$for$0<|x -x_0|$. \endproclaim \demo{Proof} If$\Delta(x_0)\equiv 0(\mod 2\pi)$and$\theta_{u_2} (x_0)\not\equiv 0(\mod \pi)$, then$\sin(\theta_{u_2}(x_0)) \sin(\theta_{u_1}(x_0))>0$so$u_1(x_0)u_2(x_0) >0$for$0<|x-x_0|$sufficiently small, and thus by (2.2),$\frac{dW}{dx}(x_0)>0$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$(pu_1')(x_0) (pu_2') (x_0)>0$and so since$u(x_0)=v(x_0)=0$, we see that it is still true that$\frac{dW}{dx}(x)>0$a.e.~for$0<|x-x_0|$sufficiently small. \qed \enddemo \remark{Remarks} (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$c_0>0$. 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$c_{1}>0$. 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$c_2>0$. 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 3.3 and 3.4 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, $$\frac{d\theta_{\psi_\pm}}{dE}\,(E,x) = -\frac{\int\limits^{\ell_\pm}_x \psi_\pm(E,t)^2 \,dt} {\rho_{\psi_\pm}(E,x)} \tag 3.3$$ proves that$\mp\theta_{\psi_\pm}(E,x)$is strictly increasing with respect to$E$. In fact, from Proposition 2.3 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 (3.3). \endremark \vskip 0.3in \flushpar{\bf{\S 4. The Hare and the Tortoise ($\boldkey N_{\bold 0} \boldsymbol{\leq}\boldkey W_{\bold 0}$in the Regular Case)}} \medpagebreak Our goal in this section is to prove Theorem 1.5 in the regular case with opposite boundary conditions, that is, \proclaim{Proposition 4.1} Let$H$be a regular SL operator and suppose$E_10$. If there are$m$eigenvalues below$E_1$and$n_0+m$below$E_2$, then, by standard oscillation theory (essentially Proposition 3.3),$\theta_{\psi_-}(E_1,b) \in (m\pi,(m+1)\pi)$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$\Gamma_{\pm}-\theta_{\psi_\pm}\in \Bbb Z\pi$. 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 3.3 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 3.4 holds). Since$\Gamma_-(E_1,a)=0$and$\Gamma_+(E_2,a)>0$, the hare starts out ahead of the tortoise. Since$\Gamma_-(E_1,c)<\pi$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$b$,$\Gamma_-(E_1,b)>\Gamma_-(E_2,b)$, 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 $$W_0=n_0-1. \tag 4.2$$ Suppose$E_1E_4>E$and$u$is any solution of$\tau u=E u$, then \rom{(4.3)} holds. \endproclaim \demo{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. \qed \enddemo \proclaim{Lemma 4.5} Lemma \rom{4.4} remains true if every$\psi_-$is replaced by a$\psi_+$. \endproclaim \demo{Proof} Reflect at some point$c \in (a,b)$implying an interchange of$\psi_+$and$\psi_-$. \qed \enddemo \demo{Proof of Proposition {\rom{4.1}}} 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 4.3. By Lemma 4.4, $$W_0(\psi_-(e_s-\epsilon), \psi_+(e_\ell+\epsilon))\leq W_0(\psi_-(E_1), \psi_+(e_\ell+\epsilon))$$ and then by Lemma 4.5, this is no larger than$W_0(\psi_-(E_1), \psi_+(E_2))$. \qed \enddemo \newpage %\vskip 0.3in \flushpar{\bf{\S 5. Strong Limits ($\boldkey N_{\bold 0} \boldsymbol{\leq}\boldkey W_{\bold 0}$in the General Case)}} \medpagebreak Using the approach of Weidmann ([22], Chapter 14) to control some limits, we prove Theorem 1.5 in this section. Fix functions$u_1,u_2 \in D_{\text{\rom{loc}}}$. 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})$$$W(u_1,\eta)(c_{n})=0=W(u_2,\eta)(d_{n}). \tag 5.1$$ 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$H_{n} =\alpha\Bbb I\oplus\tilde{H}_{n}\oplus\alpha\Bbb I$with$\alpha$a fixed real constant. Then Weidmann proves: \proclaim{Lemma 5.1} 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$n\to\infty$. \endproclaim 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$n_0 \in \Bbb N$with$\eta\in D_0$for$n \geq n_0$and$H_{n}\eta\to H\eta$as$n$tends to infinity (see [21], Theorem 9.16(i)). If$H$is limit point at both ends, take$\eta\in D_0\equiv\{u\in D_{\text{\rom{loc}}}\mid \text{supp}(u)\text{ compact in }(a,b)\}$. 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$H$([22], Chapter 14). Secondly we note: \proclaim{Lemma 5.2} Let$A_{n}\to A$in strong resolvent sense as$n \to \infty$. Then $$\dim\text{\rom{Ran}}\,P_{(E_1,E_2)}(A)\leq \varliminf\limits_{n \to \infty} \dim\text{\rom{Ran}}\, P_{(E_1,E_2)}(A_{n}). \tag 5.2$$ \endproclaim \demo{Proof} Fix$m\leq\dim\text{\rom{Ran}}\,P_{(E_1,E_2)}(A)$with$m<\infty$. We'll prove$m\leq${\eightpoint{RHS}} of (5.2). Suppose first that$(E_1,E_2)$aren't eigenvalues of$A$. Then by Theorem VIII.24 of [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$\text{\rom{Ran}}\,P_{(E_1,E_2)}(A)$, we see that $$\varliminf\limits_{n \to \infty} \text{Tr}(P_{(E_1,E_2)}(A_{n})) \geq \varliminf\limits_{n \to \infty} \sum_j \langle\varphi_j, P_{(E_1,E_2)}(A_{n})\varphi_j\rangle=m$$ as required. If$E_{1,2}$are arbitrary, we can always find a$\delta > 0$such that$E_1+\delta, E_2-\delta$are not eigenvalues of$A$and such that$\dim\text{\rom{Ran}}\, P_{(E_1+\delta, E_2-\delta)}(A)\geq m$. Thus, $$\varliminf\limits_{n \to \infty} \dim\text{\rom{Ran}}\, P_{(E_1,E_2)}(A_{n})\geq \varliminf\limits_{n \to \infty} \dim\text{\rom{Ran}}\, P_{(E_1+\delta, E_2-\delta)}(A_{n})\geq m. \qed$$ \enddemo \demo{Proof of Theorem {\rom{1.5}}} Let$c_{n}\downarrow a$,$d_{n} \uparrow b$and$H_n$be as in Lemma 5.1 with$\alpha\notin [E_1, E_2]$. Proposition 4.1 implies$W_0(u_1,u_2)\geq\dim\text{\rom{Ran}}\, P_{(E_1, E_2)}(\tilde H_{n})= \dim\text{\rom{Ran}}\,P_{(E_1, E_2)} (H_{n})$since$\alpha\notin [E_1,E_2]$. Thus by Lemmas 5.1 and 5.2, $$W_0(u_1,u_2)\geq\dim\text{\rom{Ran}}\,P_{(E_1, E_2)}(H)$$ as was to be proven. \qed \enddemo \newpage %\vskip 0.3in \flushpar{\bf{\S 6. A Variational Argument ($\boldkey N_{\bold 0} \boldsymbol{\geq}\boldkey W_{\bold 0}$)}} \medpagebreak In this section, we'll prove Theorem 1.6. Let$E_1x_j \endcases, \quad 1 \le j \le m. $$If E_1 is an eigenvalue of H, we define in addition \eta_0 = u_2 = -\tilde{\eta}_0, x_0=a and if E_2 is an eigenvalue of H, \eta_{m+1} = u_1 = \tilde{\eta}_{m+1}, x_{m+1}=b. \proclaim{Lemma 6.1} \langle\eta_j, \eta_{k}\rangle =\langle \tilde{\eta}_j, \tilde{\eta}_{k}\rangle for all j,k where \langle \,\cdot\, ,\,\cdot\,\rangle is the L^{2}((a,b);r\,dx) inner product. \endproclaim \demo{Proof} Let jx_j \endcases, \quad 1 \le j \le m.$$ If $E_1$ is an eigenvalue of $H$, we define in addition $\eta_0 = u_2 = -\tilde{\eta}_0$, $x_0=b$ and if $E_2$ 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 6.1, we need $$\int\limits^{x_j}_{a}u_1(x)u_2(x)r(x)\, dx =0.$$ But by (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 1.1. The cases $u_1=\psi_+(E_1)$, $u_2=\psi_\pm(E_2)$ can be obtained by reflection. \vskip 0.3in \flushpar{\bf{\S 7. Extensions, Comments, and Examples}} \medpagebreak The following includes Theorem 1.7: \proclaim{Theorem 7.1} 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 \rom(in the sense that there is exactly one zero of one function in between two zeros of the other\rom). In particular, $|W_0(u_1,u_2)-W_0(u_1,v_2)| \leq 1$. \endproclaim \demo{Proof} We'll suppose $E_1\Gamma_{u_1}(E_1,b-\epsilon)$), the number of zeros of $u_2$ equals the number of laps plus the number of zeros of $u_1$ plus one. \qed \enddemo Next we want to see how Hartman's theorem (Theorem 1.2) follows from Theorem 1.4. We start by assuming $\tau$ to be oscillatory at $E_2$. By Theorem 1.4, $W_0(u_1,u_2)=N_0$ since $H$ in Theorem 1.2 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 (1.5a). Suppose first that $W_0(u_1,u_2)=m<\infty$. Let $x_m$ be the last zero of $W(u_1,u_2)(x)$ (set $x_m=a$ if $m=0$). Measure Pr\"ufer angles starting at the same $\theta (a)\in [0,\pi)$. At $x_m$, $$\theta_{u_2}(x_{m})=\theta_{u_1}(x_{m})+m\pi \tag 7.1$$ and then $$\Gamma_{u_2}(x_{m}+\epsilon)>\Gamma_{u_1}(x_{m}+\epsilon). \tag 7.2$$ Let $N_{u_j}(x)$ be the number of zeros of $u_j$, $j=1,2$ in $(a,x)$. By (7.1) and Proposition 3.3, $$N_{u_2}(x_{m})=N_{u_1}(x_{m})+m.$$ As $x$ increases, (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 $c>x_{m}$, $N(c)\equiv N_{u_2}(c)-N_{u_1}(c)$ alternates between $m$ and $m+1$ and since $\tau$ is assumed to be oscillatory at $E_2$, 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 (7.1) and (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 $E_2$, 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 $u_{1,2}(x)>0$ for $x$ near $b$ and claim in addition that $u_1 u_2$ is not $L_1$ near $b$. If $u_1 u_1^2$ or $u_1 u_2 > u_2^2$ for $x$ near $b$ and $u_j \not\in L^2$ near $b$. In fact, by hypothesis, $u_j \in L^2$ near $a$ and since $E_j$ are not eigenvalues and $\tau$ is limit point at $b$, $u_j$ cannot be $L^2$ 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 (2.1). But $u_1 u_2$ not $L_1$ near $b$ together with (2.1) implies that $u_2'/u_2 > u_1'/u_1$ 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 $E' < E''$ 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 2.4) shows that the number of zeros of $u(E'')$ equals the number of eigenvalues below $E''$ (compare Corollary 2.4), that is, equals $\dim\text{\rom{Ran}}\, P_{(-\infty, E'')}$ if $E''$ is not an eigenvalue. Theorem 2.1 then covers the case where $E''$ is an eigenvalue. Applying this to $E''=E_1$ and $E''=E_2$ proves (1.5b) since clearly $$\dim\text{\rom{Ran}}\, P_{(-\infty, E'')} - \dim\text{\rom{Ran}}\, P_{(-\infty,E')} = \dim\text{\rom{Ran}}\, P_{[E', E'')}.$$ \vskip 0.1in Finally, we want to consider the relation to the density of states. Given an SL operator $H$, let $H^{(L)}_D$ 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: $$k(E)=\lim\limits_{L\to\infty}\,(2L)^{-1}\dim\text{\rom{Ran}}\, P_{(-\infty, E)} (H^{(L)}_D).$$ \proclaim{Theorem 7.5} Suppose $H$ is such that the ids exists for all $E$. Let $E_10$ a.e.~on } (a,b). \tag A.1 $$Next, define in L^2((a,b);rdx) the following linear operators$$ \gather (H^0_{\alpha,\beta} u)(x) = -r(x)^{-1}(p(x)u'(x))', \\ \matrix \format\l&\l\\ D(H^0_{\alpha,\beta}) = \{u \in L^2((a,b);r\,dx) \mid\, & 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\}, \endmatrix \\ \qquad\qquad\qquad \qquad\qquad\qquad \qquad\qquad\qquad \alpha,\beta \in {\Bbb R} \cup \{ \infty \} \endgather $$(here \alpha=\infty denotes a Dirichlet boundary condition u(a) =0 and similarly at b),$$ \gather S_{\alpha,\beta} u = s\, u, \quad (s\, u)(x) = (p(x)/r(x))^{1/2}u'(x), \quad \alpha,\beta \in \{0,\infty\}, \\ \matrix \format\l&\l\\ D(S_{\alpha,\beta}) = \{u \in L^2((a,b);r\,dx) \mid\, & 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 \}, \endmatrix\\ 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\},\\ \matrix \format\l&\l\\ D(S^+_{\alpha,\beta}) = \{u \in L^2((a,b);r\,dx) \mid\, & (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 \}, \endmatrix \endgather $$and the form$$ R^0_{\alpha,\beta}(u,v) = \langle S_{\alpha,\beta}u, S_{\alpha,\beta}v \rangle, \quad {\Cal D}(R^0_{\alpha,\beta}) = D(S_{\alpha,\beta}), \quad \alpha,\beta \in \{0,\infty\} $$(\langle\,.\, ,\,.\,\rangle the scalar product in L^2((a,b);r\,dx)). \proclaim{Lemma A.1} {\rom{(i)}} S_{\alpha,\beta} = (S_{\alpha,\beta}^+)^* and S_{\alpha,\beta}^+ = S^*_{\alpha,\beta} for all \alpha,\beta \in \{0, \infty\}. {\rom{(ii)}} H^0_{\alpha,\beta} = S^*_{\alpha,\beta} S_{\alpha,\beta}, \alpha,\beta \in \{0, \infty\}. \endproclaim \demo{Proof} Define$$ \matrix \format\l&\l\\ K: & L^2((a,b);r\,dx) \to D(S_{\infty,0})\\ & g \mapsto \int\limits_a^x \frac{g(y)r(y)\,dy}{(p(y)r(y))^{1/2}} \endmatrix, \quad \matrix \format\l&\l\\ K^+: & L^2((a,b);r\,dx) \to D(S^+_{0,\infty})\\ & g \mapsto (pr)(x)^{-1/2} \int\limits_a^x g(y)r(y)\,dy \endmatrix \, . $$A straightforward calculation verifies (K\,g)(a)=0, sK\,g = g and ((pr)^{1/2}K^+\,g)(a)=0, s^+ K^+\,g = g. We only show S_{\alpha,\beta}^* = S^+_{\alpha,\beta}, the case (S^+_{\alpha,\beta})^* = S_{\alpha,\beta} being analogous. Moreover, since S_{\infty,\infty} \subseteq S_{\alpha,\beta} implies S_{\alpha,\beta}^* \subseteq S_{\infty,\infty}^* we only concentrate on proving S_{\infty,\infty}^* = S_{\infty,\infty}^+, the rest following from an additional integration by parts. An integration by parts proves S^+_{\infty,\infty}\subseteq S^*_{\infty,\infty}. Conversely, let f \in D(S_{\infty,\infty}^*) and set g = K^+ S_{\infty,\infty}^* f. Then$$ \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 $$for all h\in D(S_{\infty,\infty}). Thus, \text{\rom{Ran}} (S_{\infty,\infty}) is a subset of the kernel of the linear functional k\mapsto \langle f-g,k \rangle. But \text{\rom{Ran}} (S_{\infty, \infty})= \{(pr)^{-1/2}\}^\perp (since g\in \text{\rom{Ran}} (S_{\infty,\infty}) is equivalent to (Kg)(b)=0) and hence f=g+c (pr)^{-1/2} \in D(S^+_{\infty,\infty}) for some constant c proving S_{\infty,\infty}^* \subseteq S^+_{\infty,\infty} and hence (i). By inspection, we obtain D(S^+_{\alpha,\beta} S_{\alpha,\beta}) = \{u\in S_{\alpha,\beta}\mid S_{\alpha,\beta}u \in D(S^+_{\alpha, \beta})\} = D(H^0_{\alpha,\beta}) since pu'\in AC([a,b]) implies (p/r)^{1/2} u'= (pr)^{-1/2}(pu')\in L^2((a,b);r\,dx) and S^+_{\alpha,\beta} S_{\alpha,\beta}u = H^0_{\alpha,\beta}u. This fact together with (i) proves (ii). \qed \enddemo Furthermore, we introduce the forms$$ \gather Q_{q/r}(u,v) = \int\limits_a^b q(x)r(x)^{-1}\, \overline{u(x)}\, v(x) r(x)\, dx, \\ \Cal 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)\}, \endgather $$and$$ \gather q_{\alpha,\beta}(u,v)= \tilde{\beta}\, \overline{u(b)}\, v(b) - \tilde{\alpha}\, \overline{u(a)}\, v(a), \quad \Cal D (q_{\alpha,\beta}) = AC([a,b]),\\ \tilde{\alpha} = \cases \alpha, & \alpha\in\Bbb R \\ 0, & \alpha=\infty \endcases, \quad \tilde{\beta} = \cases \beta, & \beta\in\Bbb R \\ 0, & \beta=\infty \endcases, \quad \alpha,\beta \in \Bbb R \cup \{\infty\}. \endgather $$Finally, we set$$ \gather Q^0_{\alpha,\beta} = R^0_{\widehat{\alpha},\widehat{\beta}} + q_{\alpha,\beta}, \quad \Cal D (Q^0_{\alpha,\beta}) = D(S_{\widehat{\alpha},\widehat{\beta}}), \\ \widehat{\alpha} = \cases 0, & \alpha\in\Bbb R \\ \infty, & \alpha=\infty \endcases, \quad \widehat{\beta} = \cases 0, & \beta\in\Bbb R \\ \infty, & \beta=\infty \endcases, \quad \alpha,\beta \in\Bbb R \cup \{\infty\}. \endgather $$and$$ Q_{\alpha,\beta} = Q^0_{\alpha,\beta} + Q_{q/r}, \quad \Cal D (Q_{\alpha, \beta}) = D(S_{\widehat{\alpha}, \widehat{\beta}}), \quad \alpha,\beta \in {\Bbb R} \cup \{\infty\}. \tag A.2 $$\proclaim{Lemma A.2} {\rom{(i)}} q_{\alpha,\beta} is infinitesimally form bounded with respect to Q^0_{0,0}. {\rom{(ii)}} Q_{q/r} is relatively form compact with respect to Q^0_{\alpha,\beta}, \alpha,\beta \in\Bbb R \cup \{\infty\}. \endproclaim \demo{Proof} (i) Since for arbitrary c \in [a,b] and u \in D(S_{0,0}),$$ |u(c)|^2 = \biggl| u(x)^2 - 2 \int\limits_c^x u(t) u'(t)\,dt\biggr| \leq |u(x)|^2 + \int\limits_a^b |u(t) u'(t)|\,dt, $$one infers (after integrating from a to b) for any \epsilon>0,$$ \align \| u \|_\infty^2 &\leq (b-a)^{-1}\| u \|^2_2 + 2 \int\limits_a^b \frac{|u(t)|}{(\epsilon p(t)/2)^{1/2}}\, (\epsilon p(t)/2)^{1/2} |u'(t)|\,dt \\ & \leq (b-a)^{-1}\| u \|^2_2 + \int\limits_a^b \biggl(\frac{2} {\epsilon}\, \frac{|u(t)|^2}{p(t)} + \frac{\epsilon}{2}\, p(t) |u'(t)|^2 \biggr)\, dt. \tag A.3 \endalign  Since \$0