\input amstex \documentstyle{amsppt} \magnification=1200 \baselineskip=15 pt \loadbold \TagsOnRight \topmatter \title Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schr\"odinger operators \endtitle \rightheadtext{Bounded eigenfunctions and a.c.~spectra} \author Barry Simon$^{*}$ \endauthor \leftheadtext{B.~Simon} \affil Division of Physics, Mathematics, and Astronomy \\ California Institute of Technology, 253-37 \\ Pasadena, CA 91125 \endaffil \date March 27, 1995 \enddate \thanks $^{*}$ This material is based upon work supported by the National Science Foundation under Grant No.~DMS-9401491. The Government has certain rights in this material. \endthanks \thanks To be submitted to {\it{Proc.~Amer.~Math.~Soc.}} \endthanks \abstract We provide a short proof of that case of the Gilbert-Pearson theorem that is most often used: That all eigenfunctions bounded implies purely a.c.~spectrum. Two appendices illuminate Weidmann's result that potentials of bounded variation have strictly a.c.~spectrum on a half-axis. \endabstract \endtopmatter \document \flushpar{\bf{\S 1. Introduction and Reduction to $\boldkey m$-functions}} \medpagebreak In this note, I want to consider Schr\"odinger operators and Jacobi matrices on a half-line. Specifically, we'll consider the operator $h$ on $\ell^{2}(\Bbb Z_{+})$ (with $\Bbb Z_{+}=\{1,2,\dots\}$) given by \align (hu)(n) &= u(n+1)+u(n-1)+v(n)u(n) \tag 1.1a \\ u(0)&= 0 \tag 1.1b \endalign and the self-adjoint operator on $L^{2}(0,\infty)$ \align (Hu)(x) &= -u''(x)+V(x)u(x) \tag 1.2a \\ u(0) &= 0 \tag 1.2b \endalign where we suppose $$\Gamma(V)\equiv\sup\limits_{x}\, \biggl(\int\limits^{x+1}_{x-1} |V(y)|^{2}\biggr) <\infty. \tag 1.3$$ For any $E\in\Bbb C$, define two solutions $u_{1}, u_{2}$ of the formal difference (resp.~differential) equation $hu=Eu$ (resp.~$Hu=Eu$) with boundary conditions: \xalignat 2 u_{1}(0,E) &= 0 & u_{1}(1,E) & =1 \\ u_{2}(0,E) &= 1 & u_{2}(1,E) & =0 \endxalignat in the discrete case and \xalignat 2 u_{1}(0,E) & =0 & u'_{1}(0,E) & =1 \\ u_{2}(0,E) & =1 & u'_{2}(0,E) & =0 \endxalignat in the continuous case. Let $S=\{E\in\Bbb R\mid u_{1}\text{ and$u_{2}$are bounded on$[0,\infty)$}\}$. Then our purpose here is to prove \proclaim{Theorem 1} On $S$, the spectral measure $\rho$ for $h$ \rom(resp.~$H$\rom) is purely absolutely continuous in the sense that \roster \item"\rom{(i)}" $\rho_{\text{\rom{ac}}}(T)>0$ for any $T\subset S$ with $|T|>0$ \rom(where $|\,\cdot\,|=$ Lebesgue measure\rom) \item"\rom{(ii)}" $\rho_{\text{\rom{sing}}}(S)=0$. \endroster \endproclaim This theorem is not new. In [9,8,11,13], Gilbert, Khan, and Pearson proved a complete characterization of the essential support of $\rho_{\text{\rom{ac}}}$ in terms of mutually subordinate solutions. Their approach has the advantage of not requiring (1.3). Behncke [2] and Stolz [16] have noted that $V$ uniformly $L^{1}_{\text{\rom{loc}}}$ with bounded eigenfunctions allows one to use the Gilbert-Pearson theory. Virtually all applications of [16,12] use the weaker Theorem 1. There seems to be some point in the short proof I'll present here which avoids some of their tricky calculations and which makes the result transparent. In addition, we'll obtain explicit bounds on $m$-functions. I should mention earlier work of Carmona [4] (which is weaker than Theorem 1) and related work of Briet-Mourre [3]. As with Gilbert-Pearson, our proof uses the theory of Weyl $m$-functions. For $E\in\Bbb C_{+}=\{z\mid\text{Im}\, z>0\}$, we can find a unique solution $u_{+}(n,E)$ (resp.~$u_{+}(x,E)$) of (1.1a)/(1.2a) with $u_{+}\in\ell^{2}$ (resp.~$L^2$) at infinity, normalized by $$u_{+}(0,E)=1. \tag 1.4$$ Then one defines the $m$ function by $$m_{+}(E)=u_{+}(1,E) \tag 1.5$$ in the discrete case and $$m_{+}(E)=u'_{+}(0,E) \tag 1.6$$ in the continuous case. By looking at the Wronskian of $u_+$ and $\bar u_{+}$, one gets the well-known formula: $$\text{Im}\, m_{+}(E)=\text{Im}\, E\sum^{\infty}_{n=1} |u_{+} (n,E)|^{2} \tag 1.7$$ in the discrete case and $$\text{Im}\, m_{+}(E)=\text{Im}\, E\int\limits^{\infty}_{0} |u_{+}(x, E)|^{2}\, dx \tag 1.8$$ in the continuous case. It is known (see [5,15]) that $$d\rho(E)=\frac{1}{\pi}\,\lim\limits_{\epsilon\downarrow 0}\, \text{Im}\, m_{+}(E+i\epsilon)\, dE. \tag 1.9$$ It follows [1,7] by the de la Vall\'ee-Poussin theorem that $$\rho_{\text{\rom{sc}}}\text{ is supported on } \biggr\{E\mid\lim\limits_{\epsilon\downarrow 0}\,\text{Im}\, m_{+}(E+i\epsilon)=\infty\biggl\}$$ and $$d\rho_{\text{\rom{ac}}}(E)=\frac{1}{\pi}\,\text{Im}\, m_{+} (E+i0)\, dE.$$ Thus, Theorem 1 is an immediate consequence of \proclaim{Theorem 2} If $E\in S$, then \alignat2 &\text{\rom{(i)}} &&\qquad \varliminf\,\text{\rom{Im}}\, m_{+} (E+i0)> 0 \tag 1.10 \\ &\text{\rom{(ii)}} &&\qquad \varlimsup\, |m_{+}(E+i0)|<\infty. \tag 1.11 \endalignat \endproclaim \remark{Remark} While the results are stated for the half-line with Dirichlet boundary conditions, Theorem 2 immediately implies the result for any fixed boundary condition and for the whole line. For it is known [1,15] that the essential support $d\rho_{\text{\rom{ac}},\theta}$ for $\theta$ boundary conditions (given by $\sin(\theta)u'(0)+\cos(\theta)u(0)=0$) is $\theta$ independent and that $d\rho_{\text{\rom{sc}},\theta}$ is supported on the set where $m_{+}(E+i0)=-\cot(\theta)$, which cannot happen if (1.10)/(1.11) holds. For the whole line, we can define $S$ via the right half-line condition from which (1.10)/(1.11) and the formula (for the continuous case; the discrete case is similar) \align d\rho_{1}(E) &= -\lim\limits_{\epsilon\downarrow 0}\, \frac{1}{\pi}\, \text{Im} \biggl(\frac{1}{m_{+}(E+i\epsilon)+m_{-}(E+i\epsilon)} \biggr)\,dE \\ d\rho_{2}(E) &= \lim\limits_{\epsilon\downarrow 0}\, \frac{1}{\pi}\, \text{Im}\biggl(\frac{1}{m_{+}(E+i0)^{-1}+m_{-}(E+i0)^{-1}}\biggr)\, dE \endalign imply $\rho_{i,\text{\rom{sc}}}(S)=0$. \endremark \medpagebreak It is a pleasure to thank F.~Gesztesy, A.~Kiselev, and G.~Stolz for useful discussions. \newpage \flushpar{\bf{\S 2. The Jacobi Matrix Case}} \medpagebreak In this section, we'll prove Theorem 2 in the discrete case. Define the fundamental or transfer matrix by $$T(E,n,0)= \pmatrix u_{1}(n+1, E) & u_{2}(n+1, E) \\ u_{1}(n,E) & u_{2}(n,E) \endpmatrix$$ and then $$T(E,n,m) = T(E,n,0)T(E,m,0)^{-1} \tag 2.1$$ $T$ is defined so that if $u$ obeys $hu=Eu$, then $\Phi(n)=\binom {u(n+1)}{u(n)}$ obeys $$\Phi(n)=T(E,n,m)\Phi(m).$$ Constancy of the Wronskian implies $\det\,T=1$ so $\|T^{-1}\|= \|T\|$ and thus by (2.1) $$C(E)=\sup\limits_{n,m}\,\|T(E,n,m)\|\leq \sup\limits_{n}\, \|T(E,n,0)\|^{2} \tag 2.2$$ is finite if and only if $E\in S$. We'll prove Theorem 2 in the following explicit form: \proclaim{Theorem 2J} If $E\in S$, then \align \varliminf\,\text{\rom{Im}}\, m_{+}(E+i\epsilon) &\geq \frac{1}{4}\, C^{-3} \tag 2.3 \\ \varlimsup\, |m_{+}(E+i\epsilon)| &\leq 4C^{3} \tag 2.4 \endalign where $C(E)$ is given by {\rom{(2.2)}}. \endproclaim \demo{Proof} Let $$A(E,n)\equiv \pmatrix E-V(n) & -1 \\ 1 & 0 \endpmatrix$$ so $T(E,n,0)=A(E,n)T(E,n-1,0)$. It follows (as a telescoping sum) that $$T(E+i\epsilon, n,0)=T(E,n,0)+\sum^{n-1}_{j=0}(i\epsilon) T(E,n,j+1)\pmatrix 1 & 0 \\ 0 & 0 \endpmatrix T(E+i\epsilon, j, 0)$$ so by iteration, we get $$\|T(E+i\epsilon, n,0)\| \leq \sum^{n}_{k=0}\binom{n}{k} C^{k+1} \epsilon^{k}= C(1+C\epsilon)^{n}\leq Ce^{\epsilon Cn} \tag 2.5$$ By $\|T^{-1}\|=\|T\|$, we see that $$\biggl\|\binom{u_{+}(E+i\epsilon, n+1)}{u_{+}(E+i\epsilon, n)}\biggr\| \geq C^{-1}e^{-\epsilon Cn} (|m_{+}(E+i\epsilon)|^{2}+1)^{1/2}$$ since $\binom{u_{+}(E+i\epsilon,1)}{u_{+}(E+i\epsilon, 0)} = \binom{m_{+}(E+i\epsilon)}{1}$. Squaring and summing over $n=1,3,\dots$ we see that $$\sum^{\infty}_{n=1} |u_{+}(E+i\epsilon, n)|^{2} \geq C^{-2} e^{-2\epsilon C} (1-e^{-4\epsilon C})^{-1} (|m_{+}(E+i\epsilon)|^{2} +1).$$ Thus by (1.7) $$\text{Im}\, m_{+}(E+i\epsilon)\geq \frac{1}{4}\, C^{-3} e^{-2\epsilon C}(4\epsilon C)(1-e^{-4\epsilon C})^{-1} [1+|m_{+}(E+i\epsilon)|^{2}]$$ or $$\varliminf\,\biggl[\text{Im}\, m_{+}(E+i\epsilon)\big/[1+|m_{+} (E+i\epsilon)|^{2}]\biggr] \geq \frac{1}{4}\, C^{-3} \tag 2.6$$ Noting that $(1+|m_{+}|^{2})^{-1}\leq 1$, we see that (2.6) immediately implies (2.3). And since $(1+|m_{+}|^{2})/\text{Im}\, m_{+}\geq |m_{+}|$, it also implies (2.4). \qed \enddemo With only minor changes, the theorem extends to the general Jacobi matrix (tridiagonal self-adjoint) matrix: $$(hu)(n)=a_{n+1} u(n+1)+a_{n}u(n-1)+b_{n}u(n) \tag 2.7$$ so long as there is $\alpha$ finite with $$\alpha^{-1}< |a_{n}|<\alpha \tag 2.8$$ for all $n$. If $d\rho$ is the spectral measure for $u(n)=\delta_{1n}$, then $$\int\frac{d\rho (E)}{z-E}=m_{+}(z)$$ where $m_{+}(z)$ is defined to be $a^{-1}_{1}u_{+}(1)$ (if $u_+$ is normalized by $u_{+}(0)=1$). (1.7) becomes $$\text{Im}\, m_{+}(E)=a^{-2}_{1}(\text{Im}\, E) \sum^{\infty}_{n=1} |u_{+}(n,E)|^{2}.$$ It is no longer true that $\|T(E,n,0)^{-1}\|=\|T(E,n,0)\|$ since $\det(T(E,n,0))$ may not be $1$. Rather $\det(T(E,n,0))=\frac{a_1} {a_{n}+1}$ so using (2.8), (2.2) becomes $C(E)\leq \alpha^{2} \sup\limits_{n}\,\mathbreak \|T(E,n,0)\|^{2}$. (2.5) becomes $$\|T(E+i\epsilon, n,0)\|\leq C(1+C\alpha\epsilon)^{n} \leq Ce^{\epsilon Cn\alpha}$$ and (2.6) becomes $$\varliminf\, [\text{Im}\,m_{+}(E+i\epsilon)]\big/ [1+a^{2}_{1} |m_{+}(E+i\epsilon)|^{2}] \geq\frac{1}{4}, a^{-2}_{1} C^{-3} \alpha^{-1}.$$ \newpage \flushpar{\bf{\S 3. The Schr\"odinger Case}} \medpagebreak To carry the proof through from the discrete case, we must use (1.3) to bound $u'$ locally by $u$. This is a standard Sobolev-type estimate; we haven't tried to optimize constants. \proclaim{Lemma 3.1} If $u$ obeys $-u''+Vu=Eu$, then $$|u'(x)|^{2}\leq\biggl[ 4+\frac{3}{4}\,\Gamma(|V-E|)\biggr] \int\limits^{x+1}_{x-1} |u(y)|^{2}\, dy \tag 3.1$$ where $\Gamma$ is given by {\rom{(1.3)}}. \endproclaim \demo{Proof} By Taylor's theorem with remainder, $$f'(0)=\frac{1}{2}\,\frac{f(x)-f(-x)}{x}-\frac{1}{2x} \int\limits^{x}_{0} (x-y)[f''(y)+f''(-y)]\,dy.$$ Integrate this from $\frac{1}{2}$ to $1$ to get $$|f'(0)|\leq\int\limits^{1}_{-1} |f(x)|\,dx +\frac{3}{8}\int \limits^{1}_{-1} |f''(0)|\, dx.$$ Let $f(y)=u(y+x)$ and use $u''=(V-E)u$ and the Schwarz inequality to get (3.1). \qed \enddemo By (3.1), if $E\in S$, $u'$ is also bounded and thus the transfer matrix $T(E,x,y)$ defined by $$T(E,x,y)\binom{u'(y)}{u(y)}=\binom{u'(x)}{u(x)}$$ is bounded. Let $$C(E)\equiv\sup\limits_{x,y}\,\|T(E,x,y)\|.$$ \proclaim{Theorem 2S} Let $E\in S$ and define $A(E)=\frac{1}{2} C(E)^{-3}/(9+\frac{3}{2}\Gamma(|E-V|))$. Then \align \varliminf\,\text{\rom{Im}}\, m_{+}(E+i\epsilon) &\geq A \\ \varlimsup\,|m_{+}(E+i\epsilon)| &\leq A^{-1}. \endalign \endproclaim \demo{Proof} By mimicking the proof of (2.5), using integrals in place of sums $$\|T(E+i\epsilon, x,0)\|\leq Ce^{\epsilon C|x|} \tag 3.2$$ By (3.1) $$\int\limits^{\infty}_{1} |u'(x)|^{2}\,dx \leq\biggl[ 8+\frac{3}{2}\, \Gamma (|V-E-i\epsilon|)\biggr] \int\limits^{\infty}_{0} |u(y)|^{2}\, dy$$ so if $\beta=1/(9+\frac{3}{2}\Gamma)$, then \align \int\limits^{\infty}_{0} |u(y)|^{2}\,dy &\geq \beta\int \limits^{\infty}_{1} [|u(x)|^{2}+|u'(x)|^{2}]\, dx \\ &\geq C^{-2}\beta (1+|m_{+}|^{2})\int\limits^{\infty}_{1} e^{-2\epsilon Cx}\,dx \endalign so by (1.8), $$\text{Im}\,m_{+}\geq\frac{1}{2}\, C^{-3}\beta e^{-\epsilon C} (1+|m_{+}|^{2})$$ and the result follows as in the discrete case. \qed \enddemo \vskip 0.3in \flushpar{\bf{Appendix 1: A Discrete Version of Weidmann's Theorem}} \medpagebreak One of the more interesting applications of Theorem 2 is the result of Weidmann [17,18,19] that if $V=V_{1}+V_2$ where $V_{1}\in L^1$ and $V_2$ is of bounded variation with $V_{2}(x)\to 0$ at infinity, then $-\frac{d^2}{dx^2}+V(x)$ has purely a.c.~spectrum on $(0, \infty)$. A key to his argument is a proof that for any $E>0$, solutions are bounded. He does this by noting one can suppose $V_2$ is $C^1$ with $V'_{2}\in L^1$ (by adjusting the breakup) and that if $K(x)=(u')^{2}+ (E-V_{2})u^2$, then $K'(x)=2V_{1}u'u- 2V'_{2}u^{2}\leq C(|V_{1}|+ |V'_{2}|)K(x)$ for $x$ large. Here we'll prove a discrete analog: \proclaim{Theorem A.1} Let $v_n$ be a sequence on $\{1,2,\dots\}$ so that $v_{n}\to 0$ and $$\sum^{\infty}_{n=1} |v_{n+1}-v_{n}|<\infty. \tag A.1$$ Then, the operator $h$ of {\rom{(1.1)}} has purely absolutely continuous spectrum on $(-2, 2)$. \endproclaim \remark{Remarks} 1. (A.1) implies $\lim\, v_n$ exist so by adding a constant, it is no loss to suppose $v_{n}\to 0$. 2. If $v_{n}\in\ell^1$, then (A.1) holds so we don't need to consider sums as Weidmann does in the continuous case. \endremark \demo{Proof} Given a solution of $hu=Eu$, let $$K_{n}=u^{2}_{n+1}+u^{2}_{n}+(v_{n}-E) u_{n}u_{n+1}.$$ Then $$(K_{n+1}-K_{n}) =(u_{n+2}-u_{n})(u_{n+2}+u_{n}+(v_{n+1}-E)u_{n+1}) +(v_{n}-v_{n+1})u_{n}u_{n+1}$$ so $$|K_{n+1}-K_{n}|\leq |v_{n}-v_{n+1}|\,|u_{n}u_{n+1}|. \tag A.2$$ Suppose now $E\in (-2,2)$. Then for $n\geq\text{some }N_0$, $2-|v_{n}- E|\geq\delta >0$. For such $n$, \align K_{n} &\geq\frac{\delta}{2}\,({u_{n+1}}^{2}+{u_{n}}^{2})+\biggl(1- \frac{\delta}{2}\biggr) (|u_{n+1}|-|u_{n}|)^{2} \\ &\geq\frac{\delta}{2}\, ({u_{n+1}}^{2}+{u_{n}}^{2}) \endalign so (A.2) becomes $$K_{n+1}\leq \biggl( 1+\frac{2}{\delta}\,|v_{n}-v_{n+1}|\biggr) K_{n}$$ and for all $n\geq N_0$: $$K_{n}\leq\prod\limits^{\infty}_{m=N_{0}} \biggl(1+\frac{2}{\delta}\, |v_{m}-v_{m+1}|\biggr) K_{N_0}.$$ The product is convergent by (A.1). \qed \enddemo By using the remark at the end of Section 1, Theorem A.1 extends to the operator (2.7) so long as (2.8) holds and \alignat2 b_{n} &\to 0, \qquad && \sum^{\infty}_{n=1} |b_{n+1}-b_{n}|<\infty \\ a_{n} &\to 1, \qquad && \sum^{\infty}_{n=1} |a_{n+1}-a_{n}|<\infty. \endalignat We merely define $K_n$ by $$K_{n}=a_{n+1}u^{2}_{n+1}+a_{n}u^{2}_{n}+(b_{n}-E)e_{n}u_{n+1}.$$ This is related to results of [6]. \vskip 0.3in \flushpar{\bf{Appendix 2: Eigenfunctions for Weidmann's Theorem}} \medpagebreak We want to further elucidate Weidmann's theorem by showing how to actually find the asymptotics of the eigenfunctions. We'll suppose $V(x)=V_{1}(x)+V_{2}(x)$ with $V_{1}\in L^1$ and $V_{2}$ a $C^1$ function with $V'_{2}\in L^1$ and $V_{2}\to 0$ at infinity. We claim: \proclaim{Theorem B.1} Fix $E=k^{2}>0$ with $k>0$. Then every solution of $(-\frac{d^2}{dx^2}+V(x))u=Eu$ is bounded; indeed, there exist $a,b$ so that \align |u(x)-au_{+}(x)-bu(x)| &\to 0 \\ |u'(x)-iaku_{+}(x)+ibku_{-}(x)| &\to 0 \endalign where $$u_{\pm}(x)=\exp\biggl(\pm i\int\limits^{x}_{x_0} \sqrt{k^{2}- V_{2}(x)}\, dx\biggr) \tag B.1$$ where $x_0$ is chosen so large that $V_{2}(x)x_0$. \endproclaim \remark{Remarks} 1. Since $(k^{2}-V_{2}(x))^{-1/4}\to k^{-1/2}$, we could use the {\eightpoint{WKB}} form instead of (B.1), but the form (B.1) is what enters naturally. 2. This theorem and proof can be regarded as specializations of arguments in Hinton-Shaw [10]. \endremark \demo{Proof} Define $u_\pm$ by (B.1). Note that $u_\pm$ are $C^2$ and $$-u''_{\pm}+(V(x)-E)u_{\pm}=F_{\pm}u_{\pm} \tag B.2a$$ where $$F_{\pm}(x)=V_{1}(x)\pm\frac{i}{2}\,V'_{2}(x) (k^{2}-V_{2}(x))^{-1/2} \tag B.2b$$ is in $L^1$ near infinity. Let $W(x)$ be the Wronskian of $u_+$ and $u_-$. Clearly, $W(x)=2ik +o(1)$. Define $a(x), b(x)$ by the equations (variation of parameters) \align u(x) &= a(x)u_{+}(x)+b(x)u_{-}(x) \\ u'(x) &= a(x)u'_{+}(x)+b(x)u'_{-}(x). \endalign A straightforward and standard calculation (see prob.~98 on pg.~395 of [14]) shows that $a,b$ obey the equations $$\binom{a(x)}{b(x)}' = M(x)\binom{a(x)}{b(x)}$$ where $$M(x)=W(x)^{-1} \pmatrix -F_{+} & -u^{2}_{-}F_{-} \\ F_{+}u^{2}_{+} & F_{-} \endpmatrix.$$ Since this is in $L^1$, standard arguments show that $\lim\limits _{k\to\infty}\binom{a(x)}{b(x)}=\binom{a}{b}$ exists. \qed \enddemo If, moreover, $V$ obeys (1.3) (a mild restriction), this and Theorem 1 implies that $\sigma_{\text{\rom{ac}}}(H)=[0,\infty)$, $\sigma_{\text{\rom{sing}}}\cap (0,\infty)=\emptyset$. \vskip 0.3in \Refs \widestnumber\key{19} \ref\key 1 \by N.~Aronszajn \paper On a problem of Weyl in the theory of Sturm-Liouville equations \jour Am.~J.~Math. \vol 79 \yr 1957 \pages 597--610 \endref \ref\key 2 \by H. Behncke \paper Absolute continuity of Hamiltonians with von Neumann Wigner potentials, II \jour Manu-scripta Math. \vol 71 \yr 1991 \pages 163--181 \endref \ref\key 3 \by P.~Briet and E.~Mourre \paper Some resolvent estimates for Sturm-Liouville operators \paperinfo preprint \endref \ref\key 4 \by R.~Carmona \paper One-dimensional Schr\"odinger operators with random or deterministic potentials: New spectral types \jour J.~Funct.~Anal. \vol 51 \yr 1983 \pages 229--258 \endref \ref\key 5 \by E.~Coddington and N.~Levinson \book Theory of Ordinary Differential Equations \publ McGraw-Hill \publaddr New York \yr 1955 \endref \ref\key 6 \by J.~Dombrowksi and P.~Nevai \paper Orthogonal polynomials, measures, and recurrence relations \jour Siam J.~Math. \vol 17 \yr 1986 \pages 752--759 \endref \ref\key 7 \by W.~Donoghue \paper On the perturbation of the spectra \jour Commun.~Pure Appl.~Math. \vol 18 \yr 1965 \pages 559--579 \endref \ref\key 8 \by D.J.~Gilbert \paper On subordinacy and analysis of the spectrum of Schr\"odinger operators with two singular endpoints \jour Proc.~Roy.~Soc.~Edinburgh Sect.~A \vol 112 \yr 1989 \pages 213--229 \endref \ref\key 9 \by D.J.~Gilbert and D.B.~Pearson \paper On subordinacy and analysis of the spectrum of one-dimensional Schr\"odinger operators \jour J.~Math.~Anal.~Appl. \vol 128 \yr 1987 \pages 30--56 \endref \ref\key 10 \by D.B.~Hinton and J.K.~Shaw \paper Absolutely continuous spectra of second order differential operators with short and long range potentials \jour Siam J.~Math.~Anal. \vol 17 \yr 1986 \pages 182--196 \endref \ref\key 11 \by S.~Kahn and D.B.~Pearson \paper Subordinacy and spectral theory for infinite matrices \jour Helv.~Phys.~Acta \vol 65 \yr 1992 \pages 505--527 \endref \ref\key 12 \by A.~Kiselev \paperinfo in preparation \endref \ref\key 13 \by D.B.~Pearson \book Quantum Scattering and Spectral Theory. Techniques of Physics \vol 9 \publ Academic Press \publaddr London \yr 1988 \endref \ref\key 14 \by M.~Reed and B.~Simon \book Methods of Modern Mathematical Physics, III.~Scattering Theory \publ Academic Press \publaddr New York \yr 1979 \endref \ref\key 15 \by B.~Simon \paper Spectral analysis of rank one perturbations and applications \inbook Proc.~Mathematical Quantum Theory II: Schr\"odinger Operators \eds J.~Feldman, R.~Froese, and L.M.~Rosen \publ Amer.~Math.~Soc. \publaddr Providence, RI \toappear \endref \ref\key 16 \by G.~Stolz \paper Bounded solutions and absolute continuity of Sturm-Liouville operators \jour J.~Math. Anal. Appl. \vol 169 \yr 1992 \pages 210--228 \endref \ref\key 17 \by J.~Weidmann \paper Zur Spektraltheorie von Sturm-Liouville-Operatoren \jour Math.~Z. \vol 98 \yr 1967 \pages 268--302 \endref \ref\key 18 \bysame \paper Absolut stetiges Spektrum bei Sturm-Liouville-Operatoren und Dirac-Systemen \jour Math.~Z. \vol 180 \yr 1982 \pages 423--427 \endref \ref\key 19 \bysame \book Spectral Theory of Ordinary Differential Operators \bookinfo Lecture Notes in Mathematics Vol.~1258 \publ Springer-Verlag \publaddr Berlin/Heidelberg \yr 1987 \endref \endRefs \enddocument