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%References
\def \beh {1} % Behncke
\def\br {2} % Birkhoff-Rota
\def\dms {3} % del Rio-Makarov-Simon
\def\dss {4} % del Rio-Simon-Stolz
\def\egg {5} % Eggarter
\def\gora {6} % Gordon
\def\gor {7} % Gordon
\def\grp {8} % Gredeskul-Pastur
\def\kmp {9} % Kirsch-Molchanov-Pastur
\def\kls {10} % Kiselev-Last-Simon
\def\ls {11} % Last-Simon
\def\pf {12} % Pastur-Figotin
\def\rue {13} % Ruelle
\def\svan {14} % Simon (Vancouver school)
\def\ss {15} % Simon-Spencer
\def\sw {16} % Simon-Wolff
\def\sz {17} % Simon-Zhu
\def\sto {18} % Stolz
\topmatter
\title Modified Pr\"ufer and EFGP Transforms and Deterministic
Models With Dense Point Spectrum
\endtitle
\rightheadtext{Modified Pr\"ufer and EFGP Transforms}
\author Yoram Last and Barry Simon$^{*}$
\endauthor
\leftheadtext{ Y.~Last and B.~Simon}
\affil Division of Physics, Mathematics, and Astronomy \\
California Institute of Technology \\ Pasadena, CA 91125
\endaffil
\thanks$^*$ This material is based upon work supported by the
National Science Foundation under Grant No.~DMS-9401491. The
Government has certain rights in this material.
\endthanks
\date June 10, 1997
\enddate
\abstract We provide a new proof of the theorem of Simon and Zhu
that in the region $|E| < \lambda$ for a.e.~energies,
$-\frac{d^2}{dx^2}+\lambda \cos (x^\alpha)$, $0<\alpha <1$ has
Lyapunov behavior with a quasi-classical formula for the
Lyapunov exponent. We also prove Lyapunov behavior for a.e.~$E
\in [-2,2]$ for the discrete model with $V(j^2) = e^j$, $V(n)=0$
if $n\notin \{1,4,9,\dots \}$. The arguments depend on a direct
analysis of the equations for the norm of a solution.
\endabstract
\endtopmatter
\document
\vskip 0.1in
\flushpar{\bf \S 1. Introduction}
\vskip 0.1in
In this paper, we will consider half-line Schr\"odinger operators
$$
H_\theta = -\frac{d^2}{dx^2} + V(x) \tag 1.1
$$
on $L^2 (0,\infty)$ with $u(0) \cos (\theta) + u'(0)\sin(\theta)=
0$ boundary conditions and the discrete analog on $\ell^2
(\Bbb Z^+)$, $\Bbb Z^+=\{1,2,3,\dots \}$,
$$
(h_\alpha u)(n) = \cases u(n+1) + u(n-1) + V(n)u(n) &\quad n\leq 2 \\
u(2) + (V(1) + \alpha)u(1) &\quad n=1
\endcases \tag 1.2
$$
where $\alpha$ plays the role of a boundary condition.
We are interested in models where $H_\theta$ has dense point
spectrum in some interval $[a,b]$. By general instability results
[\dms,\gor], this cannot happen for all $\theta$ but can and does
for a.e.~$\theta$ if $[a,b]\subset\text{spec}(H_\theta)$, and if
for a.e.~$E\in [a,b]$, there is a solution $-u''+Vu =Eu$ which is
$L^2$ at infinity [\dss,\ss,\sw]. The first examples of such
operators involved random $V$'s where one proves dense point
spectrum for a.e.~$V$\!.
Examples which are deterministic were first found by Gordon [\gora]
(also see [\kmp]), who showed it for problems with very high and
sparse but not too sparse barriers. Simon-Zhu [\sz] proved a
similar result for slowly oscillating potentials like $V(x) =
\lambda \cos (x^\alpha)$; $0<\alpha<1$. Attention on the first
class was focused by work of Simon-Spencer [\ss], and on the
second by work of Behncke [\beh] and Stolz [\sto] --- these
authors showed the absence of a.c.~spectrum.
Our goal here is to obtain dense pure point spectrum by direct
control of the asymptotics of the transfer matrix $T(0,x)$,
defined by $T(0,x) \binom{u'(0)}{u(0)} = \binom{u'(x)}{u(x)}$
for solutions of
$$
-u'' + Vu = Eu \tag 1.3
$$
in the continuum case, and $T(0,n) \binom{u(1)}{u(0)} =
\binom{u(n+1)}{u(n)}$ for solutions of
$$
u(n+1) + u(n-1) + V(n) u(n) = Eu(n) \tag 1.4
$$
in the discrete case. It follows from results of Ruelle [\rue]
that if $\lim_{n\to\infty} \frac{1}{n} \ln \|T(0, n)\| >0$, then
there is an $L^2$ solution (here we include existence and
finiteness of the limit). The same idea works for other
situations, for example, if $\lim_{n\to\infty} \frac{1}{n^\gamma}
\ln \|T(0,n)\| >0$ for any $\gamma >0$; see [\ls].
For the case $V(x) = \lambda \cos (x^\alpha)$, that
$\lim_{n\to\infty} \frac{1}{n} \ln \| T(0,n)\|$ exists for
a.e.~$E$ and an explicit formula for the limit was found by
Simon and Zhu [\sz]. In Section~6, we will prove
\proclaim{Theorem 1.1} Let $V(x) = 1 + \cos (x^\alpha)$;
$0<\alpha <1$. Let $x_n = (2\pi n)^{1/2}$; let $a(n)=
n^{(1-\alpha)/2\alpha}$, and let $\{E^{(n)}_j\}^\infty_{j=1}$
be the eigenvalues of
$$
-\frac{d^2}{dx^2}+V(x); \qquad u(x_{n-1}) = u(x_n)=0
$$
on $L^2 (x_{n-1}, x_n; dx)$ and let
$$
\bar A = \bigcap^\infty_{k=1}\, \bigcup^\infty_{m=k}\,
\bigcup^\infty_{j=1} (E^{(n)}_j - e^{-a(n)}, E^{(n)}_j + e^{-a(n)})
$$
\rom(so that $\bar A$ is a $G_\delta$ dense in $[0,\infty)$ of
Lebesgue measure zero\rom).
Let $E\in (0,2)\backslash\bar A$. Then,
$$
\lim_{n\to\infty} \frac{1}{n} \, \ln \|T_E (0,n)\| =
\frac{1}{2\pi} \int_{\{y\mid 1+\cos (y) \geq E\}}
(1+\cos (y)-E)^{1/2}\, dy.
$$
\endproclaim
\remark{Remarks} 1. The forbidden set $\bar A$ in [\sz] is larger;
they conjecture that our $\bar A$ is the ``right" one. One should
be able to use WKB methods to describe $\bar A$ more completely.
2. It is known [\beh,\sto] that for $E>2$, the limit exists and
is zero.
3. $V(x)$ can be replaced by any $f(x^\alpha)$ where $f$ is any
$C^2$ periodic function with a finite number of critical points
in each period.
\endremark
Unlike Simon-Zhu [\sz], we will directly attack the transfer
matrix by using a transformation idea. In the continuum case, we
transform from $u'(x), u(x)$ to $R(x), \theta(x)$ defined by
$(k=\sqrt E)$:
$$\align
u'(x) &= kR(x) \cos (\theta(x)) \tag 1.5a \\
u(x) &= R(x) \sin (\theta(x)). \tag 1.5b
\endalign
$$
Then (1.3) is equivalent to
$$\align
\frac{d(\theta(x))}{dx} &= k - \frac{V(x)}{k}\,
\sin^2 (\theta(x)) \tag 1.6 \\
\frac{d\ln R}{dx}\, (x) &= \frac{1}{2k} \, V(x)
\sin(2\theta(x)). \tag 1.7
\endalign
$$
In [\kls], together with A.~Kiselev, we have shown how to exploit
these formulas in a variety of spectral situations, and our main
goal here is to show that they are useful in many tunnelling
calculations.
In the classically forbidden region where $V(x) > k^2$, (1.6)
tends to drive $\theta$ toward values where the left side
vanishes, that is,
$$
\sin(\theta) = \pm \sqrt{\frac{k^2}{V(x)}}\, . \tag 1.8
$$
At such points,
$$
\frac{1}{2k}\, V(x) \sin (2\theta(x)) =
\pm \sqrt{V(x) -k^2}\, . \tag 1.9
$$
The solutions of (1.8) where (1.9) has the plus sign are
attracting, which is why $R$ grows like $\exp\big(+\int
\sqrt{V(x)-k^2}\,\big)$.
In the classically allowed region where $V(x) 0$ (resp.~$k\in (0,\pi)$),
$$
C_1 (k) R(n,\theta_1) \leq \| T(0,n)\| \leq C_2
(k,\theta_1, \theta_2) \max (R(n,\theta_1), R(n,\theta_2)),
$$
where $C_1$ and $C_2$ are constants independent of $n$ and $V$. In
particular,
\proclaim{Proposition 1.2} If for $\theta_1 \neq \theta_2$
\rom(both in $[0,\pi)$\rom) we have
$$
\lim_{n\to\infty} \frac{1}{n}\, \ln R(n,\theta_1) =
\lim_{n\to\infty} \frac{1}{n}\, \ln R(n, \theta_2) = \gamma,
$$
then
$$
\lim_{n\to\infty} \frac{1}{n} \, \ln \|T(n,0)\| = \gamma.
$$
\endproclaim
As already mentioned, in the classically forbidden region, the
basic equations push $R$ to want to grow as $\exp\big(+\int
\sqrt{V(y)-E}\, dy\big)$ or else to decay as $\exp\big( -\int
\sqrt{V(y)-E}\, dy\big)$. In examples like $\cos(x^\alpha)$,
forbidden and allowed regions alternate. Our strategy will be
to prove one of three possibilities occurs:
\roster
\item"\rom{(i)}" All forbidden regions are decay regions for
$x$ sufficiently large. In that case, $u$ will be in $L^2$.
\item"\rom{(ii)}" All forbidden regions are growth regions for
$x$ sufficiently large. In that case, $R$ grows in the expected
WKB manner.
\item"\rom{(iii)}" Arbitrarily far out, there will be a growing
region followed by a decaying region. In that case, we can cut
off $u$ at the centers of those forbidden regions and get a
very good approximate eigenfunction, and so see that $E\in\bar A$.
\endroster
So if $E\notin\bar A$, either $R$ grows in the expected way or
$u$ is $L^2$. Since at most one $\theta_0$ can lead to an
$L^2$ solution, we can always find two $\theta$'s with the
expected growth and so use Proposition~1.2.
In Section~2, we discuss a discrete model with sparse growing
barriers for which $\lim_{n\to\infty} \mathbreak \frac{1}{n}
\lim \ln \|T(0,n)\|>0$. This shows the use of EFGP variables.
In Section~3, we discuss a model like $\cos(x^\alpha)$ but
where $\cos$ is replaced by a periodized step function.
Sections~4--6 present our proof of Theorem~1.1. An appendix
discusses bounded transfer matrices in the region $E>2$ in the
$1+\cos(x^\alpha)$ model if $\alpha <\frac12$.
B.S.~would like to thank M.~Ben-Artzi for the hospitality of
the Hebrew University where some of this work was done.
Y.L.~would like to thank J.~Avron for the hospitality of the
ITP at the Technion where some of this work was done.
\vskip 0.3in
\flushpar {\bf \S 2. A Model of Simon-Spencer Type}
\vskip 0.1in
In this section, we will study the following model on $\ell^2
([1,\infty))$,
$$\alignat2
(H_\alpha u)(n) &= u(n+1) + u(n-1) + V(n) u(n) &&\qquad n\geq 2 \\
&= u(n+1) + \alpha u(n) &&\qquad n=1
\endalignat
$$
where
$$\alignat2
V(j^2) &= e^{\beta j} &&\qquad j \geq 2 \\
V(n) &=0 &&\qquad n\neq 4,9,16,\dots
\endalignat
$$
$\alpha$ plays the role of a boundary condition. $\beta$ is a
parameter, $\beta >0$.
Define
$$
A_m = \bigcup^{m-1}_{j=1} (2\cos (\pi j/m) -e^{-\sqrt m}\, ,
2\cos \bigl(\pi j/m) + e^{-\sqrt m}\, \bigr)
$$
and let
$$
\bar A = \bigcap^\infty_{k=1} \, \bigcup^\infty_{m=k} \, A_{2m+1}
$$
so $\bar A$ is a dense $G_\delta$ in $[-2,2]$ of Lebesgue measure
zero.
We will the prove the following theorem:
\proclaim{Theorem 2.1} Suppose $E\notin\bar A$ is in $[-2,2]$.
Then,
$$
\lim_{n\to\infty} \frac{1}{n} \, \ln \| T(n,0)\| =
\frac{\beta}{2}\, . \tag 2.1
$$
For a.e.~$\alpha$, $H_\alpha$ has dense point spectrum in
$[-2,2]$ with eigenfunctions decaying as $e^{-\beta n/2}$.
\endproclaim
\remark{Remarks} 1. By ``eigenfunctions decaying as
$e^{-\beta n/2}$," we mean $\lim_{n\to\infty} \ln (|u(n)|^2
+ |u(n+1)|^2)^{1/2} / n = -\frac{\beta}{2}$.
2. Since $\varlimsup |V(n)|= \infty$, the results of
Simon-Spencer [\ss] imply $\sigma_{\text{\rom{ac}}}(H_\alpha)=
\emptyset$. Gordon [\gora] and Kirsch-Molchanov-Pastur [\kmp]
proved that for some potentials of Simon-Spencer type (where
the distances between the bumps are not too large), $H$ has
dense point spectrum for a.e.~boundary condition. Their methods
apply to the problem discussed here. Our method is different
and identifies the set $\bar A$ and the Lyapunov exponent
$\gamma (E) = \frac{\beta}{2}$.
3. A similar result holds if $V ([j^\beta]) = e^{\nu j^\mu}$
for any $\mu >0$ and $\beta >1$ (here $[j^\beta]$ is the
greatest integer less than $j^\beta$). Then, $\lim_{n\to\infty}
\frac{\ln \|T(n,0)\|}{n^\zeta}=\frac{\nu}{\mu+1}$ where $\zeta
= (\mu+1)/\beta$. Where we use Ruelle's result [\rue] in the
argument below, one instead uses its extension in [\ls].
\endremark
\demo{Proof} It obviously suffices to prove that for $E\notin
\bar A$, $\lim \frac{1}{n} \ln \| T(n,0) u_{\theta_0} \| =
\frac{\beta}{2}$ for $u_{\theta_0} = \mathbreak (\cos(\theta_0),
\sin(\theta_0))$ and at least two out of three values of
$\theta_0$, say, $\theta_0 =0$, $\frac{\pi}{4}$, and
$\frac{\pi}{2}$. Pick $\theta_0 =0$ and let $k$ be defined by
$E = 2\cos (k)$ and let $\theta(n), R(n)$ be the EFGP variables
for this value of $\theta_0$ and $E$.
Assume that for $j\geq j_0$,
$$
|\bar\theta (j^2)| \geq \exp(-j^{2/3}). \tag 2.2
$$
We then have that by (1.18), $R(n)^2$ is constant for $n=j^2 + 1,
\dots, (j+1)^2$ and jumps from $n=j^2$ to $n=j^2 +1$. By (1.16)
and (1.18),
$$\align
\frac{R(j^2 +1)^2}{R(j^2)^2} &\leq 1 + \frac{e^{\beta j}}
{|\sin k|} + \frac{e^{2\beta j}}{\sin^2 (k)} \\
&\geq 1 + \frac{\sin^2 \exp(-j^{2/3})}{\sin^2 (k)} \,
e^{2\beta j} - \frac{e^{\beta j}}{\sin(k)}
\endalign
$$
for $j\geq j_0$. From these inequalities and $\sum^m_{j=1}
j = \frac{m(m+1)}{2}$, one easily sees that $\lim_{n\to\infty}
\mathbreak \frac{\ln R(j^2 +1)}{j^2} = \frac{\beta}{2}$ and then
$$
\lim_{n\to\infty} \frac{\ln R(n)}{n} = \frac{\beta}{2}\, . \tag 2.3
$$
We need to examine (2.2). We will prove that at least one of
the following holds for $E,\theta_0$ fixed:
\roster
\item"\rom{(i)}" (2.2) holds; or
\item"\rom{(ii)}" $E\in\bar A$; or
\item"\rom{(iii)}"
$$
\sum_n \| T(n,0) u_{\theta_0}\|^2 < \infty. \tag 2.4
$$
\endroster
If we prove this and $E\notin\bar A$, then for each of $\theta_0
=0$, $\theta_0 = \frac{\pi}{4}$, and $\theta_0 = \frac{\pi}{2}$,
one of (i) or (iii) must hold. Since (2.4) can hold for at most
one $\theta_0$ (by constancy of the Wronskian), (2.2) must hold
for at least two $\theta_0$'s and so (2.3) holds for two
$\theta_0$'s, and thus (2.1) holds.
Once (2.1) holds, application of Ruelle's theorem [\rue] implies
that for $E\notin\bar A$, there exists an initial $u_E$ so that
$\lim \ln \|T(0,n) u_E\| /n = -\frac{\beta}{2}$, and then the
Simon-Wolff [\sw] method proves point spectrum for a.e.~$\alpha$
(see, e.g., [\dss,\svan]).
Thus, we need only prove that one of the three alternatives
(i)--(iii) above holds. Suppose neither (i) nor (ii) holds. We
will prove that (iii) holds.
Since (ii) is assumed false, there exists $j_0$ large so that
Lemma~2.2 holds and so that $E\notin\bar A_{2j-1}$ for $j\geq
j_0$. In particular, alternative (a) of Lemma~2.2 does not hold.
Suppose $j_1\geq j_0$ and $|\bar\theta (j^2_1)|\leq
\exp(-j^{2/3}_1)$. Since alternative (b) of Lemma~2.2 holds,
we can iterate and see that (2.5) holds for $j=j_0 + 1, \dots,
j_1$.
If alternative (i) fails, there are $j_1$'s going to infinity
with $|\bar\theta (j^2_1)| \leq \exp (-j^{2/3}_1)$, so (2.7)
holds for all $j\geq j_0$, and thus $|R(n)|\leq
Ce^{-(\alpha-\epsilon)n/4}$ so (2.4) holds. \qed
\enddemo
\proclaim{Lemma 2.2} There is a $j_0$ \rom(depending only on
$k$ and $\beta$\rom) so that if $j\geq j_0$ and $|\bar\theta
(j^2)|\leq \exp (-j^{2/3})$, then either
\roster
\item"\rom{(a)}" For some $\ell\in \{1, \dots, 2j-2\}$,
$|E-2\cos (\frac{\pi\ell}{2j-1})| \leq e^{-\sqrt{2j-1}}\, $,
or
\item"\rom{(b)}" $|\bar\theta (j-1)^2)| \leq \exp(-(j-1)^{2/3})$
and
$$
R(j^2) \leq e^{-\beta j/2} R((j-1)^2). \tag 2.5
$$
\endroster
\endproclaim
\demo{Proof} By (1.19),
$$
|u(j^2)| = \biggl| \frac{R(j^2)}{\sin k} \,
\sin(\bar\theta (j^2)) \biggr| \leq CR(j^2) \exp(-j^{2/3}),
\tag 2.6
$$
where $C$ will be used to indicate a constant depending only on
$k$. $C$ can vary from formula to formula!
Let $q=(j-1)^2$. If (recall $R(n) = R(j^2)$ if $q \frac{\pi}{2}$), we need
to deal with $\sqrt{1-E}+\epsilon$ instead of $\sqrt{1-E} -
\epsilon$. The net result is that
$$
\bigl| \ln R(y_k) - \ln R(x_k) - \sqrt{1-E}\, |Q_k|\, \bigr|
\leq \epsilon \, |Q_k| + C\ell_0 + C\, |Q_k|^{2/3}. \tag 3.6
$$
If we can show that (3.4) fails for large $k$, then for $y$
large,
$$
\bigl| \ln R(y) - \tfrac12 \sqrt{1-E}\, |y| \, \bigr| \leq
\tfrac12 \epsilon |y| + o(y). \tag 3.7
$$
So, since $\epsilon$ is arbitrary, we obtain the desired result.
Suppose (3.4) holds. Go back to $Q_{k-1} = (x_{k-1}, y_{k-1})$.
Again, for simplicity, suppose $\theta (y_{k-1}) \in (-\eta, \eta)$.
If $\theta(y_{k-1} - |Q_k|^{2/3}) \geq \eta_1$, then
$$
\ln R(y_{k-1} - |Q_k|^{2/3}) \leq \ln R(y_{k-1}) -
(\sqrt{1-E}- \epsilon)|Q_k|^{2/3}. \tag 3.8
$$
If $\theta(y_{k-1} - |Q_k|^{2/3}) \leq \eta_1$, then $\theta(x)
\leq -\eta_1$ for $x_{k-1} \leq x \leq y_{k-1} - |Q_k|^{2/3} -
\ell_0$, and (assuming $k$ is so large that $|Q_k| \geq 2
|Q_k|^{2/3}+\ell_0$) we conclude that (3.4) holds for $k-1$
replacing $k$. Moreover, $R(y_k) \leq R(x_k)\exp(-\frac12
\sqrt{1-E}\, |Q_k|)$ (again for $k$ large).
If (3.4) and (3.8) hold, we can smoothly cut off $u$ at $y_{k-1}
- \sqrt{Q_k}$ and $x_k + \sqrt{Q_k}$ and get a trial function for
$-\frac{d^2}{dx^2}+W_{L_k}$, and so we see that $|E-e_j (|L_k|)|
\leq e^{-\sqrt{Q_k}}$. As in the last section, we see that one
of the following holds:
\roster
\item"\rom{(1)}" $E\in \bar A$
\item"\rom{(2)}" (3.5) holds for all large $k$ (and so (3.7) holds)
\item"\rom{(3)}" $u\in L^2$.
\endroster
As explained at the start of the proof, this suffices. \qed
\enddemo
\vskip 0.3in
\flushpar {\bf \S 4. The Classically Allowed Region}
\vskip 0.1in
In proving Theorem~1.1, we will break up $[0,\infty)$ into three
regions where $V(x) \leq E-\epsilon_0$, where $V(x) \geq E -
\epsilon_0$, and where $|V(x)-E| \leq \epsilon_0$. Here
$\epsilon_0$ is a parameter we will take to zero eventually,
using the fact that we can show the contribution of the
$|V(x) -E| \leq \epsilon_0$ region to $\lim \frac{\ln R(x)}{x}$
is bounded by $C\epsilon_0$. In this section, we will control
the contribution of the classically allowed region where $V(x)
\leq E-\epsilon_0$. The goal will be to show that each
oscillation of $V$ contributes at most a constant $C$ to
$\ln R(x)$, so that, since $x^{-1} \#\text{ of oscillations}
\to 0$, the classically allowed region makes no contribution to
$\gamma$ (as it makes no contribution to the integral in
Theorem~1.1).
\proclaim{Theorem 4.1} Fix $0\leq A\leq B0$ because of the
$\epsilon_0$ cutoff and $E>0$. We claim that
\roster
\item"\rom{(a)}" If $|\theta - \eta(x)|<\epsilon$, then
$|\frac{1}{2k} V(x) \sin (2\theta) - k(x)| \leq D\epsilon$
\item"\rom{(b)}" $k-\frac{V(x)}{k} \sin^2 (\eta(x) -\epsilon)
\geq Y>0$ uniformly in $x$
\item"\rom{(c)}" $k-\frac{V(x)}{k} \sin^2 (\eta(x) + \epsilon)
\leq -Y<0$ uniformly in $x$.
\endroster
Here $D$ and $Y$ are fixed $\epsilon$ independent (but they are
$\epsilon_0$ dependent) non-zero constants. (a) holds by (5.2).
(b), (c) follow from the monotonicity of $\sin^2$ in $(0,
\frac{\pi}{2})$ and the condition $\epsilon \leq \epsilon_1$.
We claim in any interval where $V(x) \geq E+\epsilon_0$ and
$|x|$ is sufficiently large, as $x$ increases, once $x\in
(\eta(x) - \epsilon, \eta(x)+\epsilon)\equiv I_1$, it remains in
that interval. For $\frac{d}{dx} [\theta(x) - \eta(x)] \geq E -
Cx^{1-\alpha}$ at $\theta = \eta-\epsilon$ and $\leq -E +
Cx^{1-\alpha}$ at $\theta = \eta+\epsilon$. Similarly, once
$\theta$ leaves $(-\eta - \epsilon, \eta + \epsilon) \equiv I_2$,
it stays outside it; and in a finite distance $\ell_0$, it moves
from anywhere outside $I_2$ into $I_1$ (or the interval $(\pi +
\eta-\epsilon, \pi + \eta+\epsilon)$).
By mimicking the arguments in Section~3, we see that either
$E\in\bar A$ or $\| T(x,0) u_{\theta_0}\| \in L^2$ or else
$$
\lim_{x\to\infty} \frac{1}{x} \int_{\{y\mid V(y) \geq
E+\epsilon_0; 0\leq y \leq x\}} \biggl| \frac{d}{dy}\,
\ln R(y) - k(y) \biggr| \leq D\epsilon.
$$
Since we can take $\epsilon$ to zero and
$$
\frac{1}{x} \int_{\{y\mid V(y) \geq E+\epsilon_0;
0\leq y \leq x\}} k(y) \, dy = \frac{1}{2\pi}
\int_{\{1+\cos (y) \geq E + \epsilon_0; 0\leq y \leq 2\pi\}}
(1+\cos (y) - E)^{1/2}\, dy
$$
the theorem is proven. \qed
\enddemo
\vskip 0.3in
\flushpar {\bf \S 6. Putting It Together}
\vskip 0.1in
Here we will prove Theorem~1.1. Suppose $E\notin\bar A$ and
$\theta_0$ is such that $\| T(x,0) u_{\theta_0}\| \notin L^2$.
Let $R(x)$ solve (1.7) with $\theta(x) =\theta_0$. Fix
$\epsilon_0 <0$ and consider the three regions:
$$\align
Z(1)&: \{x\mid V(x) \leq E- \epsilon_0\} \\
Z(2) &: \{x\mid |V(x)-E| \leq \epsilon_0\} \\
Z(3) &: \{x\mid V(x) \geq E+\epsilon_0\}.
\endalign
$$
In Section~4, we proved that
$$
\frac{1}{x} \int_{Z(1)\cap \{y\leq x\}}
\biggl( \frac{d}{dy}\, \ln R(y) \biggr)\, dy \to 0.
$$
In Section~5, we proved that
$$\multline
\varlimsup \, \biggl| \frac{1}{x} \int_{Z(3)\cap \{y\leq x\}}
\biggl( \frac{d}{dy}\, \ln R(y)\biggr) \, dy - \frac{1}{2\pi}
\int_{\{y\mid 1+\cos(y) \geq E; 0\leq y \leq 2\pi\}}
(1-\cos(y) - E)^{1/2} \, dy \, \biggr| \\
\leq D_0 \epsilon_0
\endmultline
$$
for a constant $D_0$.
By (1.7), $|\frac{d}{dy}\ln R(y)|\leq\frac{2}{2k}$. Moreover, it
is clear that $\varlimsup \frac{1}{x} |Z(2)\cap \{y\leq x\}|
\leq D_1 \epsilon_0$ for some constant $D_1$.
Thus, we have
$$
\varlimsup \, \biggl| \frac{1}{x} \, [\ln R(x) - \ln R(0)] -
\frac{1}{2\pi} \int_{\{y\mid 1+\cos (y) \leq E; 0\leq y \leq 2\pi\}}
(1+\cos (y) - E)^{1/2} \, dy \, \biggr| \leq D_2 \epsilon_0
$$
with $D_2 = D_0 + \frac{D_1}{2k}$. Taking $\epsilon_0$ to zero,
we prove that
$$
\frac{1}{x}\, \ln R(x) \to \frac{1}{2\pi}
\int_{\{y\mid 1+\cos (y) \geq E; 0\leq y \leq 2\pi\}}
(1+\cos (y) - E)^{1/2} \, dy.
$$
Since at most one $\theta_0$ has $\| T(x,0) u_{\theta_0}\|
\in L^2$, we see that $\frac{1}{x} \ln \|T(x,0)\|$ has the
required limit. \qed
\vskip 0.3in
\flushpar{\bf Appendix: WKB Pr\"ufer Variables and Bounded
Transfer Matrices}
\vskip 0.1in
In this appendix, we will show how to use WKB-Pr\"ufer variables
to show for $E>1$, the transfer matrix for $\cos (x^\alpha)$
potentials is bounded. This is a result of Behncke [\beh] and
Stolz [\sto] whose proof is not unrelated. Their method is
basically a variation of parameters, and this appendix reiterates
the idea of [\kls] that modified Pr\"ufer variables are often a
suitable replacement for variation of parameters.
Recall the definition (1.13) for $\tilde R_w(x)$ and
$\tilde\theta_w(x)$. They obey
$$\align
\frac{d\theta_w}{dx} &= k(x) + \frac12 \, \frac{k'}{kx} \,
\sin (2\theta_2 (x)) \tag A.1 \\
\frac{d\ln \tilde R_w}{dx} &= -\frac{k'}{2k}\,
\cos (2\theta_w (x)). \tag A.2
\endalign
$$
Let $V(x) = \cos (x^\alpha)$, with $\frac12 < \alpha <1$ and
$E>1$. Then $k(x) = \sqrt{E-V(x)} > \sqrt{|E-1|}$ is bounded
away from zero. Moreover, we have for $j=0,1,2,\dots$ and
$|x|\geq 1$:
$$\align
\biggl| \frac{d^j k(x)}{dx^j}\biggr| &\leq
C_j (1+|x|)^{-j(1-\alpha)} \tag A.3 \\
\biggl| \frac{d^j}{dx^j} \biggl(\frac{k'}{k}\biggr) \biggr| &\leq
D_j (1+|x|)^{-(j+1)(1-\alpha)}. \tag A.4
\endalign
$$
In particular, for $x$ large, $\frac{d\theta}{dx} \geq
\sqrt{|E-1|} - D_0 (1+|x|)^{-(1-\alpha)} >0$. By (A.1) and
(A.3/A.4), we see
$$
\biggl| \frac{d^2}{dx^2}\, \theta_w (x) \biggr| \leq
x^{-(1-\alpha)}. \tag A.5
$$
Integrate (A.2) to get (where $x_0$ is picked so large that
$k(x) > \delta >0$ for $x>x_0$)
$$
\ln \tilde R_w (x) - \ln \tilde R_w (x_0) = \int^{x_0}_x -
\frac{k'}{4k} \, \frac{1}{\frac{d\theta_w}{dx}}\,
\frac{d}{dx}\, (\sin (2\theta_w(x)))\, dx
$$
and integrate by parts. By (A.4) and (A.5), the integrand
bounded by $(1+|x|)^{-2(1-\alpha)}$ is integrable, so
$\tilde R_w (x)$ is bounded.
\remark{Remarks} 1. One doesn't gain anything by iterating the
integration by parts because $\frac{d^j}{dx^j} \theta_w(x)$ only
falls as $(1+|x|)^{-2(1-\alpha)}$.
2. One also sees by integrating by parts that $\theta_w(x) -
\int^x_{x_0} k(y)\, dy$ has a limit, and so one can prove there
are WKB-type solutions.
3. By using higher-order modifications, it should be possible
to accommodate $0<\alpha \leq \frac12$.
4. All this proof requires, if one keeps track of the derivatives,
is that $V$ is $C^2$ and
\roster
\item"\rom{(i)}" $\sup_x V(x) = V_+ <\infty$, $\inf_x V(x) >-\infty$
\item"\rom{(ii)}" $V'(x) \to 0$ as $x\to\infty$
\item"\rom{(iii)}" $V'\in L^2$, $V''\in L^1$.
\endroster
One obtains a bounded transfer matrix if $E >V_+$.
5. The point of this is that bounded transfer matrices imply
purely a.c.~spectrum [\beh,\ls,\sto].
\endremark
\vskip 0.3in
\Refs
\endRefs
\vskip 0.1in
\item{\beh.} \ref{H.~Behncke}{Absolute continuity of Hamiltonians
with von Neumann-Wigner potentials II}{Manuscripta Math.}{71}{1991}
{163--181}
\gap
\item{\br.} G.~Birkhoff and G.C.~Rota, {\it{Ordinary Differential
Equations}}, 3rd ed., Wiley, New York, 1978.
\gap
\item{\dms.} \ref{R.~del Rio, N.~Makarov, and B.~Simon}{Operators
with singular continuous spectrum, II. Rank one operators}
{Commun.~Math.~Phys.}{165}{1994}{59--67}
\gap
\item{\dss.} \ref{R.~del Rio, B.~Simon, and G.~Stolz}{Stability
of spectral types for Sturm-Liouville operators}
{Math.~Research Lett.}{1}{1994}{437--450}
\gap
\item{\egg.} \ref{T.~Eggarter}{Some exact results on electron
energy levels in certain one-dimensional random potentials}
{Phys.~Rev.}{B5}{1972}{3863--3865}
\gap
\item{\gora.}\ref{A.Ya.~Gordon}{Deterministic potential with a
pure point spectrum}{Math.~Notes}{48}{1990}{1197--1203}
\gap
\item{\gor.} \ref{A.~Gordon}{Pure point spectrum under
1-parameter perturbations and instability of Anderson
localization}{Commun.~Math.~Phys.}{164}{1994}{489--505}
\gap
\item{\grp.} \ref{S.A.~Gredeskul and L.A.~Pastur}{Behavior of
the density of states in one-dimensional disordered systems near
the edges of the spectrum}{Theor.~Math.~Phys.}{23}{1975}{132--139}
\gap
\item{\kmp.} \ref{W.~Kirsch, S.A.~Molchanov, and L.A.~Pastur}
{One-dimensional Schr\"odinger operator with high potential
barriers}{Operator Theory: Advances and Applications}{57}{1992}
{163--170}
\gap
\item{\kls.} A.~Kiselev, Y.~Last, and B.~Simon, {\it{Modified
Pr\"ufer and EFGP transforms and the spectral analysis of
one-dimensional Schr\"odinger operators}}, preprint.
\gap
\item{\ls.} Y.~Last and B.~Simon, {\it{Eigenfunctions, transfer
matrices, and absolutely continuous spectrum of one-dimensional
Schr\"odinger operators}}, preprint.
\gap
\item{\pf.} L.~Pastur and A.~Figotin, {\it{Spectra of Random
and Almost-Periodic Operators}}, Springer, Berlin, 1992.
\gap
\item{\rue.} \ref{D.~Ruelle}{Ergodic theory of differentiable
dynamical systems}{Publ.~Math.~IHES}{50}{1979}{275--306}
\gap
\item{\svan.} B.~Simon, {\it{Spectral analysis and rank one
perturbations and applications}}, CRM Lecture Notes Vol.~8
(J.~Feldman, R.~Froese, L.~Rosen, eds.), pp.~109--149,
Amer.~Math.~Soc., Providence, RI, 1995.
\gap
\item{\ss.} \ref{B.~Simon and T.~Spencer}{Trace class
perturbations and the absence of absolutely continuous
spectrum}{Commun.~Math.~Phys.}{125}{189}{113--126}
\gap
\item{\sw.} \ref{B.~Simon and T.~Wolff}{Singular continuous
spectrum under rank one perturbations and localization for
random Hamiltonians}{Commun.~Pure Appl.~Math.}{39}{1986}
{75--90}
\gap
\item{\sz.} \ref{B.~Simon and Y.F.~Zhu}{The Lyapunov exponents
for Schr\"odinger operators with slowly oscillating potentials}
{J.~Funct.~Anal.}{140}{1996}{541--556}
\gap
\item{\sto.} \ref{G.~Stolz}{Bounded solutions and absolute
continuity of Sturm-Liouville operators}{J. Math.~Anal.~Appl.}
{169}{1992}{210--228}
\gap
\enddocument