Content-Type: multipart/mixed; boundary="-------------0002071708419" This is a multi-part message in MIME format. ---------------0002071708419 Content-Type: text/plain; name="00-64.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-64.keywords" Random Operators, Scattering Theory, Localization, Displacement Model ---------------0002071708419 Content-Type: application/x-tex; name="newvers.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="newvers.tex" \documentclass{amsart} \date{\today} \def\M{\|} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\p}{\mathbb{P}} \newcommand{\Cal}{\mathcal} \def\pd#1#2{\dfrac{\partial#1}{\partial#2}} \def\deriv#1#2{\dfrac{d#1}{d#2}} \newtheorem{Theorem}{Theorem}[section] \newtheorem{Proposition}[Theorem]{Proposition} \newtheorem{Corollary}[Theorem]{Corollary} \newtheorem{Lemma}[Theorem]{Lemma} \newtheorem{Definition}[Theorem]{Definition} \newtheorem{Remark}[Theorem]{Remark} \newtheorem{Example}[Theorem]{Example} \newcommand{\dist}{\mbox{\rm dist}} \def\wlim{\mathop{\rm w-lim}} \renewcommand{\epsilon}{\varepsilon} \begin{document} %\baselineskip=20pt \setcounter{section}{0} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \newcounter{letters} \title[Localization: A Scattering Theoretic Approach] {Localization in one dimensional random media: \\ A Scattering Theoretic Approach} \author[Robert Sims and G\"unter Stolz]{ Robert Sims and G\"unter Stolz\\ Department of Mathematics \\ University of Alabama at Birmingham \\ Birmingham, AL 35294-1170.} \thanks{Both authors partially supported by NSF grant DMS-9706076} \date{February 2000} \begin{abstract} We use scattering theoretic methods to prove exponential localization for random displacement models in one dimension. The operators we consider model both quantum and classical wave propagation. Our main tools are the reflection and transmission coefficients for compactly supported single site perturbations. We show that randomly displaced, non-reflectionless single sites lead to localization. \end{abstract} \maketitle \section{Introduction} Electron localization in disordered media is generally understood as a multiscattering phenomenon in physics: If scattering at each single site is non-trivial and a global potential is constructed by dispersing these single sites in a sufficiently random manner (to avoid resonance effects as in the periodic case), then states should be localized. Our main goal here is to provide a proof of localization for certain one dimensional random operators which makes this idea rigorous. Specifically, we will work with random displacement models, which for the case of Schr\"{o}dinger operators are given as follows: Let \begin{equation} \label{pot} v_{\omega}(x) = \sum_{n \in Z}g(x-n-d_n( \omega)), \end{equation} where $g \in L^2$ is compactly supported and $\text{supp}(g) \subset [-s,s]$, $s>0$. The displacements $\{ d_n( \omega) \}$ are independent, identically distributed random variables which take values in $[-d_{max},d_{max}]$. We assume that their distribution has a non-trivial absolutely continuous component, and that $d_{max} +s<1/2$ to avoid overlap between adjacent sites in $( \ref{pot})$. The random operator \begin{equation} \label{rSo} H_{\omega}^S = - \frac{d^2}{dx^2} + v_{\omega} \end{equation} ("S" for Schr\"{o}dinger) is essentially self-adjoint on $C_0^{\infty}( \R)$ for every $\omega$ and the following result was recently proven by Buschmann and Stolz: \begin{Theorem}[\cite{B.S.}] \label{bst} If $g \neq 0$, then $H_{\omega}^S$ almost surely has dense pure point spectrum with exponentially decaying eigenfunctions. \end{Theorem} The main technical difficulty in proving Theorem $\ref{bst}$ is that $H_{\omega}^S$ does not depend monotonically (in form sense) on the random parameters $d_n$. This problem was overcome in \cite{B.S.} by combining results from inverse spectral theory with the method of {\it two parameter spectral averaging}. Here, while also using two parameter averaging, we provide an alternate proof of Theorem $\ref{bst}$ wherein the inverse spectral results used in \cite{B.S.} are replaced by scattering theoretic ideas which are more directly motivated by physics. It is possible to apply our strategy of proof to other models. As an example, we prove the following new result: Let \begin{equation} \label{den} \frac{1}{a_{\omega}(x)} = 1 + \sum_{n \in Z}f(x-n-d_n(\omega)), \end{equation} where $\text{supp}(f) \subset [-s,s], f \in L^1, 1+f>0$, and $\{ d_n( \omega) \}$ are i.i.d. random variables as above, in particular having an a.c. component and $d_{max} + s<1/2$. Sturm-Liouville theory shows that the operator \begin{equation} \label{rWo} H_{\omega}^W = - \frac{d}{dx} a_{\omega} \frac{d}{dx} \end{equation} ("W" for wave equation) with $$ D(H_{\omega}^W) = \left\{ u \in L^2( \R): u, a_{\omega}u^{\prime} \ \ \text{are absolutely continuous with } -(a_{\omega}u^{\prime})^{\prime} \in L^2(\R) \right\} $$ is self-adjoint. Alternatively, $H_{\omega}^W$ may be defined as the self-adjoint operator corresponding to the non-negative form $\int a_{\omega}u^{\prime} \overline{v^{\prime}}dx$, $u,v \in C_0^{\infty}( \R)$. We obtain a result similar to Theorem $\ref{bst}$: \begin{Theorem} \label{conj} If $f \neq 0$, then $H_{\omega}^W$ almost surely has dense pure point spectrum with exponentially decaying eigenfunctions. \end{Theorem} We prove Theorems $\ref{bst}$ and $\ref{conj}$ by showing: \begin{equation} \label{mtSc} \begin{array}[t]{l} \mbox{If the single site equation } -u^{\prime \prime} +gu = k^2u \mbox{ is not reflectionless, then} \\ H_{\omega}^S \mbox{ almost surely has dense pure point spectrum with exponentially} \\ \mbox{decaying eigenfunctions.} \end{array} \end{equation} and \addtocounter{equation}{-1} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}'} \begin{equation} \label{mtWc} \begin{array}[t]{l} \mbox{If the single site equation } - \left( \frac{u^{\prime}}{1+f} \right)^{\prime} = k^2u \mbox{ is not reflectionless, then} \\ H_{\omega}^W \mbox{ almost surely has dense pure point spectrum with exponentially} \\ \mbox{decaying eigenfunctions.} \end{array} \end{equation} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} Here we call \begin{equation} \label{Ssseqn} -u^{\prime \prime} +gu = k^2u, \end{equation} respectively \begin{equation} \label{Wsseqn} - \left( \frac{u^{\prime}}{1+f} \right)^{\prime} = k^2u, \end{equation} not reflectionless if its reflection coefficient does not vanish simultaneously for all $k>0$, for details see Section 2. By showing ($\ref{mtSc}$) and ($\ref{mtWc}$), we give a rigorous justification for the physical heuristics described above, where non-trivial scattering at a single site is interpreted as non-vanishing of the reflection coefficient. Theorem $\ref{bst}$ follows immediately from ($\ref{mtSc}$) and the well known fact that $( \ref{Ssseqn})$ is never reflectionless for {\it compactly supported}, non-trivial $g$, see Section 3, in particular Corollary $\ref{Borg}$. Theorem $\ref{conj}$, however, requires more work. We employ the same method to prove Theorem $\ref{conj}$, but in this case we show that there are no non-trivial, compactly supported $f$'s for which ($\ref{Wsseqn}$) has vanishing reflection coefficient, which to the best of our knowledge is a new result: \begin{Theorem} \label{refps} If $f \in L^1$ is compactly supported with $1+f>0$, then $- \left( \frac{u^{\prime}}{1+f} \right)^{\prime} = k^2u$ is reflectionless if and only if $f=0$. \end{Theorem} \noindent The key observation which enables us to prove Theorem $\ref{refps}$ is an asymptotic formula for the Weyl-Titchmarsh m-function of ($\ref{Wsseqn}$), see Proposition $\ref{mprop}$ in Section 4. Initiated by Everitt's work \cite{Everitt}, the m-function asymptotics of differential expressions has been intensely studied. While ($\ref{Wsseqn}$) can be transformed to Schr\"{o}dinger-type, i.e. ($\ref{Ssseqn}$), if $f$ is smooth, we need a new method of proof here to cover non-smooth $f$. Asymptotic formulas as provided in Proposition $\ref{mprop}$ and their applications to inverse spectral theory will be further studied in \cite{GSS}. Combining the asymptotic formula with Herglotz function formalism, we are able to determine that a reflectionless equation must have a smooth coefficient $f$, see Section 4. As mentioned above, for smooth coefficients one can reduce an equation of type ($\ref{Wsseqn}$) to one of form ($\ref{Ssseqn}$) via a Liouville-Green transformation, see Theorem $\ref{pdq}$. An application of Corollary $\ref{Borg}$ proves Theorem $\ref{refps}$; hence Theorem $\ref{conj}$ as in the Schr\"{o}dinger case. Our interest in the model $H_{\omega}^W$ is motivated by a number of recent works on localization phenomena involving classical waves, e.g. \cite{FK,FK2,Combes/Hislop/Tip,Stollmann}, or the recent review \cite{Kuchment}. $H_{\omega}^W$ is a one-dimensional version of the random operators which arise in the study of acoustic or electro-magnetic waves in disordered media. The coefficient $f$ in $( \ref{Wsseqn})$ will be a step function in many applications, modeling for example the varying density and speed of sound in a two-component acoustic medium. Examples of this nature illustrate why it is not sufficient to only prove results for "smooth" cases. We point out that "mixed" equations $- \left( \frac{u^{\prime}}{1+f} \right)^{\prime} + gu=k^2u$ where $f$ and $g$ are both compactly supported and non-trivial may very well be reflectionless, see the example in Section 3. An interesting feature of the displacement models $H_{\omega}^S$ and $H_{\omega}^W$ is that even in the case of a non-reflectionless single site, they may exhibit a discrete set of energies where the (analytic) reflection coefficient vanishes, without vanishing identically. In the case of simple square well potentials, calculations leading to this kind of resonance energies are found in most elementary quantum mechanics texts. At such energies, the unreflected waves yield "extended states". This fact does not hinder the proof of localization though, since a discrete exceptional set can not support continuous spectrum. Discrete exceptional energies with zero reflection also appear in the so-called random dimer model, where localization, in fact dynamical localization away from exceptional energies, has recently been shown by de Bi\`{e}vre and Germinet \cite{Bi/Ger2}. We now outline the basic ideas in our proof of ($\ref{mtSc}$) and ($\ref{mtWc}$) which, after some preparations in Section 5, is completed in Section 6. Compact support of the single site allows the use of Kotani's theory \cite{Ko1,Ko2}, respectively its extensions by Minami \cite{Mi1,Mi2}, to conclude positivity of the Lyapunov exponent for almost every energy. Given this, the other main ingredient into the proof of localization is spectral averaging, e.g. \cite{Simon/Wolff,Kotani/Simon,Stolz,B.S.}. To understand the relation between non-trivial scattering and spectral averaging, we for simplicity concentrate on the Schr\"{o}dinger case $( \ref{rSo})$ and start with the "one bump" problem: For any $a \in [-d_{max},d_{max}]$ define $g_a(x)=g(x-a)$, the displaced bump with support in $[-s+a,s+a] \subset [- \frac{1}{2}, \frac{1}{2}]$. For $k>0$, we consider the solution $u$ of $- u^{\prime \prime} + g_au=k^2u$ with $$ u(x) = \left\{ \begin{array}{cc} \sin(kx + \vartheta), & \mbox{for } x \leq - \frac{1}{2} \\ R \sin(kx+ \vartheta + \delta), & \mbox{for } x \geq \frac{1}{2}. \end{array} \right. $$ Scattering at a single site is described by the amplitude $$ R = R( \vartheta, k , a) = \left( \left( \frac{u^{\prime}(1/2)}{k} \right)^2+u(1/2)^2 \right)^{1/2} $$ and the phase shift $\delta = \delta( \vartheta, k , a)$. Two basic facts (Lemma $\ref{local}$ and Lemma $\ref{dmpp}$ below) relate trivial reflection at the single site $g$ with stationarity of the phase shift under translation: \newline \indent (1) Fix $a$ and $k$, then the reflection coefficient at $k$ is zero if and only if $R=1$ for all $\vartheta$. \newline \indent (2) Fix $k$ and $\vartheta$, then $\partial_a \delta = k \left( \frac{1}{R^2} - 1 \right)$. \newline Thus, at an energy $E=k^2$ with non-zero reflection coefficient, we have $R \neq 1$ and therefore $\partial_a \delta \neq 0$ for at least one $\vartheta$. Analyticity implies $\partial_a \delta \neq 0$ for all but a discrete set $N$ of exceptional $\vartheta$'s (Lemma $\ref{combo}$). To handle the $\vartheta \in N$ for which the phase shift over one bump {\it is stationary}, we introduce a second, adjacent bump, $g_{1+b}(x) = g(x-1-b)$ parametrized by $b \in [-d_{max},d_{max}]$, for which $\partial_b \delta \neq 0$ for all but a discrete set of $b$'s, where now $\delta$ is the total phase shift for both bumps (Lemma $\ref{nzp}$). Once non-stationarity of the phase shift in the random variables $a,b$ has been established, one can use the Carmona formalism \cite{Carmona,Carmona/Lacroix} to show that averaging of spectral measures over $a,b$ yields an absolutely continuous measure. We outline this is Section 6, referring for details to \cite{B.S.}. Spectral averaging for the negative energy spectrum, which only exists in the Schr\"{o}dinger case, uses analytic continuation of reflection and transmission coefficients to the imaginary axis. Results on localization for non-monotonic random operators of type $( \ref{rSo})$ or $( \ref{rWo})$ are not restricted to the displacement model studied here. Anderson models with indefinite single site potentials as well as Poisson models were studied in \cite{Stolz2}, respectively \cite{B.S.}. A study of more general models of type $( \ref{rWo})$ will be contained in \cite{Sims}. Finally, we mention that reflection and transmission coefficients have been used successfully in the solution of other spectral problems involving multiscattering in one dimension: Kirsch, Kotani, and Simon \cite{Kirsch/Kotani/Simon} use non-vanishing of the reflection coefficient to prove absence of absolutely continuous spectrum for random Schr\"{o}dinger operators with non-compactly supported single site potentials. Molchanov \cite{Molchanov} used them in the spectral analysis of sparse potentials. A scattering theory approach to one-dimensional Anderson models is provided in \cite{Kostrykin/Schrader}, where the key ingredient is the Lifshitz-Krein shift function, and transmission and reflection coefficients appear via the S-matrix. \vspace{.3cm} {\bf Acknowledgment:} The authors of this paper are greatly indebted to Fritz Gesztesy for indicating that the asymptotic formula given in Proposition $\ref{mprop}$ should exist. In particular, he identified the relevance of such a formula to Theorem $\ref{conj}$ by linking the scattering data to m-function asymptotics through Herglotz representations. \setcounter{equation}{0} \section{Transmission and Reflection Coefficients} The most basic objects in one-dimensional scattering theory are the transmission and reflection coefficients. We start by defining them in the context of a general, "mixed" single site equation \begin{equation} \label{cseqn} -(pu^{\prime})^{\prime} + qu = k^2u \end{equation} where $p: \R \rightarrow (0, \infty)$ and $q: \R \rightarrow \R$. For notational simplicity, we take supp$(p-1)$ and supp$(q) \subset [0,D]$. (Results obtained for ($\ref{cseqn}$) apply to the single site equations ($ \ref{Ssseqn}$) and ($\ref{Wsseqn}$) by translation.) Assuming $\frac{1}{p} \in L_{loc}^1( \R)$ and $q \in L^1( \R)$, the differential expression \begin{equation} \label{csexp} \tau = - \frac{d}{dx}p \frac{d}{dx} + q \nonumber \end{equation} is a compactly supported perturbation of the Laplacian $- d^2 / dx^2$. $\tau$ defines a self-adjoint operator $L$ on $L^2( \R)$, see \cite{Codd/Lev, Weidmann}, with \begin{equation} D(L) = \bigg\{ f \in L^2( \R): \mbox{ } f \mbox{ } \text{and} \mbox{ } pf^{\prime} \mbox{ } \text{are absolutely continuous and} \mbox{ } \tau f \in L^2( \R) \bigg\}. \nonumber \end{equation} As $p$ is not necessarily smooth, e.g. step functions are allowed, we note that smoothness of $pf^{\prime}$ is not equivalent to smoothness of $f^{\prime}$. The {\it Jost solution} $u$ of ($\ref{cseqn}$) for $k \in \C \setminus \{ 0 \}$ is the solution satisfying \begin{equation} \label{jsol} u(x,k) = \left\{ \begin{array}{cc} e^{ikx} & \text{for} \ \ x \geq D, \\ a(k)e^{ikx} + b(k)e^{-ikx} & \text{for} \ \ x \leq 0. \end{array} \right. \end{equation} By the linear independence of $e^{ikx}$ and $e^{-ikx}$, the coefficients $a(k)$ and $b(k)$ are well defined. The Wronskian $W[u,v](x):= (pu^{\prime})(x)v(x)-u(x)(pv^{\prime})(x)$ for any two solutions $u$ and $v$ of $( \ref{cseqn})$ is constant. Applying this to $u(x,k)$ and $u(x,-k)$, we find \begin{equation} \label{abeqn} a(k)a(-k)-b(k)b(-k) = 1, \ \ \text{for any} \ \ k \in \C \setminus \{ 0 \}. \end{equation} For {\it real} $k$, one has $u(x,-k) = \overline{u(x,k)}$, and thus $a(-k) = \overline{a(k)}$, $b(-k) = \overline{b(k)}$, and $|a(k)|^2=1+|b(k)|^2$. Hence, in this case, $a(k) \neq 0$, and we may define $r(k):= \frac{b(k)}{a(k)}$ and $t(k):= \frac{1}{a(k)}$, the reflection and transmission coefficients. While the latter are closer to physics, describing the reflected and transmitted parts of an incoming plane wave from $- \infty$, we will work mainly with the mathematically more convenient $a(k)$ and $b(k)$. If $k$ is {\it purely imaginary}, then $u(x,k)$ is a real solution of the real equation $( \ref{cseqn})$, in particular $a(k)$ and $b(k)$ are real. Using $( \ref{jsol})$ with $k= i \alpha$, $ \alpha >0$, we also see that $L$ has the negative eigenvalue $- \alpha^2$ if and only if $a(i \alpha)=0$. One may represent $a(k)$ and $b(k)$ as solutions of integral equations. This is quite standard in the Schr\"{o}dinger case. We provide some details to include the less well known case $p \neq 1$. Variation of parameters (or direct verification) shows that $u(x,k)= a(k,x)e^{ikx} +b(k,x)e^{-ikx}$, where \begin{eqnarray} \label{a} a(k,x) = 1- \frac{1}{2ik} \int_x^De^{-iky} \left\{ q(y)u(y,k) -ik \left(p(y)- 1 \right) u^{\prime}(y,k) \right\} dy \\ \label{b} b(k,x) = \frac{1}{2ik} \int_x^De^{iky} \left\{q(y)u(y,k) +ik \left(p(y)-1 \right) u^{\prime}(y,k) \right\} dy. \end{eqnarray} Due to the supports of $q$ and $p-1$, the above are trivial for $x \geq D$ and constant for $x \leq 0$, specifically $a(k)=a(k,0)$ and $b(k)=b(k,0)$. By differentiation, via $( \ref{a})$ and $( \ref{b} )$, we arrive at $pu^{\prime}(x,k) = ik \left[ a(k,x)e^{ikx} - b(k,x)e^{-ikx} \right]$. We can now can rewrite $( \ref{a})$ and $( \ref{b} )$ as a closed system in $a$ and $b$: \begin{eqnarray} a(k,x) & = & 1 - \frac{1}{2ik} \int_x^D \bigg{\{} q(y) \left[ a(k,y) + e^{-2iky} b(k,y) \right] \nonumber \\ & & + \left. k^2 \left(1- \frac{1}{p(y)} \right) \left[ a(k,y) - e^{-2iky} b(k,y) \right] \right\} dy, \nonumber \end{eqnarray} and \begin{eqnarray} b(k,x) & = & - \frac{1}{2ik} \int_x^D \bigg{\{} q(y) \left[ -e^{2iky} a(k,y) - b(k,y) \right] \nonumber \\ & & + \left. k^2 \left(1- \frac{1}{p(y)} \right) \left[ e^{2iky} a(k,y) - b(k,y) \right] \right\} dy. \nonumber \end{eqnarray} More concisely, we have \begin{equation} \label{gamma} \gamma(k,x) = \left( \begin{array}{c} 1 \\ 0 \end{array} \right) - \frac{1}{2ik} \int_x^D \left[ q(y)A(k,y) +k^2 \left(1- \frac{1}{p(y)} \right) \tilde{A}(k,y) \right] \gamma(k,y) dy, \end{equation} where $$ \begin{array}{cc} A(k,y) = \left( \begin{array}{cc} 1 & e^{-2iky} \\ -e^{2iky} & -1 \end{array} \right), & \tilde{A}(k,y) = \left( \begin{array}{cc} 1 & -e^{-2iky} \\ e^{2iky} & -1 \end{array} \right), \end{array} $$ and $\gamma(k,x) = \left( \begin{array}{c} a(k,x) \\ b(k,x) \end{array} \right)$. For fixed $x$, iteration of the Volterra-type integral equation $( \ref{gamma})$ yields a series representation for $\gamma(x,k)$ which converges locally uniformly for $k \in \C \setminus \{ 0 \}$. This shows that $a(k)$ and $b(k)$ are analytic in $\C \setminus \{ 0 \}$. In the case of $q=0$, the singularity of $( \ref{gamma})$ at $k=0$ vanishes, thus both $a(k)$ and $b(k)$ are entire. Finally, we note that information about the transmission and reflection coefficients of $( \ref{cseqn})$ can be used to understand the periodic equation \begin{equation} \label{pext} -( \tilde{p}u^{\prime})^{\prime} + \tilde{q}u = Eu, \end{equation} where $\tilde{p}$ and $\tilde{q}$ are the $D$-periodic extensions of $p|_{[0,D]}$ and $q|_{[0,D]}$ to $\R$. The periodic self-adjoint operator defined by $( \ref{pext})$ on $L^2( \R)$ will be denoted by $L_{per}$. Its spectral properties are determined by the discriminant, $d(E)$, the trace of the transfer matrix of $( \ref{pext})$ from $0$ to $D$, see \cite{Eastham}. In particular, the spectral bands of $L_{per}$ are found from $|d(E)| \leq 2$. As the transfer matrix has determinant 1, its inverse, i.e. the transfer matrix from $D$ to $0$, has the same trace. This observation simplifies the following calculation: Consider solutions $u_1$ and $u_2$ of $( \ref{cseqn})$ with $u_1(D)=(pu^{\prime}_2)(D)=1$ and $u_2(D)=(pu^{\prime}_1)(D)=0$. We may write these in terms of the Jost solutions, \begin{equation} u_1(x,k) = \frac{1}{2}e^{-ikD}u(x,k) + \frac{1}{2}e^{ikD}u(x,-k) \nonumber \end{equation} and \begin{equation} u_2(x,k) = \frac{1}{2ik}e^{-ikD}u(x,k) - \frac{1}{2ik}e^{ikD}u(x,-k). \nonumber \end{equation} For any $k \in \C \setminus \{ 0 \}$, the transfer matrix of ($\ref{pext}$) at $E=k^2$ is given by $$ \left( \begin{array}{cc} u_1(0) & u_2(0) \\ (pu^{\prime}_1)(0) & (pu^{\prime}_2)(0) \end{array} \right). $$ Thus the discriminant at $k^2$ is \begin{equation} \label{diseqn} d(k^2)=u_1(0) + pu^{\prime}_2(0) = a(k)e^{-ikD}+a(-k)e^{ikD}, \end{equation} a relation which has been used frequently (e.g. \cite{Keller, Kirsch/Kotani/Simon}) in the case $p=1$. Note that the use of purely imaginary $k$ in $( \ref{diseqn})$ allows one to determine the negative spectral bands of $L_{per}$. \section{Equations with Zero Reflection} \setcounter{equation}{0} In this section, we prove a variety of results charaterizing equations for which the reflection coefficient is either identically zero or zero at some fixed energy. We say that $-(pu^{\prime})^{\prime}+qu=k^2u$ is reflectionless if $r(k)= r(k,p,q)$ vanishes for all real $k > 0$. This is equivalent to requiring that $b(k)=0$ for all $k >0$ and thus, by analyticity, for all $k \in \C \setminus \{ 0 \}$. We begin by noting that "reflectionless" implies "no gaps" for the spectrum of the corresponding periodic operator, a fact well known in the case of $p=1$. This observation leads to another well-known result for Schr\"{o}dinger equations: the only reflectionless, compactly supported $q$'s are trivial. We then demonstrate that equations with smooth $p$'s can be transformed into Schr\"{o}dinger type equations, in which case a similar "reflectionless implies trivial" result is true. Lastly, we relate zero reflection at a fixed energy to the amplitude of solutions. \begin{Lemma} \label{nogap} If $-(pu^{\prime})^{\prime}+qu=k^2u$ is reflectionless, then $\sigma(L_{per}) = [ 0 , \infty)$. \end{Lemma} \begin{proof} To prove this we show that, \begin{eqnarray} \label{dpos} |d(k^2)| \leq 2 & \mbox{for all } k>0, \mbox{ and} \\ \label{dneg} |d(- \alpha^2)| \geq 2 & \mbox{for all } \alpha >0. \end{eqnarray} $( \ref{dpos})$ implies $[0, \infty) \subset \sigma(L_{per})$ and $( \ref{dneg})$ implies $|d(- \alpha^2)| >2$ for all $\alpha>0$ (bands do not degenerate to points, see \cite{Eastham}), and thus $\sigma(L_{per}) \subset [0, \infty)$. Both $( \ref{dpos})$ and $( \ref{dneg})$ follow readily from $( \ref{diseqn})$: In the case that $( \ref{cseqn} )$ is reflectionless, equation $( \ref{abeqn})$ becomes \begin{equation} a(k)a(-k) = 1. \nonumber \end{equation} If $k >0$, then $a(-k)= \overline{a(k)}$, \begin{equation} d(k^2) = 2 \text{Re}[a(k)e^{-ikD}], \end{equation} and thus $|d(k^2)| \leq 2$. On the other hand, if $k = i \alpha$, $\alpha>0$, we have that $a(i \alpha)$ is real, \begin{equation} d(- \alpha^2) = a(i \alpha)e^{ \alpha D} + \frac{1}{a(i \alpha)e^{ \alpha D}}, \nonumber \end{equation} and so clearly $|d(- \alpha^2)| \geq 2$. \end{proof} Let us now consider the two special cases of $( \ref{cseqn})$, where either (i) $p=1$ or (ii) $q=0$. In case (i), one has \begin{Corollary} \label{Borg} If $-u^{\prime \prime} +qu = k^2u$ is reflectionless and $q$ is compactly supported, then $q=0$. \end{Corollary} This is well known: there are no compactly supported {\it solitons}. It follows for example from Lemma $\ref{nogap}$ and the result of Borg \cite{Borg}, extended by Hochstadt \cite{Hochstadt} to the case $q \in L^1_{loc}$, that if the periodic operator $- \frac{d^2}{dx^2} + \tilde{q}$ has spectrum $[0, \infty)$, then $\tilde{q} =0$. In case (ii), if $p$ is sufficiently smooth, one can use a Liouville-Green transform to reduce $-(pu^{\prime})^{\prime}=k^2u$ to a Schr\"{o}dinger equation and prove that reflectionless implies $p=1$. This will follow from a more general observation: \begin{Theorem} \label{pdq} Let $p>0$ and $q$ be real-valued functions such that supp$(p-1)$ and supp$(q) \subset [0,D]$ with $\frac{1}{p} \in L^1_{loc}(\R)$ and $q \in L^1(\R)$. Suppose in addition that both $p$ and $p^{\prime}$ are absolutely continuous. Then $-(pu^{\prime})^{\prime} +qu=k^2u$ is reflectionless if and only if $$ q = \frac{1}{16} \frac{ ( p^{\prime})^2}{p} - \frac{1}{4}p^{\prime \prime}. $$ \end{Theorem} \begin{proof} Let $k > 0$ and $u$ be any solution of $( \ref{cseqn})$. This equation has a Liouville-Green transform, see \cite{Eastham2}, which is defined by setting $t(x) = \int_0^x \left[ \frac{1}{p(s)} \right]^{1/2}ds$, $x(t)$ the inverse of $t(x)$, and \begin{equation} \label{lgsol} z(t): = p(x(t))^{1/4}u(x(t)). \end{equation} A short calculation shows that $-z^{\prime \prime}+Qz=k^2z,$ where $Q(t)=Q_1(t) + Q_2(t)$ with \begin{eqnarray} Q_1(t)=p(x(t))^{-1/4} \frac{d^2}{dt^2}p(x(t))^{1/4} & \text{and} & Q_2(t) = q(x(t)). \nonumber \end{eqnarray} Under the above transformation we have $t=x$ for $x \leq 0$ and $t=x + P -D$ for $x \geq D$, where $P:= \int_0^D \left( \frac{1}{p(s)} \right)^{1/2}ds$. Choosing the particular solution $u(x)=u(x,k)$ from ($\ref{jsol}$), it follows that \begin{eqnarray*} z(t) & = & \left\{ \begin{array}{cccc} u(x(t),k) & = & a(k) e^{ikt} + b(k) e^{-ikt} & \text{for} \ \ t \leq 0 \\ u(x(t),k) & = & e^{ik(t+D-P)} & \text{for} \ \ t \geq P. \end{array} \right. \end{eqnarray*} Clearly then, $e^{-ik(D-P)}z(t)$ is the Jost solution of $-z^{\prime \prime}+Qz=k^2z,$ and the corresponding reflection coefficient $r^*(k)$ is given by $$ r^*(k) = \frac{ e^{-ik(D-P)}b(k)}{e^{-ik(D-P)}a(k)} = \frac{b(k)}{a(k)} = r(k). $$ We have shown that $( \ref{cseqn})$ is reflectionless if and only if $-z^{\prime \prime}+Qz=k^2z$ is reflectionless as well. Since $Q$ is compactly supported, we conclude from Corollary $\ref{Borg}$ that $Q=0$, i.e. almost everywhere \begin{equation} \label{qeq} q(x) = -p(x)^{-1/4} \frac{d^2}{dt^2}p(x)^{1/4} = \frac{1}{16} \frac{p^{\prime}(x)^2}{p(x)} - \frac{1}{4}p^{\prime \prime}(x). \end{equation} \end{proof} The following Corollary, again with the choice $p(t) = 1/(1+f(t-s))$, is a reformulation of Theorem $\ref{refps}$ in the smooth case. \begin{Corollary} \label{psmooth} If $-(pu^{\prime})^{\prime}=k^2u$ is reflectionless, $p-1$ is compactly supported, and $p$ and $p^{\prime}$ are absolutely continuous, then $p=1$. \end{Corollary} \begin{proof} The right hand side of equation $( \ref{qeq})$ may be rewritten \begin{equation} \frac{1}{16} \frac{p^{\prime}(x)^2}{p(x)} - \frac{1}{4}p^{\prime \prime}(x) = - \frac{1}{4} p(x)^{1/4} \left( p^{-1/4}(x)p^{\prime}(x) \right)^{\prime}. \nonumber \end{equation} If $q=0$, then integration, using the compact support of $p-1$, yields $p=1$. \end{proof} \noindent {\bf Example:} We have shown that there are no compactly supported, non-trivial $q$'s for which $-u^{\prime \prime} +qu=k^2u$ is reflectionless. Accordingly, in Section 4 we will improve on Corollary $\ref{psmooth}$, i.e. remove smoothness assumptions, and show that there are no non-trivial $p$'s for which $p-1$ is compactly supported and $-(pu^{\prime})^{\prime}=k^2u$ is reflectionless. There are, however, compactly supported, non-trivial pairs $(p, q)$ for which ($\ref{cseqn}$) is reflectionless. Theorem $\ref{pdq}$, in fact, illustrates that a given smooth $p$ determines a $q$ for which $( \ref{cseqn})$ is reflectionless. For example, if $p(x):=2- \cos(x)$ on $[0, 2 \pi]$, then $q(x) = \frac{1}{16} \cdot \frac{ \sin^2(x)}{2- \cos(x)} - \frac{1}{4} \cos^2(x)$ on $[0, 2 \pi]$ defines a non-trivial, reflectionless equation $-(pu^{\prime})^{\prime} +qu=k^2u$. The corresponding $2 \pi$-periodic operator has spectrum $[0, \infty)$, which either follows from Lemma $\ref{nogap}$ or, more directly, from the fact that it is equivalent to the negative Laplacian under the unitary Liouville-Green transform $( \ref{lgsol})$. \vspace{.3cm} Finally, we prove a local result on zero reflection, i.e. a characterization of $b(k)=0$ for fixed $k$. For any solution $v$ of \begin{equation} \label{veqn} -(pv^{\prime})^{\prime} +qv=Ev, \end{equation} where $E>0$, we consider the square of the modified Pr\"{u}fer amplitude, \begin{equation} \label{sqpa} F_v(x,E) := \frac{1}{E}(pv^{\prime})^2(x) +v^2(x), \end{equation} for more on Pr\"{u}fer variables see Section 5. One easily calculates that \begin{equation} \partial_xF_v = 2vpv^{\prime} \left[ \frac{q}{E}- \left(1- \frac{1}{p} \right) \right], \nonumber \end{equation} which implies that $F$ remains constant (for all solutions of $( \ref{veqn} )$) wherever $q=0$ and $p=1$. The Lemma below shows that zero reflection at a fixed $k = \sqrt{E}$ is equivalent to the magnitude of the Pr\"{u}fer amplitude being unchanged over $[0,D]$. \begin{Lemma} \label{local} $b(k)=0$ for $k = \sqrt{E} >0$ holds if and only if $F_v(0,E)=F_v(D,E)$ for all real solutions $v$ of $( \ref{veqn} )$. \end{Lemma} \begin{proof} A given real valued solution of $( \ref{veqn} )$, normalized such that $F_v(0,E)=1$, may be parametrized with $\theta \in [0, \pi)$ by taking \begin{eqnarray} v(0)= \sin( \theta) & \text{and} & pv^{\prime}(0) = k \cos( \theta). \nonumber \end{eqnarray} Using the Jost solutions we may write $v = A u( \cdot,k) + B u( \cdot, -k)$. Applying the boundary conditions at zero, we find that \begin{eqnarray} \label{Beqn} B = \overline{A} = \frac{i}{2} \left[ a(k)e^{-i \theta} +b(k)e^{i \theta} \right]. \end{eqnarray} A straightforward calculation shows that $F_v(0,E)=F_v(D,E)$ for all real solutions of $( \ref{veqn})$ if and only if $1 = 4|B|^2$, for all $B= B( \theta)$ from ($\ref{Beqn}$). Now suppose $b(k)=0$, then $|a(k)|^2=1$ and $$ 4|B|^2 =4 \cdot \frac{i}{2}a(k)e^{-i \theta} \cdot \frac{-i}{2} \overline{a(k)}e^{i \theta} =1, \ \ \text{for all} \ \ B( \theta). $$ If, on the other hand, $1=4|B|^2$ for all $B( \theta)$, then for $\theta = 0$, $( \ref{Beqn} )$ shows that $B= \frac{i}{2} \left[ a(k) +b(k) \right]$. Direct substitution shows, $$ 1 = |a(k)|^2 + \overline{a(k)}b(k) + \overline{b(k)}a(k) + |b(k)|^2. $$ A similar argument for $\theta = \frac{\pi}{2}$ gives $$ 1 = |a(k)|^2 -\overline{a(k)}b(k) - \overline{b(k)}a(k) + |b(k)|^2. $$ Addition yields $|b(k)|^2=0,$ where $|a(k)|^2=1+|b(k)|^2$ was used. \end{proof} \section{Proof of Theorem $\ref{refps}$} \setcounter{equation}{0} Let $p>0,$ $\frac{1}{p} \in L^1_{loc}(\R)$, and supp$(p-1) \subset [0,D]$. Then Theorem $\ref{refps}$ is equivalent to \begin{Proposition} \label{trivps} $-(pu^{\prime})^{\prime}=k^2u$ is reflectionless if and only if $p=1$. \end{Proposition} Our proof of Proposition $\ref{trivps}$ uses Weyl-Titchmarsh m-functions (see e.g. \cite{Codd/Lev}). For $z \in \C \setminus [0, \infty)$ choose $z^{1/2}$ with Im$[z^{1/2}]>0$. Let $f_{\pm}(x,z)$ be the solutions of $-(pu^{\prime})^{\prime}=zu$ which are equal to $e^{\pm iz^{1/2}x}$ near $\pm \infty$. Then for each $x \in \R$ the m-functions of $-(pu^{\prime})^{\prime}=zu$ on $(x, \infty)$ and $(- \infty,x)$ can be defined as \begin{equation} \label{mfuns} m_{\pm}(z,x) = \frac{(pf_{\pm}^{\prime})(x,z)}{f_{\pm}(x,z)}. \end{equation} For fixed $x$, $m_{\pm}(z,x)$ is analytic in $\C \setminus [0, \infty)$, where the negative half-axis is included due to positivity of the operator $- \frac{d}{dx}p \frac{d}{dx}$. \begin{Proposition} \label{mprop} If $\frac{1}{p}$ is Lebesgue-continuous at $x$, then \begin{equation} \label{+masy} m_+(-\alpha^2,x)=-\alpha p(x)^{1/2} +o(\alpha) \ \ \mbox{ for real } \alpha \rightarrow + \infty, \end{equation} and \begin{equation} \label{-masy} m_-(-\alpha^2,x)= \alpha p(x)^{1/2} +o(\alpha) \ \ \mbox{ for real } \alpha \rightarrow + \infty. \end{equation} \end{Proposition} \noindent {\bf Remarks:} (i) Recall that $\frac{1}{p}$ is Lebesgue-continuous at $x$ if \begin{equation} \nonumber \lim_{\delta \downarrow 0} \frac{1}{\delta} \int_{x- \delta}^{x+ \delta} \left| \frac{1}{p(t)} - \frac{1}{p(x)} \right|dt = 0 \end{equation} and that $L^1_{loc}$-functions are a.e. Lebesgue-continuous. (ii) For smooth $p$ (i.e. $p$ and $p^{\prime}$ absolutely continuous) and $\alpha \rightarrow \infty$ in the cone $- \frac{\pi}{2} + \epsilon <$ arg( $\alpha$) $< - \epsilon$ (i.e. $- \alpha^2$ in the upper half-plane), ($\ref{+masy}$) and ($\ref{-masy}$) have been established in \cite{Everitt} by using a Liouville-Green transformation to reduce the problem to a result on m-function asymptotics for Schr\"{o}dinger operators. In fact, due to smoothness, \cite{Everitt} gets $O(1)$ instead of $o(\alpha)$. \vspace{.3cm} \noindent {\it Proof of Proposition $\ref{+masy}$:} We only prove ($\ref{+masy}$), the proof of ($\ref{-masy}$) being similar. To further simplify notation, we only treat the point $x=0$, assuming that $\frac{1}{p}$ is Lebesgue-continuous at $0$, i.e. in particular, \begin{equation} \label{pat0} \varphi(\delta):= \frac{1}{\delta} \int_0^{\delta}\left| \frac{1}{p(t)}- \frac{1}{p(0)} \right|dt \rightarrow 0 \ \ \mbox{as} \ \ \delta \downarrow 0. \end{equation} By ($\ref{mfuns}$) we need to show that \begin{equation} \label{f+asy} \lim_{\alpha \rightarrow \infty} \frac{1}{\alpha} \frac{(pf_+^{\prime})(0, - \alpha^2)}{f_+(0, - \alpha^2)}= - p(0)^{1/2}. \end{equation} Choose a function $\mu:[0, \infty) \rightarrow [0, \infty)$ satisfying: \begin{equation} \label{muprop1} \mu(\alpha) \to \infty \quad \mbox{as $\alpha\to\infty$}, \end{equation} and \begin{equation} \label{muprop2} \frac{\mu(\alpha)}{\alpha} \to 0 \quad \mbox{as $\alpha\to\infty$}. \end{equation} Additional assumptions on $\mu$ will be posed later, see Lemma $\ref{choosemu}$ below. Let $p_{\alpha}(y) := p(\frac{y}{\alpha})$ and \[ \tilde{p}_{\alpha}(y) := \left\{ \begin{array}{ll} p_{\alpha}(y), & \mbox{if $y\ge \mu(\alpha)$}, \\ p(0), & \mbox{if $0\le y<\mu(\alpha)$}. \end{array} \right.\] \noindent Also, let $w_{\alpha}$ be the solution of $-(p_{\alpha}w_{\alpha}')'=-w_{\alpha}$ with $w_{\alpha}(\alpha)=1$, $(p_{\alpha}w_{\alpha}')(\alpha)=-1$ and $\tilde{w}_{\alpha}$ the solution of the same initial value problem with $p_{\alpha}$ replaced by $\tilde{p}_{\alpha}$. In particular, one has $w_{\alpha}(y) = f_+( \frac{y}{ \alpha}, - \alpha^2)$ and therefore the proof of ($\ref{f+asy}$) is equivalent to showing that \begin{equation} \label{wasy} \lim_{\alpha\to\infty} \frac{(p_{\alpha}w_{\alpha}')(0)}{w_{\alpha}(0)} = -p(0)^{1/2}. \end{equation} We introduce the Pr\"ufer phases \[ \theta_{\alpha}(y):= \:\mbox{arccot}\frac{(p_{\alpha}w_{\alpha}')(y)}{w_{\alpha}(y)}\] and \[ \tilde{\theta}_{\alpha}(y):= \:\mbox{arccot}\frac{(\tilde{p}_{\alpha} \tilde{w}_{\alpha}')(y)}{\tilde{w}_{\alpha}(y)},\] where the branch of the arccot is determined by requiring $\theta_{\alpha}(\alpha)=\tilde{\theta}_{\alpha}(\alpha)= 3\pi/4$ and continuity of $\theta_{\alpha}$ and $\tilde{\theta}_{\alpha}$. They satisfy the first order differential equations \begin{equation} \label{dta} \theta_{\alpha}' = \frac{1}{p_{\alpha}} \cos^2 \theta_{\alpha} - \sin^2 \theta_{\alpha} \end{equation} and \begin{equation} \label{dtta} \tilde{\theta}_{\alpha}' = \frac{1}{\tilde{p}_{\alpha}} \cos^2 \tilde{\theta}_{\alpha} - \sin^2 \tilde{\theta}_{\alpha}. \end{equation} \begin{Lemma} \label{QII} $\theta_{\alpha}(y) \in (\frac{\pi}{2}, \pi)$ for all $y \geq 0$. \end{Lemma} \begin{proof} The result is obvious for $y \geq \alpha$. Suppose that it does not hold for all $y<\alpha$ and let $y_0$ be maximal such that either $\theta_{\alpha}(y_0)=\pi/2$ or $\theta_{\alpha}(y_0)=\pi$. In the first case we will show that $\theta_{\alpha}$ is decreasing near $y_0$, which yields a contradiction to $\theta_{\alpha}(y) > \pi/2$ for $y_0< y \leq \alpha$. Note that for continuous $p$ this would immediately follow from ($\ref{dta}$). For general $ \frac{1}{p} \in L^1_{loc}$ we follow an argument in \cite[Proof of Theorem~13.2]{Weidmann}: Write ($\ref{dta}$) as $\theta_{\alpha}' = (\frac{1}{p_{\alpha}} +1) \cos^2 \theta_{\alpha}-1$ and let \[ h(y) := \left| \int_{y_0}^y \left(\frac{1}{p_{\alpha}(t)}+1 \right)dt \right|. \] Then \begin{equation} \label{inteqtheta} \theta_{\alpha}(y)-\theta_{\alpha}(y_0) = y_0-y + \int_{y_0}^y \left( \frac{1}{p_{\alpha}(t)}+1 \right) \cos^2 \theta_{\alpha}(t)\,dt \end{equation} and, since $\cos^2 \theta_{\alpha}(t) = \cos^2 \theta_{\alpha}(t) - \cos^2 \theta_{\alpha}(y_0) \le 2|\cos \theta_{\alpha}(t)- \cos \theta_{\alpha}(y_0)| \le$ \noindent $2|\theta_{\alpha}(t)-\theta_{\alpha}(y_0)|$, \[ |\theta_{\alpha}(y)-\theta_{\alpha}(y_0)| \le |y_0-y| + 2M(y)h(y), \] where $M(y) := \max\{|\theta_{\alpha}(t)-\theta_{\alpha}(y_0)|:\: t\in [y_0,y] \:\mbox{or $[y,y_0]$, resp.}\}$. By monotonicity \begin{equation} \label{maxest} M(y) \le |y_0-y| + 2M(y)h(y). \end{equation} Choose $\eta>0$ so small that $h(y) \le 1/6$ for $|y-y_0| \le \eta$. For those $y$ we get from (\ref{maxest}) that $M(y) \le \frac{3}{2}|y_0-y|$ and thus from (\ref{inteqtheta}) \[ \left|\theta_{\alpha}(y)-\theta_{\alpha}(y_0) - (y_0-y)\right| \le 2M(y)h(y) \le 3|y_0-y|h(y) \le \frac{1}{2}|y_0-y|. \] This implies $\theta_{\alpha}(y)-\theta_{\alpha}(y_0) \le -\frac{1}{2} (y-y_0) <0$ if $y>y_0$ and $\theta_{\alpha}(y)-\theta_{\alpha}(y_0) \ge \frac{1}{2}(y_0-y) >0$ if $y \frac{\tilde{w}_{\alpha}'(\mu(\alpha)-)}{\tilde{w}_{\alpha}(\mu (\alpha))} = p(0)^{-1/2} \frac{C_{\alpha} e^{2p(0)^{-1/2} \mu(\alpha)}-1}{C_{\alpha} e^{2p(0)^{-1/2}\mu(\alpha)}+1} \] where $C_{\alpha} := A_{\alpha}/B_{\alpha}$. This implies $|C_{\alpha}| < e^{-2p(0)^{-1/2}\mu(\alpha)} \to 0$ as $\alpha\to\infty$, using ($\ref{muprop1}$). One concludes \[ \frac{(\tilde{p}_{\alpha}\tilde{w}_{\alpha}')(0)}{\tilde{w}_{\alpha}(0)} = p(0)^{1/2} \frac{C_{\alpha}-1}{C_{\alpha}+1} \to -p(0)^{1/2} \quad \mbox{as $\alpha\to\infty$}, \] proving (a). \hfill $\qed$ \vspace{.3cm} {\it Proof of (b):} From ($\ref{dta}$) and ($\ref{dtta}$) it follows that \begin{equation} \nonumber \theta_{\alpha}'-\tilde{\theta}_{\alpha}' = (\frac{1}{p_{\alpha}} - \frac{1}{\tilde{p}_{\alpha}}) \cos^2 \theta_{\alpha} - (\frac{1}{\tilde{p}_{\alpha}}+1)(\sin \theta_{\alpha} - \sin \tilde{\theta}_{\alpha})(\sin \theta_{\alpha} + \sin \tilde{\theta}_{\alpha}). \end{equation} Using $\theta_{\alpha}(\mu(\alpha)) = \tilde{\theta}_{\alpha}(\mu(\alpha))$, $\tilde{p}_{\alpha} = p(0)$ in $[0,\mu(\alpha))$ and $|\sin \theta_{\alpha} - \sin \tilde{\theta}_{\alpha}| \le |\theta_{\alpha} - \tilde{\theta}_{\alpha}|$ in the above we see that for $0\le x < \mu(\alpha)$ \begin{eqnarray} \nonumber |\theta_{\alpha}(x)- \tilde{\theta}_{\alpha}(x)| & \le & \int_x^{\mu(\alpha)} \left| \frac{1}{p_{\alpha}(y)} - \frac{1}{p(0)} \right|\,dy \\ & & \mbox{} + 2(\frac{1}{p(0)}+1) \int_x^{\mu(\alpha)} |\theta_{\alpha}(y)-\tilde{\theta}_{\alpha}(y)|\,dy. \nonumber \end{eqnarray} This allows an application of Gronwall's lemma and hence \begin{equation} \label{estgronwall} |\theta_{\alpha}(x)-\tilde{\theta}_{\alpha}(x)| \le \int_x^{\mu(\alpha)} \left| \frac{1}{p_{\alpha}(y)} - \frac{1}{p(0)}\right|\,dy \: e^{2(\mu(\alpha)-x)(\frac{1}{p(0)} +1)}. \end{equation} Next, we note that by ($\ref{pat0}$) \begin{eqnarray} \label{estintegral} \int_0^{\mu(\alpha)} \left| \frac{1}{p_{\alpha}(y)} - \frac{1}{p(0)}\right| \,dy & = & \alpha \int_0^{\mu(\alpha)/\alpha} \left| \frac{1}{p(t)}-\frac{1}{p(0)} \right|\,dt \\ & = & \alpha \cdot \frac{\mu(\alpha)}{\alpha} \cdot \varphi(\frac{\mu(\alpha)}{\alpha}) \nonumber \\ & = & \mu(\alpha) \varphi(\frac{\mu(\alpha)}{\alpha}). \nonumber \end{eqnarray} Inserting (\ref{estintegral}) into (\ref{estgronwall}) at $x=0$ gives \begin{equation} \label{finalest} |\theta_{\alpha}(0)-\tilde{\theta}_{\alpha}(0)| \le \mu(\alpha) e^{2\mu(\alpha)(\frac{1}{p(0)}+1)} \varphi(\frac{\mu(\alpha)}{\alpha}). \end{equation} Now part (b) of Lemma $\ref{tresult}$ follows from ($\ref{finalest}$) under a suitable choice of $\mu$, which is possible by \begin{Lemma} \label{choosemu} Let $\varphi:(0,\infty) \to (0,\infty)$ be such that $\lim_{\delta \downarrow 0} \varphi(\delta)=0$ and let $C>0$. Then there exists a function $\mu:(0,\infty) \to (0,\infty)$ with the properties ($\ref{muprop1}$), ($\ref{muprop2}$) and \begin{equation} \label{muprop3} \mu(\alpha) e^{C\mu(\alpha)} \varphi(\frac{\mu(\alpha)}{\alpha}) \to 0 \quad \mbox{as $\alpha\to\infty$}. \end{equation} \end{Lemma} {\it Proof of Lemma $\ref{choosemu}$:} We establish the lemma in several elementary reduction steps: (i) We may assume that $\varphi$ is monotone increasing and strictly positive on $(0,\infty)$ (otherwise replace $\varphi$ with $\tilde{\varphi}(\delta) := \max \{ \sup_{00$ and $x \in \R$. \end{Lemma} \begin{proof} For Im$[z]>0$, we have $f_+(x,z)=u(x,z^{1/2})$ where $u$ is the Jost solution introduced in Definition 2.2. Thus \begin{equation} \nonumber m_+(z,0)= \frac{(pu^{\prime})(0,z^{1/2})}{u(0,z^{1/2})}= \frac{iz^{1/2}(a(z^{1/2})-b(z^{1/2}))}{a(z^{1/2})+b(z^{1/2})}. \end{equation} If $-(pu^{\prime})^{\prime}=k^2u$ is reflectionless, i.e. $b(k)=0$ for all $k \in \C\setminus \{ 0 \}$, then \begin{equation} \label{refm+} m_+(z,0) = iz^{1/2}. \end{equation} Obviously, we also have that \begin{equation} \label{refm-} m_-(z,0) = -iz^{1/2}. \end{equation} We want to prove that \begin{equation} \label{Craig} \mbox{Re}[G(\lambda+i0,x,x)]=0 \ \ \mbox{for all } \lambda>0 \ \ \mbox{and} \ \ x \in \R. \end{equation} To this end, we use that \begin{equation} \label{psig} G(\lambda+i\epsilon,x,x)=\frac{\psi_+(x,\lambda+i\epsilon)\psi_-(x, \lambda+i\epsilon)}{W[\psi_+(\lambda+i\varepsilon),\psi_-(\lambda+i\varepsilon)]}, \end{equation} where \begin{equation} \label{trigpsi} \psi_{\pm}(x,z) = C(x,z)+ m_{\pm}(z,0)S(x,z) = C(x,z) \pm iz^{1/2}S(x,z), \end{equation} and $C(x,z),S(x,z)$ are solutions of $-(pu^{\prime})^{\prime}=zu$ with $C(0,z)=(pS^{\prime})(0,z)=1$ and $(pC^{\prime})(0,z)=S(0,z)=0$. ($\ref{refm+}$), ($\ref{refm-}$), and a calculation show that \begin{eqnarray*} W[\psi_+(\lambda+i\varepsilon),\psi_-(\lambda+i\varepsilon)] & = & m_-(\lambda+i\epsilon,0)-m_+(\lambda+i\epsilon,0) \\ & = & -2i(\lambda+ i \epsilon)^{1/2} \rightarrow -2i \lambda^{1/2} \mbox{ as } \epsilon \downarrow 0. \end{eqnarray*} By ($\ref{trigpsi}$) \begin{equation} \nonumber \lim_{\epsilon \downarrow 0} \psi_+(x, \lambda + i \epsilon)\psi_-(x, \lambda + i \epsilon) =C^2(x, \lambda)+\lambda S^2(x,\lambda), \end{equation} in particular, this limit is real. Inserting into ($\ref{psig}$) yields ($\ref{Craig}$). This and ($\ref{xidef}$) imply that $\xi(\lambda,x)= 1/2$. \end{proof} By Lemma $\ref{refxi}$, ($\ref{logg}$) simplifies to \begin{eqnarray} \nonumber \mbox{ln}[G(z,x,x)] & = & c(x) + \frac{1}{2} \int_0^{\infty} \left( \frac{1}{\lambda-z} - \frac{\lambda}{1+ \lambda^2} \right)d \lambda \\ & = & c(x) + \mbox{ln}[ \frac{i}{\sqrt{z}}], \nonumber \end{eqnarray} and thus \begin{equation} \label{simpleg} G(z,x,x)=e^{c(x)} \frac{i}{\sqrt{z}}, \end{equation} which extends to $z \in (-\infty,0)$ by analyticity. On the other hand, we have by Proposition $\ref{mprop}$ that in Lebesgue points $x$ of $\frac{1}{p}$ and for $z \in (- \infty, 0)$ \begin{equation} \label{1oGasy} \frac{1}{G(z,x,x)} = m_-(z,x)-m_+(z,x) = -2ip^{1/2}(x) \sqrt{z} + o(\sqrt{z}). \end{equation} Combining ($\ref{cofx}$), ($\ref{simpleg}$), and ($\ref{1oGasy}$) we find that \begin{equation} \label{expp} e^{- \mbox{Re}[ \mbox{ln}[G(i,x,x)]]}=2p^{1/2}(x). \end{equation} This implies that $p$ is absolutely continuous since ln$[G(i,x,x)]$ is absolutely continuous by ($\ref{psgf}$). Also, \begin{equation} \nonumber \frac{d}{dx} \mbox{ln}[G(i,x,x)] = \frac{1}{p(x)} \left( \frac{(pf_+^{\prime})(x,i)}{f_+(x,i)} + \frac{(pf_-^{\prime})(x,i)}{f_-(x,i)} \right), \end{equation} whose right hand side is absolutely continuous. By differentiating ($\ref{expp}$), we finally observe that $p^{\prime}$ is absolutely continuous. We can now use Corollary $\ref{psmooth}$ to conclude that $p=1$. In this section we only included those results on $m$-function asymptotics and inverse spectral and scattering theory for $-(pu')'=zu$ which are used in our applications to random operators. Many other natural questions arise in this context, e.g.\ relaxing the assumption of compact support for $p-1$ or extending Proposition~\ref{trivps} to cover a larger range for $-\alpha^2$. These and related questions will be discussed in \cite{GSS}. \section{Modified Pr\"{u}fer Amplitudes and Phases} \setcounter{equation}{0} The key result of this section is Lemma $\ref{dmpp}$ below, which provides the relation between the phase shift $\delta$ and the amplitude $R$ stated in the introduction. Mathematically, $\delta$ and $R$ arise as {\it modified Pr\"{u}fer variables}: For any real $c$, $\vartheta$, and $k >0$, take a solution $u \neq 0$ of $-(pu^{\prime})^{\prime}+qu=k^2u$ with $u(c) = \sin( \vartheta)$ and $pu^{\prime}(c) = k \cos( \vartheta)$. For such a $u$, we define $\varphi_c(x, \vartheta, k)$ and $R_c(x, \vartheta, k)$, the modified Pr\"{u}fer phase and amplitude, by setting $u = R_c \sin \varphi_c$ and $pu^{\prime} = k R_c \cos \varphi_c$. We fix a unique value of $\varphi_c$ by requiring that $\varphi_c(c, \vartheta, k) = \vartheta$ and continuity in $x$. As a consequence, we have that $\varphi_c(x, \vartheta + \pi, k) = \varphi_c(x, \vartheta, k) + \pi$. In the next three Lemmas we collect facts which are well known in the Schr\"{o}dinger case, i.e. for $p=1$. \begin{Lemma} We have for fixed $c, \vartheta,$ and $k >0$ that \begin{eqnarray} \label{dReqn} k \partial_xR_c(x) & = & \left[ q(x)-k^2 \left( 1 - \frac{1}{p(x)} \right) \right] R_c(x) \sin \varphi_c(x) \cos \varphi_c(x), \\ \label{dTeqn} k \partial_x \varphi_c(x) & = & k^2 -k^2 \left( 1 - \frac{1}{p(x)} \right) \cos^2 \varphi_c(x) - q(x) \sin^2 \varphi_c(x). \end{eqnarray} \end{Lemma} \begin{proof} We calculate \begin{eqnarray} \frac{1}{p} k R_c \cos \varphi_c = & u^{\prime} = & ( \partial_x R_c) \sin \varphi_c + R_c \cos \varphi_c ( \partial_x \varphi_c), \nonumber \\ \left( q-k^2 \right) R_c \sin \varphi_c = & (pu^{\prime})^{\prime} = & k ( \partial_x R_c) \cos \varphi_c - k R_c \sin \varphi_c ( \partial_x \varphi_c). \nonumber \end{eqnarray} Solving the above system for the left hand sides of $( \ref{dReqn})$ and $( \ref{dTeqn})$ yields the desired equations. \end{proof} We note that $( \ref{dReqn})$ may be rewritten as \begin{eqnarray} \label{dR2} k \partial_x \ln[R^2_c(x)] & = & \left[ q(x)-k^2 \left( 1 - \frac{1}{p(x)} \right) \right] \sin[2 \varphi_c(x)]. \end{eqnarray} \begin{Lemma} For every $x$ and $\vartheta$, we have \begin{equation} \label{dpteqn} ( \partial_{\vartheta} \varphi_c)(x, \vartheta ) = R_c^{-2}(x, \vartheta ). \end{equation} \end{Lemma} \begin{proof} A naive calculation shows that \begin{eqnarray*} \partial_x \partial_{ \vartheta} \varphi_c(x, \vartheta) \ \ = \ \ \partial_{ \vartheta} \partial_x \varphi_c(x, \vartheta) & = & \frac{1}{k} \partial_{\vartheta} \left( k^2 -k^2 \left( 1 - \frac{1}{p} \right) \cos^2 \varphi_c - q \sin^2 \varphi_c \right) \\ & = & - \frac{2}{k} ( \partial_{\vartheta} \varphi_c) \left[ q-k^2 \left( 1 - \frac{1}{p} \right) \right] \sin \varphi_c \cos \varphi_c, \end{eqnarray*} where the first equality above is certainly true for the distributional derivatives. In dimension one, however, the distributional derivative of an absolutely continuous function coincides with its pointwise derivative in $L^1_{loc}$; hence $( \ref{dR2})$ implies that $\partial_x ( ln[R_c^2 \partial_{\vartheta} \varphi_c]) = 0$ for almost every $(x, \vartheta)$. Since $R_c^2(c, \vartheta) ( \partial_{\vartheta} \varphi_c)(c, \vartheta) = 1$, we conclude that $$ ln[R_c^2(x, \vartheta) ( \partial_{\vartheta} \varphi_c)(x, \vartheta)] =0 \ \ \text{for a.e.} \ \ \vartheta. $$ This yields ($\ref{dpteqn}$), first for almost every $\vartheta$, and then by continuous extension the result holds for every $\vartheta$ (Pr\"{u}fer variables are analytic in $\vartheta$). \end{proof} \begin{Corollary} \label{avamps} For any $c, x, k,$ and $\vartheta$ $$ \frac{1}{ \pi} \int_{\vartheta}^{ \vartheta + \pi} R_c^{-2}(x, \beta, k) d \beta = 1. $$ \end{Corollary} \begin{proof} This follows from integrating the above result and using that $\varphi_c(x, \vartheta + \pi, k) - \varphi_c( x, \vartheta, k) = \pi$. \end{proof} We are now ready to state how the Pr\"{u}fer phase changes under translation of the coefficients $p$ and $q$. For $p$ and $q$ with supp$(q)$, supp$(p-1) \subset [0,D]$, $D<1$, and $00$, $u$ be such that $\tau_au=k^2u$, and take $\varphi( \dots, \tau_a)$ and $R( \dots, \tau_a)$ as defined above for $u$. \begin{Lemma} \label{dmpp} For fixed $k$ and $\vartheta$, we have \begin{equation} \frac{1}{k} \frac{d}{da} \varphi_0(1, \vartheta, k, \tau_a) = R^{-2}_0(1, \vartheta, k, \tau_a)-1. \nonumber \end{equation} \end{Lemma} \begin{proof} As $k$ is fixed, we suppress this dependency. One may write \begin{equation} \varphi_0(1, \vartheta, \tau_a) = \varphi_{a+D} \left(1, \varphi_0 \left(D, \varphi_0(a, \vartheta, - \frac{d^2}{dx^2}), \tau_0 \right), - \frac{d^2}{dx^2} \right). \end{equation} When $q=0$ and $p=1$, $( \ref{dTeqn})$ can be used to explicitly calculate $\varphi$. We may thereby rewrite the above as \begin{equation} \varphi_0(1, \vartheta, \tau_a) = \varphi_0(D, \vartheta +ka, \tau_0)+k(1-a-D). \end{equation} Using this, and the lemmas above we find that \begin{eqnarray} \frac{1}{k} \frac{d}{da} \varphi_0(1, \vartheta, \tau_a) & = & (\partial_{\vartheta} \varphi_0)(D, \vartheta + ka, \tau_0) -1 \nonumber \\ & = & \frac{1}{R_0^2(D, \vartheta + ka, \tau_0)}-1 \nonumber \\ & = & \frac{1}{R_0^2(1, \vartheta, \tau_a)}-1. \nonumber \end{eqnarray} \end{proof} We note that $R^2_0(x, \vartheta, k, \tau_0) = F_{u_0}(x,k^2)$ where $F_{u_0}$ is as defined by $( \ref{sqpa})$ and $u_0$ is the solution of $( \ref{cseqn})$ normalized at zero by $u_0(0)= \sin( \vartheta)$ and $pu_0^{\prime}(0)=k \cos( \vartheta)$. This observation allows us to relate non-zero reflection, i.e. $b(k) \neq 0$ for the single site equation $-(pu^{\prime})^{\prime} + qu =k^2u$, to non-stationarity of the Pr\"{u}fer phase in $a$. More precisely, we have: \begin{Lemma} \label{combo} If $k>0$, $b(k) \neq 0$, and $a_0 \in (0,1-D)$, then $\left. \frac{d}{da} \varphi_0(1, \vartheta, \tau_a) \right|_{a=a_0} \neq 0$ for all $\vartheta \in \R \setminus N(a_0)$, where $N(a_0)$ is a discrete, $\pi$-periodic subset of $\R$. \end{Lemma} \begin{proof} For any fixed $k$ satisfying $b(k) \neq 0$, Lemma $\ref{local}$ shows that $1=R_0^2(0, \vartheta, \tau_0) \neq R_0^2(D, \vartheta, \tau_0)$ for at least one $\vartheta$. Since $R_0^2$ is analytic and $\pi$-periodic in $\vartheta$, we have that \begin{equation} N:= \left\{ \alpha \in \R: \ \ R^2_0(D, \alpha, \tau_0) = 1 \right\} \nonumber \end{equation} is a discrete, $\pi$-periodic subset of $\R$. By Lemma $\ref{dmpp}$, \begin{equation} \left. \frac{d}{da} \varphi_0(1, \vartheta, \tau_a) \right|_{a=a_0}=0 \mbox{ if and only if } 1=R_0^2(1, \vartheta, \tau_{a_0})=R_0^2(D, \vartheta +ka_0, \tau_0). \nonumber \end{equation} Clearly then, we can take \begin{equation} N(a_0):= \left\{ \vartheta \in \R: \ \ \vartheta +ka_0 \in N \right\} \nonumber \end{equation} and the lemma is proven. Note that $N(a_0)=N(a_0,k)$. \end{proof} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}\alph{letters}} \setcounter{letters}{1} We now introduce the two parameter model. Let $00$. Then for every $a_0 \in (0,1-D)$ there exists a finite subset $M(a_0) \subset (0,1-D)$ with the following property: If $b_0 \in (0,1-D) \setminus M(a_0)$, then there exist closed sets $T_1,T_2 \subset \R$ such that \newline (i) $T_1 \cup T_2 = \R$, \newline (ii) $ \left. \frac{d}{da} \varphi_0(1, \vartheta, \tau_a) \right|_{a=a_0} \neq 0$ for all $ \vartheta \in T_1$, \newline (iii) $ \left. \frac{d}{db} \varphi_0(2, \vartheta, \tau_{a_0,b}) \right|_{b=b_0} \neq 0$ for all $ \vartheta \in T_2$. \end{Lemma} \begin{proof} In keeping with the notation of Lemma $\ref{combo}$, we have \begin{equation} \left. \frac{d}{da} \varphi_0(1, \vartheta, \tau_a) \right|_{a=a_0} \neq 0 \mbox{ for all } \vartheta \in \R \setminus N(a_0). \nonumber \end{equation} Let \begin{equation} M(a_0):= \left\{ b \in (0,1-D): \mbox{ } \frac{d}{db} \varphi_0(2, \vartheta, \tau_{a_0,b})= 0 \mbox{ for some } \vartheta \in N(a_0) \right\}. \nonumber \end{equation} Similar to before, we may write \begin{equation} \varphi_0(2, \vartheta, \tau_{a_0,b}) = \varphi_0(D, \varphi_0(1, \vartheta, \tau_{a_0})+kb, \tau_0) +k(1-b-D), \nonumber \end{equation} and so \begin{equation} \left. \frac{1}{k} \frac{d}{db} \varphi_0(2, \vartheta, \tau_{a_0,b}) \right|_{b=b_0} = R_0^{-2}(D, \varphi_0(1, \vartheta, \tau_{a_0}) +kb_0, \tau_0)-1. \nonumber \end{equation} Hence for fixed $\vartheta$, \begin{equation} \bigg\{ b \in \R: \mbox{ } \varphi_0(1, \vartheta, \tau_{a_0}) +kb \in N \bigg\} \nonumber \end{equation} is discrete and $\pi$-periodic, and thus $M(a_0)$ is finite. For $b_0 \in (0,1-D) \setminus M(a_0)$ one has that $\left. \frac{d}{db} \varphi_0(2, \vartheta, \tau_{a_0,b}) \right|_{b=b_0} \neq 0$ for all $\vartheta \in N(a_0)$. Continuity and periodicity of $\frac{d}{db} \varphi_0(2, \vartheta, \tau_{a_0,b})$ in $\vartheta$ implies the existence of an $ \epsilon >0$ such that $\left. \frac{d}{db} \varphi_0(2, \vartheta, \tau_{a_0,b}) \right|_{b=b_0} \neq 0$ for all $\vartheta \in T_2:= \{ \vartheta: \mbox{ } \text{dist}( \vartheta, N(a_0)) \leq \epsilon \}$. By construction, we have that $\left. \frac{d}{da} \varphi_0(1, \vartheta, \tau_a) \right|_{a=a_0} \neq 0$ for $\vartheta \in T_1:= \{ \vartheta: \mbox{ } \text{dist}( \vartheta, N(a_0)) \geq \epsilon \}$. This completes the proof since $T_1$ and $T_2$ are closed with $T_1 \cup T_2 = \R$. \end{proof} \section{Proof of ($\ref{mtSc}$) and ($\ref{mtWc}$)} \setcounter{equation}{0} The results established in the previous sections, in particular Lemma $\ref{nzp}$, will now enable us to prove a result on two parameter spectral averaging (Theorem $\ref{tpsa}$ below), which applies to both models $H_{\omega}^S$ and $H_{\omega}^W$, and thus leads to proofs of ($\ref{mtSc}$) and ($\ref{mtWc}$). The details are very close to arguments already used in \cite{B.S.}, to which we refer frequently. We work with the two parameter differential expressions $\tau_{a,b}$ from Section 5, defined by ($\ref{twops}$), ($\ref{twoqs}$), and ($\ref{mixed2bump}$), but think of them as extended to expressions of the form $- \frac{d}{dx} p \frac{d}{dx} + q$ on the whole line, such that $\tau_{a,b}$ is limit point at $+ \infty$ and $- \infty$ in the sense of Sturm-Liouville theory. Thus each $\tau_{a,b}$ defines a unique self-adjoint operator $H_{a,b}$ on $L^2(\R)$. \begin{Lemma} \label{avtpamps} Let $k_0 >0$ with $b(k_0) \neq 0$. Choose $a_0$ and $b_0$ as in Lemma $\ref{nzp}$. Then there exists $\delta>0$, $\epsilon>0$, and $C>0$ such that \begin{equation} \label{avtpR} \int_{a_0- \delta}^{a_0+ \delta} \int_{b_0- \delta}^{b_0+ \delta}R_0(N, \vartheta, k, \tau_{a,b})^{-2}dbda \leq C \end{equation} uniformly in $N>0$, $\vartheta \in [0, \pi)$, and $k \in [k_0- \epsilon, k_0+ \epsilon]$. \end{Lemma} To prove this one can closely follow the proof of \cite[Lemma~5.3]{B.S.}. Lemma $\ref{nzp}$ replaces the use of \cite[Lemma~5.2]{B.S.}. Also, Corollary $\ref{avamps}$ enters, replacing the analogous averaging formula for "ordinary" Pr\"{u}fer amplitudes used in \cite{B.S.}. We omit the details. The Weyl-Titchmarsh spectral measures of the operators $H_{a,b}$ (e.g.\ \cite{Codd/Lev}) are denoted by $\rho_{a,b}$. For these measures we get our main result on two parameter spectral averaging: \begin{Theorem} \label{tpsa} Let $E_0=k_0^2$ for $k_0>0$ with $b(k_0) \neq 0$. For $a_0,b_0, \epsilon,$ and $\delta$ as in Lemma $\ref{avtpamps}$ and Borel sets $B$ define \begin{equation} \label{asm} \rho(B):= \int_{a_0- \delta}^{a_0+ \delta} \int_{b_0- \delta}^{b_0+ \delta} \rho_{a,b}(B)dbda. \end{equation} Then the Borel measure $\rho$ is absolutely continuous in $((k_0- \epsilon)^2,(k_0+ \epsilon)^2)$. \end{Theorem} Based on Lemma $\ref{avtpamps}$, the proof of Theorem $\ref{tpsa}$ may again be taken from \cite{B.S.} with few changes. The key is to use "Carmona's formula" \begin{equation} \label{Carmona} d \rho_{a,b}(E) = \wlim_{N \rightarrow \infty} \frac{1}{ \pi} \int_0^{\pi}r_0(N, \theta, E, \tau_{a,b})^{-2} d \rho_{- \infty}^{0, \theta}d \theta, \end{equation} expressing $\rho_{a,b}$ in terms of {\it ordinary} Pr\"{u}fer variables $r$ and $\theta$, and the spectral measure $\rho_{- \infty}^{0, \theta}$ of the restriction of the operator $H_{a,b}$ to $(- \infty,0)$ with boundary condition at 0 determined by $\theta$. Use of ($\ref{Carmona}$) reduces the proof of absolute continuity of ($\ref{asm}$) to a uniform estimate of $$ \int_{a_0- \delta}^{a_0+ \delta} \int_{b_0- \delta}^{b_0+ \delta}r_0(N, \theta, E, \tau_{a,b})^{-2}dbda $$ in $N, \theta,$ and $E \in [(k_0- \epsilon)^2,(k_0+ \epsilon)^2]$, compare with the proof of \cite[Theorem~4.1]{B.S.}. But this follows from Lemma~$\ref{avtpamps}$, since on the compact subinterval $[(k_0- \epsilon)^2,(k_0+ \epsilon)^2]$ of $(0, \infty)$ the ordinary and modified Pr\"{u}fer amplitudes $r$ and $R$ can be uniformly estimated by each other. The proofs of Carmona's formula ($\ref{Carmona}$), which can be found in the literature for the Schr\"{o}dinger case (e.g. \cite{Carmona,Carmona/Lacroix}), extend with few changes to the more general differential expressions considered here. Details will be contained in \cite{Sims}. Theorem $\ref{tpsa}$ covers positive energies away from the discrete set where the reflection coefficient for a non-reflectionless single site equation ($\ref{cseqn}$) may vanish. This will suffice to prove ($\ref{mtWc}$) since the operators $H_{\omega}^W$ are non-negative. However, the Schr\"{o}dinger operators from ($\ref{mtSc}$) may have some negative spectrum. To cover this, an analogue of Theorem $\ref{tpsa}$ which works for negative energies will be needed in the Schr\"{o}dinger case. But in this case, we may directly refer to \cite[Theorem~5.4]{B.S.}, which shows that the conclusion of Theorem $\ref{tpsa}$ extends to energies $E_0 \in (- \infty,0) \setminus M_0$, where (with $q \in L^1(0,D)$ as above) \begin{eqnarray*} M_0 & := & \bigg\{ E \in (- \infty,0): \mbox{ There exists a non-trivial solution } u \mbox{ of } -u^{\prime \prime}+qu=Eu \\ & & \mbox{ with } \:\frac{u^{\prime}(0)}{u(0)} \in \big\{ \pm \sqrt{|E|} \big\} \quad \mbox{and} \quad \frac{u^{\prime}(D)}{u(D)} \in \big\{ \pm \sqrt{|E|} \big\}. \bigg\} \end{eqnarray*} That $M_0$ is discrete in $(- \infty,0)$ for $q \neq 0$ was shown in \cite[Theorem~3.1]{B.S.} by means of Floquet Theory. This may be reproven by using the properties of reflection and transmission coefficients: By observing $( \ref{jsol})$, for the case $k=i \alpha$, $\alpha >0$, we see that \begin{equation} \nonumber M_0 = \left\{ - \alpha^2 : \mbox{ } \alpha >0 \mbox{ and } b(i \alpha)b( -i \alpha)a(i \alpha) a( -i \alpha) =0 \right\}. \end{equation} We have that $a( \cdot)$ is non-zero on the positive real line and analytic in $\C \setminus \{0 \}$. Thus it has discrete zeros on the imaginary axis. The same holds for $b( \cdot)$, since it is analytic in $\C \setminus \{ 0 \}$ and does not vanish identically on the real line by Corollary $\ref{Borg}$. We conclude that $M_0$ is discrete in $(- \infty,0)$. This completes our discussion of results on spectral averaging. As mentioned in the introduction, to finish the proofs of ($\ref{mtSc}$) and ($\ref{mtWc}$), we need to know positivity of the Lyapunov exponents for the models $H_{\omega}^S$ and $H_{\omega}^W$. In the Schr\"{o}dinger case, this follows from Kotani's work (e.g. \cite{Ko1,Ko2}, extended in \cite{Kirsch/Kotani/Simon} to cover $L^2$-potentials as used here): $H_{\omega}^S$ is non-deterministic in Kotani's sense, since $g$ is compactly supported. This implies positivity of the Lyapunov exponent for almost every energy, which is sufficient for our purposes. To treat $H_{\omega}^W$ we refer to Minami's extension of Kotani's theory to more general random Sturm-Liouville operators \cite{Mi1,Mi2}. More precisely, it follows as a special case of a remark in \cite{Mi2} that ergodic differential expressions of the type $- \frac{d}{dx}a_{\omega} \frac{d}{dx}$ can be transformed into ergodic expressions of the type $-r_{\omega} \frac{d^2}{dx^2}$. For the latter type, Kotani's theory is extended in \cite{Mi1}, in particular that non-deterministic expressions have almost everywhere positive Lyapunov exponent. This applies to $H_{\omega}^W$ since $f$ is compactly supported and thus $H_{\omega}^W$ non-deterministic. At this point, we skip the remaining details of the proofs of ($\ref{mtSc}$) and ($\ref{mtWc}$) and instead refer to Section 2 of \cite{B.S.}, where it is shown how to deduce exponential localization from almost everywhere positivity of the Lyapunov exponent and two parameter spectral averaging. The argument in \cite{B.S.} uses Gilbert and Pearson's subordinacy theory. We refer to \cite{Gilbert} for the extension of the latter to general Sturm-Liouville operators. \hfill $\qed$ We close by noting that we expect the following more general result to be true: If the single site equation $-\left(\frac{u'}{1+f}\right)'+gu=k^2u$ is not reflectionless ($f$ and $g$ as in Section~1), then $H_{\omega} = -\frac{d}{dx} a_{\omega} \frac{d}{dx} +v_{\omega}$ almost surely has pure point spectrum ($v_{\omega}$ and $a_{\omega}$ defined by (\ref{pot}) resp.\ (\ref{den})). The above proof of ($\ref{mtSc}$) and ($\ref{mtWc}$) does not carry over since, as remarked in \cite{Mi1} it is not understood how to extend Kotani's theory to operators with this kind of randomness. A problem arises from the existence of reflectionless "mixed" equations as provided by the example in Section~3. However, in the non-reflectionless case it should be possible to prove positivity of the Lyapunov exponent for sufficiently many energies by other means. \setcounter{equation}{0} \baselineskip=12pt \begin{thebibliography}{99} \bibitem{A/D} N. Aronszajn and W. Donoghue: On exponential representations of analytic functions in the upper half-plane with positive imaginary part. {\it J.Anal.Math.} {\bf 5}, 321 -- 388 (1957) \bibitem{Bi/Ger2} De Bi\`{e}vre and Germinet: Dynamical Localization for the Random Dimer Schr\"{o}dinger Operator. Preprint, available from mp\_arc 99-250 \bibitem{Borg} G. Borg: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. {\it Acta Math.} {\bf 78}, 1 -- 96 (1946) \bibitem{B.S.} D. Buschmann and G. Stolz: Two-Parameter Spectral Averaging and Localization for Non-Monotoneous Random Schr\"odinger Operators, to appear in {\it Trans. Amer. Math. Soc.} Preprint available from mp\_arc 98-617 \bibitem{Carmona} R. Carmona: Exponential localization in one-dimensional disordered systems. {\it Duke Math. J.} {\bf 49}, 191 -- 213 (1982) \bibitem{Carmona/Lacroix} R. Carmona and J. Lacroix: Spectral theory of random Schr\"odinger operators. Birkh\"auser, Basel--Berlin 1990 \bibitem{Clark/Hinton} S. Clark and D. Hinton: Strong nonsubordinacy and absolutely continuous spectra for Sturm-Liouville equations. {\it Diff. and Int. Equ.} {\bf 6}, 573 -- 586 (1993) \bibitem{Codd/Lev} E.A. Coddington and N. Levinson: Theory of ordinary differential equations. McGraw-Hill, New York 1955 \bibitem{Combes/Hislop/Tip} J.M. Combes, P.D. Hislop and A. Tip: Band edge localization and the density of states for acoustic and electromagnetic waves in random media. {\it Ann. Inst. Henri Poincar\'e, Physique Th\'eorique} {\bf 70}, 381 -- 428 (1999) \bibitem{CraigW} W. Craig: The trace formula for Schr\"{o}dinger operators on the line. {\it Commun. Math. Phys.} {\bf 126}, 379--407 (1989) \bibitem{Eastham} M.S.P. Eastham: The Spectral Theory of Periodic Differential Equations. Scottish Academic Press, Edinburgh--London 1973 \bibitem{Eastham2} M.S.P. Eastham: The asymptotic solution of linear differential systems. Clarendon Press, Oxford 1989 \bibitem{Everitt} W. Everitt: On a property of the m-coefficient of a second-order linear differential equation. {\it J. London Math. Soc. (2)} {\bf 4}, 443 -- 457 (1972) \bibitem{FK} A. Figotin and A. Klein: Localization of classical waves I: Acoustic waves. {\it Commun. Math. Phys.} {\bf 180}, 439 -- 482 (1996) \bibitem{FK2} A. Figotin and A. Klein: Localization of classical waves II: Electromagnetic waves. {\it Commun. Math. Phys.} {\bf 184}, 411 -- 441 (1997) \bibitem{Gesztesy/Holden/Simon} F. Gesztesy, H. Holden, and B. Simon: Absolute summability of the trace relation for certain Schr\"odinger operators. {\it Commun. Math. Phys.} {\bf 168}, 137 -- 161 (1995) \bibitem{Gesztesy/Simon} F. Gesztesy and B. Simon: The $\xi$ function. {\it Acta Math.} {\bf 176}, 49 -- 71 (1996) \bibitem{GSS} F. Gesztesy, R. Sims, and G. Stolz: in preparation \bibitem{Gilbert} D.J. Gilbert: On subordinacy and spectral multiplicity for a class of singular differential operators. {\it Proc. Roy. Soc. Edinburgh} {\bf A 128}, 549 -- 584 (1998) \bibitem{Hochstadt} H. Hochstadt: On the Determination of a Hill's Equation from its Spectrum. {\it Arch. Rational Mech. Anal.} {\bf 19}, 353 -- 362 (1965) \bibitem{Keller} J. Keller: Discriminant, transmission coefficients and stability bands of Hill's equation. {\it J. Math. Phys.} {\bf 25}, 2903 -- 2904 (1984) \bibitem{Kirsch/Kotani/Simon} W. Kirsch, S. Kotani and B. Simon: Absence of absolutely continuous spectrum for some one dimensional random but deterministic potentials. {\it Ann. Inst. Henri Poincar\'e} {\bf 42}, 383 -- 406 (1985) \bibitem{Kostrykin/Schrader} V. Kostrykin and R. Schrader: Scattering Theory Approach to Random Schr\"odinger Operators in One Dimension. {\it Rev. Math. Phys.} {\bf 11}, 187 -- 242 (1999) \bibitem{Ko1} S.\ Kotani: Lyapunov indices determine absolutely continuous spectra of stationary one dimensional Schr\"odinger operators. In: Stochastic Analysis, ed.\ K.\ Ito, 225 -- 247, North Holland, Amsterdam 1984 \bibitem{Ko2} S. Kotani: One-Dimensional Random Schr\"odinger Operators and Herglotz Functions. In: Probabilistic Methods in Mathematical Physics, eds.\ K.\ Ito and N.\ Ikeda, 219 -- 250, Academic Press, Boston 1987 \bibitem{Kotani/Krishna} S. Kotani and M. Krishna: Almost periodicity of some random potentials. {\it J. Funct. Anal.} {\bf 78}, 390 -- 405 (1988) \bibitem{Kotani/Simon} S. Kotani and B. Simon: Localization in General One-Dimensional Random Systems. {\it Commun. Math. Phys.} {\bf 112}, 103 -- 119 (1987) \bibitem{Kuchment} P. Kuchment: The Mathematics of Photonic Crystals. In: Mathematical Modeling in Optical Science, eds.\ G.\ Bao, L.\ Cowsar, and W.\ Masters, SIAM, to appear \bibitem{Mi1} N. Minami: An Extension of Kotani's Theorem to Random Generalized Sturm-Liouville Operators. {\it Commun. Math. Phys.} {\bf 103}, 387 -- 402 (1986) \bibitem{Mi2} N. Minami: An extension of Kotani's theorem to random generalized Sturm-Liouville operators II. In: Stochastic processes in classical and quantum systems, 411 -- 419, {\it Lecture Notes in Physics} {\bf 262}, Springer 1986 \bibitem{Molchanov} S. Molchanov: Multiscattering on sparse bumps. {\it Contemporary Math.} {\bf 217}, 157 -- 181 (1998) \bibitem{Simon/Wolff} B. Simon and T. Wolff: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. {\it Commun. Pure Appl. Math.} {\bf 39}, 75 -- 90 (1986) \bibitem{Sims} R. Sims, PhD-Thesis, in preparation \bibitem{Stollmann} P. Stollmann: Localization for random perturbations of anisotropic periodic media. {\it Israel J. Math.} {\bf 107}, 125 -- 139 (1998) \bibitem{Bsols} G. Stolz: Bounded Solutions and Absolute Continuity of Sturm-Liouville Operators. {\it J. Math. Anal. Appl.} {\bf 169}, 210 -- 228 (1992) \bibitem{Habil} G. Stolz: Spectral theory of Schr\"odinger operators with potentials of infinite barriers type. Habilitationsschrift, Frankfurt 1994 \bibitem{Stolz} G. Stolz: Localization for random Schr\"odinger operators with Poisson potential. {\it Ann. Inst. Henri Poincar\'e} {\bf 63}, 297 -- 314 (1995) \bibitem{Stolz2} G. Stolz: Non-Monotonic Random Schr\"odinger Operators: The Anderson Model. Preprint, available from mp\_arc 99-259 \bibitem{Weidmann} J. Weidmann: Spectral Theory of Ordinary Differential Operators. Springer-Verlag, New York 1987 \end{thebibliography} \end{document} ---------------0002071708419--