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Random Operators, Scattering Theory, Localization, Displacement Model
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\begin{document}
%\baselineskip=20pt
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\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 $y<y_0$, i.e.\ $\theta_{\alpha}$ is decreasing near $y_0$.

For the case $\theta_{\alpha}(y_0) = \pi$ a similar argument,
based on writing (\ref{dta}) as $\theta_{\alpha} =
\frac{1}{p_{\alpha}} - \left( \frac{1}{p_{\alpha}}+1 \right)
\sin^2 \theta_{\alpha}$, allows to show that $\theta_{\alpha}$ is
increasing near $y_0$, a contradiction to $\theta_{\alpha}(y)
<\pi$ for $y_0<y\le \alpha$. We skip the details, which in fact
are a special case of what is shown in \cite{Weidmann}.
\end{proof}

We may now rephrase ($\ref{wasy}$) as
\begin{equation} \nonumber
\lim_{\alpha\to\infty} \theta_{\alpha}(0) =
\:\mbox{arccot}(-p(0)^{1/2}).
\end{equation}

\noindent Thus, Proposition $\ref{mprop}$ is implied by the following
results:

\begin{Lemma} \label{tresult}
(a) $\lim_{\alpha\to\infty} \tilde{\theta}_{\alpha}(0) =
\:\mbox{\rm{arccot}}(-p(0)^{1/2})$,

(b) $\lim_{\alpha\to\infty}
(\theta_{\alpha}(0)-\tilde{\theta}_{\alpha}(0)) = 0$.
\end{Lemma}

{\it Proof of (a):} For $y\in [0,\mu(\alpha))$ one has
$\tilde{w}_{\alpha}''(y)=p(0)^{-1} \tilde{w}_{\alpha}(y)$ and thus
\[ \tilde{w}_{\alpha}(y)=A_{\alpha} e^{p(0)^{-1/2}y} + B_{\alpha}
e^{-p(0)^{-1/2}y}\] for suitable constants $A_{\alpha}$ and
$B_{\alpha}$. By Lemma $\ref{QII}$,
$\tilde{\theta}_{\alpha}(\mu(\alpha)) = \theta_{\alpha}(\mu(\alpha)) \in (\pi/2,\pi)$
(note that $\tilde{p}_{\alpha}(y)=p_{\alpha}(y)$ for $y \geq \mu(\alpha)$) and
therefore
\[ 0 >
\frac{\tilde{w}_{\alpha}'(\mu(\alpha)-)}{\tilde{w}_{\alpha}(\mu
(\alpha))} = p(0)^{-1/2} \frac{C_{\alpha} e^{2p(0)^{-1/2}
\mu(\alpha)}-1}{C_{\alpha} e^{2p(0)^{-1/2}\mu(\alpha)}+1} \] 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 $\tilde{\varphi}(\delta) := \max \{ \sup_{0<s\le \delta}
\varphi(s), \delta\}$).

(ii) It is sufficient to prove the existence of $\mu_1$ with
properties (\ref{muprop1}), (\ref{muprop2}) and
\begin{equation} \nonumber
e^{C\mu_1(\alpha)} \varphi(\frac{\mu_1(\alpha)}{\alpha}) \to 0
\quad \mbox{as $\alpha\to\infty$},
\end{equation}
(replace $C$ by $2C$).

(iii) It is sufficient to prove the existence of $\mu_2$ with
properties (\ref{muprop1}), (\ref{muprop2}) and
\begin{equation} \label{muprop3b}
\mu_2(\alpha) \varphi(\frac{\mu_2(\alpha)}{\alpha}) \to 0 \quad
\mbox{as $\alpha\to\infty$},
\end{equation}
(choose $\mu_1(\alpha) := \frac{1}{C} \log \mu_2(\alpha)$ for
$\alpha$ sufficiently large).

A function with properties (\ref{muprop1}), (\ref{muprop2}) and
(\ref{muprop3b}) is given by
\begin{equation} \nonumber
\mu_2(\alpha) := \min \left\{
\frac{1}{\sqrt{\varphi(\alpha^{-1/2})}}, \alpha^{1/2} \right\},
\end{equation}
as is easily verified. \hfill $\qed$

This proves Lemma $\ref{choosemu}$ and thereby concludes the proof
of Proposition $\ref{mprop}$.  \hfill $\qed$

\vspace{.3cm}

Continuing the proof of Proposition $\ref{trivps}$, we note that
for $z \in \C \setminus [0, \infty)$ the diagonal Green's function of the
operator $- \frac{d}{dx}p \frac{d}{dx}$ is given by

\begin{equation} \label{psgf}
G(z,x,x)= \frac{f_+(x,z)f_-(x,z)}{W[f_+,f_-]},
\end{equation}
where the Wronskian, $W[f_+,f_-] = f_+pf_-^{\prime}-pf_+^{\prime}f_-$, does
not depend on $x$. It follows that

\begin{equation} \nonumber
\frac{1}{G(z,x,x)} = m_-(z,x)-m_+(z,x),
\end{equation}
in particular, $G(z,x,x)$ is analytic in $\C \setminus [0,
\infty)$ for fixed $x$. Also, $m_+( \cdot,x)$ and $-m_-( \cdot,
x)$ are Herglotz functions, i.e. they map the open upper
half-plane analytically into itself. Thus, $G(z,x,x)$ is Herglotz.
Taking the standard branch of the logarithm, ln$[G(z,x,x)]$ is
also Herglotz with
\begin{equation} \nonumber
\mbox{Im}[ \mbox{ln}[G(z,x,x)]]= \mbox{arg}(G(z,x,x)) \in (0, \pi)
\end{equation}
for $z$ in the open upper half-plane. Thus
\begin{equation} \label{xidef}
\xi(\lambda,x):= \frac{1}{\pi} \lim_{\epsilon \downarrow 0} \mbox{Im}[ \mbox{ln}[G(\lambda + i \epsilon,x,x)]]
\end{equation}
exists for a.e. $\lambda$ with $0 \leq \xi(\lambda,x) \leq 1$ and one has the
logarithmic Herglotz representation (e.g. \cite{A/D})
\begin{equation} \label{logg}
\mbox{ln}[G(z,x,x)] = c(x) + \int_0^{\infty} \left( \frac{1}{\lambda-z} - \frac{\lambda}{1+ \lambda^2}
\right) \xi( \lambda,x)d\lambda.
\end{equation}
Here it was also used that, for $\lambda <0$, $\xi(\lambda,x)
= \frac{1}{\pi} \mbox{Im}[ \mbox{ln}[G( \lambda,x,x)]] =0$ by ($\ref{psgf}$). Note that
\begin{equation} \label{cofx}
c(x) = \mbox{Re}[ \mbox{ln}[G(i,x,x)]].
\end{equation}
We note that ($\ref{xidef}$) and ($\ref{logg}$) are the key
ingredients in proving trace formulas for Schr\"odinger operators
in \cite{Gesztesy/Simon}.

We include a proof of the following fact, which is well-known, at
least for Schr\"{o}dinger operators (see e.g. \cite{CraigW}[p.391]
for the observation that reflectionless implies ($\ref{Craig}$),
and also \cite{Gesztesy/Holden/Simon} and \cite{Kotani/Krishna}):
\begin{Lemma} \label{refxi}
If $-(pu^{\prime})^{\prime}=k^2u$ is reflectionless, then $\xi(\lambda,x)= 1/2$
for all $\lambda >0$ 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 $0<a<1-D$, we consider the following differential expression:

\begin{equation}
\tau_a := - \frac{d}{dx}p( \cdot - a) \frac{d}{dx} + q( \cdot - a).
\end{equation}
Let $k>0$, $u$ 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 $0<D<1$, $\frac{1}{p}$ and $q \in L^1(0,D)$ be real valued and such
that
$-(pu^{\prime})^{\prime}+qu=k^2u$ is not reflectionless. Take
$a,b \in (0,1-D)$ and define
\begin{equation} \label{twops}
p_{a,b}(x) := \left\{ \begin{array}{ll} p(x-a) & \mbox{for } a \leq x
\leq a+D, \\
p(x-1-b) & \mbox{for } 1+b \leq x \leq 1+b+D, \\
1 & \mbox{otherwise in } [0,2], \end{array} \right.
\end{equation}
\addtocounter{equation}{-1}
\addtocounter{letters}{1}
\begin{equation} \label{twoqs}
q_{a,b}(x) := \left\{ \begin{array}{ll} q(x-a) & \mbox{for } a \leq x
\leq a+D, \\
q(x-1-b) & \mbox{for } 1+b \leq x \leq 1+b+D, \\
0 & \mbox{otherwise in } [0,2], \end{array} \right.
\end{equation}

\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}

and
\begin{equation} \label{mixed2bump}
\tau_{a,b} := - \frac{d}{dx}p_{a,b} \frac{d}{dx} + q_{a,b}.
\end{equation}
\begin{Lemma} \label{nzp}
Suppose $b(k) \neq 0$ for some fixed $k>0$. 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.

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\baselineskip=12pt
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