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\begin{document}
\title{The absolutely continuous spectrum of one-dimensional
Schr\"odinger operators with decaying potentials}
\author{Christian Remling}
\maketitle
\begin{center}
(submitted to {\it Comm.\ Math.\ Phys.})
\end{center}
\vspace{0.5cm}
\noindent
Universit\"at Osnabr\"uck,
Fachbereich Mathematik/Informatik,
49069 Osnabr\"uck, GERMANY
\\[0.2cm]
E-mail: cremling@mathematik.uni-osnabrueck.de\\[0.3cm]
\begin{abstract}
We investigate one-dimensional Schr\"odinger operators
with asymptotically small potentials. It will follow
from our results that if $|V(x)|\le C(1+x)^{-\alpha}$
with $\alpha > 1/2$, then $\Sigma_{ac}=(0,\infty)$
is an essential support of the absolutely continuous
part of the spectral measure. We also prove that
if $C:=\limsup_{x\to\infty}x\, |V(x)|<\infty$, then the
spectrum is purely absolutely continuous on $((2C/\pi)^2,\infty)$.
These results are optimal.
\end{abstract}
%\newpage
\section{Introduction}
In this paper, I am interested in one-dimensional
Schr\"odinger equations,
\begin{equation}
\label{se}
-y''(x)+V(x)y(x)=Ey(x),
\end{equation}
with asymptotically small potentials $V(x)$. We will treat
only the half-line problem $x\in [0,\infty)$ explicitly
(of course, the results below extend easily to whole-line
problems). So, we are interested in the spectral properties
of the operators $H_{\beta}=-\frac{d^2}{dx^2}+V(x)$ on
$L_2(0,\infty)$. The index $\beta\in [0,\pi)$ refers
to the boundary condition $y(0)\cos\beta + y'(0)\sin\beta=0$.
Although the emphasis will be on the continuous case,
we will also occasionally discuss the discrete analogue
of (\ref{se}),
\[
y(n-1)+y(n+1)+V(n)y(n)=Ey(n).
\]
The properties of the corresponding classical system very
naturally lead to the question of whether suitable smallness
assumptions on $V(x)$ at large $x$ imply absence of singular spectrum
on $(0,\infty)$ or, at least, existence of absolutely
continuous spectrum. Indeed, one can make the elementary
remark that the spectrum is purely absolutely continuous
on $(0,\infty)$ if $V\in L_1$. On the other hand, the
classical von Neumann-Wigner example \cite{vNW} shows that
potentials $V(x)=O(1/x)$ can have embedded eigenvalues.
Moreover, the constructions of \cite{Na,Spp} show that there are
potentials with decay arbitrarily close to $O(1/x)$ and
dense point spectrum in $(0,\infty)$.
These phenomena cannot occur if one prevents the potential
from oscillating too heavily. In this case, one can use
WKB methods to prove absence of singular spectrum. The
first result in this spirit was obtained in
\cite{WM}; for generalizations, see \cite{B1,B2}.
Recently, Kiselev proved that $\Sigma_{ac}=
(0,\infty)$ is an essential support of the absolutely
continuous part of the spectral measure if $V(x)=O(x^{-\alpha})$
with $\alpha> 3/4$ \cite{Kis1}. (Recall that,
by definition, $S$ is called an essential support of the measure
$\mu$ if $\mu({\Bbb R}\setminus S)=0$ and $\mu(T)>0$
for every subset $T\subset S$ of positive Lebesgue measure.)
He subsequently weakened this
assumption to $\alpha> 2/3$ \cite{Kis2}. Later, Molchanov
found an elegant alternative proof of the same result
\cite{Mop}.
These statements are remarkable in that,
in striking contrast to the WKB methods,
no conditions other than smallness conditions are imposed on
the potential. However, the question of the optimal exponent
on the power scale
was left open. The work on decaying random
potentials \cite{Del,Deletal,KLS,KU,S} has shown that there
are potentials $V(x)=O(x^{-1/2})$ with purely singular spectrum.
(A method of constructing {\it non-random}
power-decaying potentials with
pure point spectrum was recently discovered by myself \cite{Rempp}.)
So the critical exponent has to lie in the range
$[1/2,2/3]$.
Our first aim in this paper is to solve this problem.
We will prove:
\begin{Theorem}
\label{T11}
Suppose there exists an increasing sequence $x_n\to\infty$,
such that $\sum_{n=1}^{\infty}\|V\chi_{(x_{n-1},x_n)}\|_2<\infty$ and
$\sup_n \|V\chi_{(x_{n-1},x_n)}\|_1
\|V\chi_{(x_{n-1},x_n)}\|_2^N < \infty$
for some $N\in {\Bbb N}$. Then $\Sigma_{ac}=(0,\infty)$
is an essential support of the absolutely continuous
part of the spectral measure.
\end{Theorem}
As a corollary, we get the following sharp result:
\begin{Theorem}
\label{T12}
If $|V(x)|\le C(1+x)^{-\alpha}$ with $\alpha > 1/2$, then
$\Sigma_{ac}=(0,\infty)$ is an essential support of the
absolutely continuous part of the spectral measure.
\end{Theorem}
{\it Proof.} Let $x_n=n^{2/(2\alpha -1)}$ (say). Theorem
\ref{T11} applies. $\Box$
Note that the conclusion $\Sigma_{ac}=(0,\infty)$ is equivalent
to each of the following two statements:
1. The absolutely continuous spectrum satisfies
$\sigma_{ac}=[0,\infty)$, and, moreover, $H_{\beta}$ is
purely absolutely continuous on $(0,\infty)$ for almost
every boundary condition $\beta\in [0,\pi)$.
2. The absolutely continuous parts of $H^{(0)}_{\beta}=-d^2/dx^2$
and $H_{\beta'}=-d^2/dx^2 +V(x)$ are unitarily equivalent.
For the first statement, this follows from \cite{dRSS}.
Clearly, Theorem \ref{T11} is considerably more general
than its corollary, Theorem \ref{T12}. In fact, the first
condition is always satisfied for sufficiently large $x_n$
if $V\in L_2$. Then the second condition imposes a certain
asymptotic relationship between $L_1$ and $L_2$ norms.
On the other hand, Theorem \ref{T11} is not strong enough to deal with
general $L_p$ potentials. That is to say, for any $p>1$,
there are functions $V\in L_p\cap L_2$ which do not satisfy
the hypotheses of Theorem \ref{T11}.
The proof of Theorem \ref{T11} will be given in the
next section. We will use ideas from Kiselev's and
Molchanov's proofs of the $2/3$ result as well as
a number of new ideas.
Actually, our methods will yield several extensions
and generalizations of Theorem \ref{T11} with relatively
little extra effort. In fact, we will prove in Sect.\ 3
that the assumptions of Theorem \ref{T11} also imply that
for almost every $E\in (0,\infty)$, the solutions satisfy
the WKB asymptotic formulae (= Theorem \ref{T331}).
In Sect.\ 4 we will establish an analogue
of Theorem \ref{T11} for the discrete Schr\"odinger equation
(= Theorem \ref{T41}).
Finally, in Sect.\ 5, we will use the generalized Pr\"ufer transformation
introduced in \cite{KRS} to prove a result on perturbations of
general Schr\"odinger operators $H_0=-d^2/dx^2+U$ (where not
necessarily $U=0$).
The proofs of all these extensions are relatively minor
variations on the proof of Theorem \ref{T11}.
Therefore, we will give a thorough discussion of
this proof in Sect.\ 2, in order to present the ideas
as clearly as possible. The representation in Sections
3--5 will then be rather sketchy.
Christ and Kiselev \cite{ChrKis} have independently developed
a completely different approach to these problems, which gives
very similar results. In particular,
their results also imply Theorem \ref{T12}.
We will also prove a new result on the absence of singular
spectrum, assuming only decay conditions. Namely, on
the power scale, we can improve the elementary result
on $L_1$ potentials to
\begin{Theorem}
\label{T13}
If $C:=\limsup_{x\to\infty} x\, |V(x)|<\infty$,
then $H_{\beta}$ is purely
absolutely continuous on $((2C/\pi)^2,\infty)$. In particular,
if $V(x)=o(1/x)$, then $H_{\beta}$ is purely absolutely
continuous on $(0,\infty)$.
\end{Theorem}
The point of this result is
the absence of singular
{\it continuous}
spectrum. That $E=(2C/\pi)^2$ is a (sharp) bound for
possible positive eigenvalues appears already in
\cite[Section 3.2]{EK}. See also \cite[Theorem 4.1]{KLS}
for further information on
the point spectrum.
We will prove Theorem \ref{T13} in Sect.\ 6. This proof is not
hard; it relies on a classical result on the divergence
set of a Fourier series.
Molchanov has informed me \cite{Mop} that he can prove absence of
embedded singular spectrum under the stronger assumption
$V(x)=O(x^{-1}\ln^{-\epsilon}x)$. His proof is based on ideas
developed by himself in \cite{Mo}.
The proof of Theorem \ref{T13} will automatically yield
the following result, which seems to be of independent
interest.
\begin{Theorem}
\label{T14}
If $|V(x)|\le Cx^{-\alpha}$ for all large enough $x$,
then there is a set $S\subset (0,\infty)$ of Hausdorff dimension
$\dim S\le 4(1-\alpha)$, such that for every boundary condition
$\beta\in [0,\pi)$, the singular part of the spectral measure
(restricted to $(0,\infty)$) is
supported on the set $S$: $\rho_s^{\beta}((0,\infty)\setminus
S)=0$.
\end{Theorem}
We will also briefly mention discrete versions of these
results at the end of Sect.\ 6.
Unfortunately, we do not know of any potentials $V(x)=O(x^{-\alpha}),
\alpha > 1/2$, with embedded singular continuous spectrum.
While potentials
of this type presumably exist, an explicit construction seems to
be extremely difficult. It appears, however,
that such examples are needed to gain further insight into the
issues dealt with in Theorems \ref{T13}, \ref{T14}.
Note also that in recently constructed
examples with embedded singular continuous spectrum \cite{Mo,Remsc},
$\Sigma_{ac}$ does not have full measure in the absolutely
continuous spectrum $\sigma_{ac}$.
Some of the results of this paper (along with some of the
results of \cite{ChrKis}) have been announced in
\cite{CKR}.
{\bf Acknowledgments:}
I would like to thank A.\ Kiselev for stimulating discussions
on most topics of this paper,
S.\ Molchanov for showing me his alternate proof of the
result of \cite{Kis2}, and B.\ Simon for useful comments on
an earlier version of this work.
I would like to thank the hospitality
of Caltech, where most of this work was done.
I would like to thank the Deutsche Forschungsgemeinschaft
for financial support.
\section{Proof of Theorem 1.1}
Fix once and for all a boundary $\beta \in [0,\pi)$ at
$x=0$. For $E>0$, let $y(x,E)$ be the solution of (\ref{se}) with
the initial values $y(0,E)=-\sin\beta, y'(0,E)=\cos\beta$.
It will be convenient to work with $k=\sqrt{E}$; using
the physicist's notation, we will denote the solution $y$
from above by $y(x,k)$ (instead of the more careful, but
too clumsy notation $\tilde{y}(x,k)=y(x,k^2)$).
Similar conventions apply to other quantities.
The modified Pr\"ufer variables $R(x,k),\psi(x,k)$ are defined by
the formulae $y=R\sin\psi/2, y'=kR\cos\psi/2$ and by requiring
that $R>0$ and $\psi$ be continuous in $x$. $R,\psi$ obey the
equations
\[
(\ln R)'=\frac{V}{2k}\sin\psi,\quad \psi'=
2k-\frac{V}{k}+\frac{V}{k}\cos\psi.
\]
The main part of the proof of Theorem 1.1 will
consist of a careful
analysis of the integrated form of these equations:
\begin{eqnarray}
\label{eqr}
2k(\ln R(y,k)-\ln R(x,k)) & = & \int_x^y V(t) \sin\psi(t,k)\, dt, \\
\label{eqpsi}
\psi(x,k) & = & \omega(x,k)+\theta(x,k).
\end{eqnarray}
Here,
\begin{equation}
\label{om}
\omega(x,k):= 2kx -\frac{1}{k}\int_0^x V(t)\, dt
\end{equation}
is the ``good'' part of $\psi$ (this will be made precise
in Proposition \ref{P22} below), and $\theta(x,k):=
\psi(0,k)+k^{-1}\int_0^x V(t)\cos\psi(t,k)\, dt$ will be
treated as a perturbation. The crucial property of $\theta$
is that its derivative is a small, oscillatory function:
\begin{equation}
\label{theta}
\frac{d\theta}{dx}= \frac{1}{k} V(x)\cos\psi(x,k).
\end{equation}
We will use the following relation between $\Sigma_{ac}$ and
the asymptotic behavior of the solutions of (\ref{se}). This
result is not new (see also \cite{LS} for
related results). However, the proof is easy, so
we give it.
\begin{Proposition}[\cite{Mo}]
\label{P21}
Let $x_n\to\infty$ be a fixed sequence. Then
\[
S:=\{ E=k^2: \limsup_{n\to\infty} R(x_n,k) <\infty \}
\subset \Sigma_{ac}.
\]
\end{Proposition}
{\it Proof.} Let $r^2(x,E)=y^2(x,E)
+y'^2(x,E)$. The measures $d\rho_x(E)=(\pi r^2(x,E))^{-1}
dE$ converge weakly (i.e., when smeared with continuous
functions of compact support) to the spectral measure $d\rho(E)$
as $x\to\infty$ \cite{Car,P}.
Let $f_n(E):=\min\{ 1,(\pi r^2(x_n,E))^{-1}\}$.
For fixed $[a,b]\subset (0,\infty)$, the sequence $f_n(\cdot )$
is bounded in $L_2(a,b)$, so there exists a weakly convergent
subsequence (which we denote again by $f_n$). That is, there
exists $f\in L_2(a,b)$ so that $\int_a^b f_n(E) g(E)\,dE
\to \int_a^b f(E) g(E)\, dE$ for all $g\in L_2(a,b)$.
Thus, if $\rho(\{E_0\pm\epsilon\})=0$,
\begin{eqnarray*}
\rho(E_0-\epsilon,E_0+\epsilon)
& = & \frac{1}{\pi}\lim_{n\to\infty}\int_
{E_0-\epsilon}^{E_0+\epsilon} \frac{dE}{r^2(x_n,E)}\\
&\ge & \lim_{n\to\infty}
\int_{E_0-\epsilon}^{E_0+\epsilon} f_n(E)\, dE
=\int_{E_0-\epsilon}^{E_0+\epsilon} f(E)\, dE.
\end{eqnarray*}
Dividing by $2\epsilon$ and letting $\epsilon\to 0$ shows that
$d\rho(E_0)/dE \ge f(E_0)$ for almost every
$E_0\in (a,b)$. Notice that $r(x,E)\le \max\{ E^{1/2},1\} R(x,E)$.
Hence $\liminf_{n\to\infty} f_n(E)>0$ for all $E\in S$, and thus also
$f>0$ almost everywhere on $S\cap (a,b)$. $\Box$
\begin{Proposition}[\cite{Kis2}]
\label{P22}
Assume that $V\in L_p$ for some $p\in [1,\infty)$, and
fix $[a,b]\subset (0,\infty)$. Then there is a constant
$C=C(a,b)$ so that
\[
\int_a^b \left|\int_0^{\infty} f(x)e^{i\omega(x,k)}\, dx
\right|^2\, dk \le C \int_0^{\infty} |f(x)|^2\, dx
\]
for all $f\in L_1\cap L_2$.
\end{Proposition}
{\it Proof.} Pick $g\in C_0^{\infty}$ with $0\le g\le 1$ and
$g=1$ on $(a,b)$. We want to estimate the quantity
\begin{eqnarray*}
&& \int dk\, g(k)\int_0^{\infty}dx\, f(x) \int_0^{\infty}dy\,
\overline{f(y)} e^{2ik(x-y)}e^{\frac{i}{k}\int_x^y V(t)\, dt}\\
&=& \int_0^{\infty}dx\, f(x) \int_0^{\infty}dy\,
\overline{f(y)} \int dk\, e^{2ik(x-y)}
g(k) e^{\frac{i}{k}\int_x^y V(t)\, dt}.
\end{eqnarray*}
In the $k$-integral, we integrate by parts
$[2p]+1$ times ($[a]$ is the largest integer $\le a$), integrating
the factor $e^{2ik(x-y)}$ and differentiating the rest.
Since, by H\"older's inequality,
\[
\left| \int_x^y V(t)\, dt \right| \le
C |x-y|^{1-1/p},
\]
we gain a factor $|x-y|^{-1/p}$ with each integration by
parts. So, it suffices to estimate integrals of the form
\begin{equation}
\label{a1}
\int_0^{\infty}dx\int_0^{\infty} dy\, |f(x)f(y)| K(x,y),
\end{equation}
where the kernel $K$ is non-negative and can be bounded by
\[
K(x,y)\le \min\left\{ C_0, \frac{C_1}{(x-y)^2} \right\}
\le \frac{C}{1+(x-y)^2}.
\]
Now, the Cauchy-Schwarz inequality
(in $L_2({\Bbb R}^2)$) yields the desired bound
on (\ref{a1}):
\[
\left( \int_0^{\infty} dx \int_0^{\infty} dy\, |f(x)|^2 K(x,y)
\int_0^{\infty} dx \int_0^{\infty} dy\, |f(y)|^2 K(x,y)
\right)^{1/2} \le C\pi \|f\|^2_2.
\]
$\Box$
We now have all the tools for the {\it proof of Theorem 1.1.}
First of all, note that we have $\sigma_{ess}=
[0,\infty)$ (see, e.g., \cite[Theorem 15.1]{WMLN}).
So we only need to show $(0,\infty)\subset \Sigma_{ac}$.
Let $x_n$ be as in the hypothesis. We will show that
\[
\sum_{n=1}^{\infty}|\ln R(x_n,k)-\ln R(x_{n-1},k)| < \infty
\quad \mbox{for a.e. } k>0.
\]
Then the assertion will follow by Proposition \ref{P21}.
Of course, we may restrict our attention to $k\in (a,b)$,
where $[a,b]\subset (0,\infty)$ is fixed, but otherwise
arbitrary. According to
the Pr\"ufer equations (\ref{eqr}), (\ref{eqpsi}),
it thus suffices to show that
\begin{equation}
\label{s3a}
\sum_{n=1}^{\infty}\left| \int_{x_{n-1}}^{x_n} V(x)e^{i\omega(x,k)}
e^{i\theta(x,k)}\, dx\right| < \infty \quad
\mbox{for a.e. } k\in (a,b).
\end{equation}
We integrate by parts; also, since $k$ is fixed in this and
in subsequent calculations, we will usually drop this argument.
Using (\ref{eqpsi}) and (\ref{theta}), we get
\begin{eqnarray*}
&&\int_{x_{n-1}}^{x_n} V(x)e^{i\omega(x)}
e^{i\theta(x)}\, dx = e^{i\theta(x_n)}
\int_{x_{n-1}}^{x_n} V(x)e^{i\omega(x)}
\, dx\\
&& -\frac{i}{k} \int_{x_{n-1}}^{x_n} dx\, V(x)\cos(\omega(x)
+\theta(x))e^{i\theta(x)} \int_{x_{n-1}}^x dt\, V(t) e^{i\omega(t)}.
\end{eqnarray*}
The first summand on the right-hand side (call it $S$) has
already the desired properties. In fact, the Cauchy-Schwarz
inequality together with Proposition \ref{P22} imply
\begin{eqnarray*}
\int_a^b dk\, |S(n,k)| & \le & (b-a)^{1/2}
\left( \int_a^b dk\, \left| \int_{x_{n-1}}^{x_n} dx\,
V(x) e^{i\omega(x,k)}\right|^2 \right)^{1/2} \\
& \le & C \|V_n\|_2
\end{eqnarray*}
(using the notation $V_n=V\chi_{(x_{n-1},x_n)}$).
By hypothesis, this is summable, hence indeed $\sum_n |S(n,k)|
<\infty$ for almost every $k\in (a,b)$ by monotone convergence.
Using $\cos\psi=(e^{i\psi}+e^{-i\psi})/2$, we thus see
that it suffices to prove that sums of the form
\begin{equation}
\label{int1}
\sum_{n=1}^{\infty} \left| \int_{x_{n-1}}^{x_n} dx\,
V(x) e^{i\sigma \omega(x)}e^{im\theta(x)}\int_{x_{n-1}}^x
dt\, V(t) e^{i\omega(t)} \right|,
\end{equation}
with $\sigma\in \{ -1,1\} , m\in {\Bbb N}_0$, converge for almost
every $k\in (a,b)$. To handle these terms, we will use an
iterative procedure. Rather than formulate immediately the
basic step in a general setting,
we will discuss the first step in detail and only then
give the general result (see Lemma \ref{L21} below).
We may
assume that $\|V_n\|_2 >0$ for all $n$.
Define $N_n\in {\Bbb N}$ by $N_n=\max \{1,[1/\|V_n\|_2]\}$.
Since $\|V_n\|_2 \to 0$, there is an $n_0\in {\Bbb N}$ so that
\begin{equation}
\label{b2}
\frac{1}{2} \le \|V_n\|_2 N_n \le 1 \quad\quad \forall n\ge n_0.
\end{equation}
We subdivide the interval $[x_{n-1},x_n]$ into $N_n$ subintervals:
For $n\in {\Bbb N}$,
pick numbers $y_1(n,l)$ such that $x_{n-1}=y_1(n,0)0$, there
are solutions $y,\overline{y}$ of the Schr\"odinger
equation (\ref{se}) with the asymptotic form
\[
\left( \begin{array}{c} y(x,E) \\ y'(x,E) \end{array} \right)
= \left( \left( \begin{array}{c} 1 \\ ik \end{array} \right)
+o(1) \right)
e^{i\omega(x,E)/2}
\quad (x\to\infty).
\]
\end{Theorem}
{\it Remark.} The term ``WKB'' is justified because if $V\to 0$, then
$\omega(x)/2=kx-(2k)^{-1}\int_0^x V(t)\, dt
= \int_0^x \sqrt{E-V(t)}\, dt +c+o(1)$.\\
{\it Proof.} First of all, we observe that the assumptions
of Theorem \ref{T11} imply that $V\in L_p$ for some $p<2$.
To prove this, let $p_0=1$, $p_{n+1}=1+(p_n/2)$, and note
that by the Cauchy-Schwarz inequality
\begin{equation}
\label{s3cs}
\|f\|_{p_{n+1}}^{2p_{n+1}} \le \|f\|_2^2\, \|f\|_{p_n}^{p_n}.
\end{equation}
Moreover, we have that $p_n\in (1,2)$ for all $n\in {\Bbb N}$.
By hypothesis, there exists $r\in {\Bbb N}$ so that
\[
\|V_m\|_1\|V_m\|_2^{2+2^2+\ldots +2^r} \le C \quad \forall m.
\]
Now, repeatedly applying (\ref{s3cs}) with $f=V_m$, we get
\begin{eqnarray*}
C & \ge & \|V_m\|_1\|V_m\|_2^{2+2^2+\ldots +2^r} \\
& \ge & \|V_m\|_{p_1}^{2p_1}\|V_m\|_2^{2^2+2^3+\ldots +2^r}
\ge \ldots \\
& \ge & \|V_m\|_{p_r}^{2^r p_r} \ge \|V_m\|_{p_{r+1}}^{2^{r+1}p_{r+1}}
\|V_m\|_2^{-2^{r+1}},
\end{eqnarray*}
hence $\|V_m\|_{p_{r+1}}^{p_{r+1}} \le \tilde{C} \|V_m\|_2$.
This shows $V\in L_{p_{r+1}}$, as claimed.
It follows from (\ref{s3a}) that $\lim_{n\to\infty}
R(x_n,k)$ and $\lim_{n\to\infty} \theta(x_n,k)$ already
exist for almost every $k>0$. We will show that the limits
$\lim_{x\to\infty}R(x,k), \theta(x,k)$ exist, too. It is
easy to see that this implies the claimed asymptotic
formulae.
Thus it suffices
to show that
\begin{equation}
\label{s3max}
\lim_{n\to\infty}\max_{\xi\in [x_{n-1},x_n]} \left| \int_{x_{n-1}}^{\xi}
dx\, V(x) e^{i\omega(x,k)} e^{i\theta(x,k)} \right|=0
\end{equation}
for almost every $k$.
We need one additional tool, namely a norm estimate on the
maximal function
\[
M_n(k)=\max_{\xi\in [x_{n-1},x_n]} \left| \int_{x_{n-1}}^{\xi}
dx\, V(x) e^{i\omega(x,k)} \right| .
\]
Fix $p<2$ such that $V\in L_p$, and let $q$ be the
conjugate index (i.e. $1/p+1/q=1$). Then it follows
by interpolation (see \cite[Theorem 2.1]{Kis2}) that
\begin{equation}
\label{s3normest}
\left( \int_a^b M_n(k)^q\, dk \right)^{1/q} \le C
\left( \int_{x_{n-1}}^{x_n}
|V(x)|^p\, dx \right)^{1/p}.
\end{equation}
Now Theorem \ref{T331} is proved by basically repeating the
steps of the proof of Theorem \ref{T11}.
Pick a function $\xi(k)$, so that
the maximum in (\ref{s3max}) is attained for $\xi=\xi(k)$.
Integration by parts shows that we can estimate (\ref{s3max}) by
\begin{equation}
\label{s3a2}
\left|\int_{x_{n-1}}^{\xi(k)} dx\, V(x) e^{i\omega(x)}
\right| + \frac{1}{k} \left| \int_{x_{n-1}}^{\xi(k)} dx\, V(x)
\cos\psi(x) e^{i\theta(x)} \int_{x_{n-1}}^x dt\, V(t)
e^{i\omega(t)}\right|.
\end{equation}
Call the first term $S(n,k)$. Then,
by (\ref{s3normest}), $\int_a^b S(n,k)^q\, dk\le
C\|V_n\|_p^q$ which is summable because of $q>p$. Thus
$S(n,k)\in l_q$ for almost every $k\in (a,b)$, and, in
particular, $\lim_{n\to\infty}S(n,k)=0$ for these $k$,
as desired.
We now study the second term
of (\ref{s3a2}). The fact that the upper limit is
$\xi(k)$ instead of $x_n$ forces a slight modification
of the reasoning of Sect.\ 2. However, since we only need
to show convergence to zero (rather than summability), the
argument becomes, if anything, easier. Let $y_1(n,l)$ be as in Sect.\ 2,
and define $L_1=L_1(n,k)$ by $y_1(n,L_1-1)<\xi(k)\le
y_1(n,L_1)$. Then we have to analyze expressions of the
form
\begin{eqnarray*}
&& \sum_{l=1}^{L_1-1}\left| \int_{y_1(n,l-1)}^{y_1(n,l)}
dx\, V(x) e^{i\sigma\omega(x)}e^{im\theta(x)}
\int_{x_{n-1}}^x dt\, V(t) e^{i\omega(t)}\right| + \\
&& \left| \int_{y_1(n,L_1-1)}^{\xi(k)}
dx\, V(x) e^{i\sigma\omega(x)}e^{im\theta(x)}
\int_{x_{n-1}}^x dt\, V(t) e^{i\omega(t)}\right| .
\end{eqnarray*}
The first contribution can obviously be estimated by
the corresponding sum over the whole range $l=1,\ldots,N_n$,
and this term has already been dealt with in Sect.\ 2. It
remains to investigate the second term.
Proceed as in Sect.\ 2; again, we need to control terms of three
different types:
\begin{equation}
\label{s3a3}
\left| \int_{x_{n-1}}^{y_1(n,L_1-1)} dx\, V(x) e^{i\omega(x)}\right|
\left| \int_{y_1(n,L_1-1)}^{\xi(k)} dx\, V(x) e^{i\omega(x)}\right|,
\end{equation}
\begin{equation}
\label{s3a4}
\left| \int_{x_{n-1}}^{y_1(n,L_1-1)} dx\, V(x) e^{i\omega(x)}\right|
\left| \int_{y_1(n,L_1-1)}^{\xi(k)} dx\, V(x) e^{i\sigma_1\omega(x)}
e^{im'\theta(x)} \int_{y_1(n,L_1-1)}^x dt\, V(t)
e^{i\sigma_2\omega(t)}\right|,
\end{equation}
\begin{equation}
\label{s3a5}
\left| \int_{y_1(n,L_1-1)}^{\xi(k)} dx\, V(x) e^{i\sigma\omega(x)}
e^{im\theta(x)} \int_{y_1(n,L_1-1)}^x dt\, V(t)
e^{i\omega(t)}\right|.
\end{equation}
As above, it is easy to see, using (\ref{s3normest}), that
(\ref{s3a3}) and the first integral of (\ref{s3a4}) tend to
zero for almost every $k\in (a,b)$. Thus it suffices to show
that expressions of the form of (\ref{s3a5}) go
to zero for almost every $k$.
Applying this whole argument $j$ times, we see that it is enough
to prove that
\begin{equation}
\label{s3a6}
\int_{y_j(n,L_1,\ldots,L_{j-1},L_j-1)}^{\xi(k)} dx\,
V(x) e^{i\sigma_1\omega(x)}
e^{im\theta(x)} \int_{y_j(n,L_1,\ldots,L_{j-1},L_j-1)}^x dt\, V(t)
e^{i\sigma_2\omega(t)}
\end{equation}
tends to zero
for almost every $k\in (a,b)$. Here, the $L_i=L_i(n,k)$ are defined in
a way analogous to the definition of $L_1$. To conclude the
proof, we note that (\ref{s3a6}) can be estimated by
\begin{eqnarray*}
&&\int_{y_j(n,L_1,\ldots,L_{j-1},L_j-1)}^{\xi(k)} dx\,
|V(x)|
\left| \int_{y_j(n,L_1,\ldots,L_{j-1},L_j-1)}^x dt\, V(t)
e^{i\omega(t)}\right|\\
& & \le
2 M_n(k)
\max_{l_1,\ldots,l_j}
\int_{y_j(n,l_1,\ldots,l_{j-1},l_j-1)}^{y_j(n,l_1,\ldots,l_{j-1},l_j)}
dx\, |V(x)|.
\end{eqnarray*}
We know already that $M_n$ tends to zero for almost every
$k$, and the second
factor is equal to $\|V_n\|_1 N_n^{-j}$ by (\ref{n1}). This
expression is bounded,
provided $j\ge N$ .
$\Box$
{\it Remarks.} 1. Note that we could afford using several crude estimates in
this proof. This is due to the fact that we needed to establish
only convergence to zero, while in the preceding section, we
proved that a closely related quantity is actually summable.
2. In particular, Theorem \ref{T331} implies that for almost every
$E>0$, the Schr\"odinger equation (\ref{se}) has only bounded
solutions. This provides a way of proving
Theorem \ref{T11} without using
Proposition \ref{P21} \cite{Sbdd,Stolz}.
3. The maximal function estimate (\ref{s3normest}) can also be
used to control what we called $M_n$ in Sect.\ 2.
\section{The discrete case}
In this section, we want to obtain analogues of Theorems
\ref{T11}, \ref{T331} for the discrete Schr\"odinger equation
\begin{equation}
\label{dse}
y(n-1)+y(n+1)+V(n)y(n)=Ey(n)\quad (n\ge 1).
\end{equation}
The corresponding operator on $l_2({\Bbb N})$ is given by
\[
(Hy)(n)=\left\{ \begin{array}{lr} y(2)+V(1)y(1) & (n=1) \\
y(n-1)+y(n+1)+V(n)y(n) & (n\ge 2) \end{array} \right. .
\]
In this formulation, the role of the boundary condition is
now taken by the parameter $V(1)$.
We will again work with Pr\"ufer type variables
(cf.\ \cite{KLS,KRS}): So,
let $y(n,E)$ be the solution of (\ref{dse}) with
$y(0,E)=0, y(1,E)=1$. For $E\in (-2,2)$, write
$E=2\cos k$ with $k\in (0,\pi)$. Then, if $V\equiv 0$,
the motion of the vector
\[
Y(n,k)=\frac{1}{\sin k} \left( \begin{array}{cc} \sin k & 0\\
-\cos k & 1 \end{array} \right) \left( \begin{array}{c}
y(n-1,k) \\ y(n,k) \end{array} \right)
\]
is simply rotation by $k$. Thus it is natural to
introduce Pr\"ufer variables $R,\psi$ by writing
\[
Y(n,k)=R(n,k) \left( \begin{array}{c} \sin((\psi(n,k)/2)-k) \\
\cos((\psi(n,k)/2)-k) \end{array} \right).
\]
We need not worry about the non-uniqueness of $\psi$, since
only the value modulo $2\pi$ will matter in the sequel.
$R,\psi$ obey the equations
\begin{eqnarray}
\label{eqrdiscr}
\frac{R(n+1,k)^2}{R(n,k)^2} & = & 1 - \frac{V(n)}{\sin k}
\sin\psi(n,k) + \frac{V(n)^2}{\sin^2 k} \sin^2 \frac{\psi(n,k)}{2},\\
\label{eqpsidiscr}
\cot\left( \frac{\psi(n+1,k)}{2}-k
\right) & = & \cot\frac{\psi(n,k)}{2}-
\frac{V(n)}{\sin k}.
\end{eqnarray}
There is no problem with the singularities of the $\cot$, because
we may as well use a similar equation with $\tan$ instead of
$\cot$.
\begin{Theorem}
\label{T41}
Suppose there exists an increasing sequence $x_n\in {\Bbb N}$,
$x_n\to\infty$, such that $\sum_{n=1}^{\infty} \|V_n\|_2
<\infty$ and $\sup_{n\in {\Bbb N}} \|V_n\|_1 \|V_n\|_2^N
<\infty$ for some $N\in {\Bbb N}$ (writing
$\|V_n\|_p^p = \sum_{m=x_{n-1}}^{x_n-1} |V(m)|^p$). Then
$\Sigma_{ac}=(-2,2)$ is an essential support of the
absolutely continuous part of the spectral measure. Moreover,
for almost every $E\in (-2,2)$, there are solutions $y,
\overline{y}$ of the Schr\"odinger equation (\ref{dse})
with the asymptotic form
\[
y(n,E)=(1+o(1))e^{i\omega(n,k)/2}
\quad (n\to\infty)
\]
(where $\omega(n,k)= 2kn +\frac{1}{\sin k}\sum_{s=1}^{n-1} V(s)$).
\end{Theorem}
{\it Sketch of the proof.} We will discuss only the proof
of $\Sigma_{ac}=(-2,2)$. This proof is, of course, modelled
on the proof of Theorem \ref{T11}. The assertion on the WKB asymptotics
is established by a similar adaption of the proof of Theorem
\ref{T331}.
A Taylor expansion of the basic equations (\ref{eqrdiscr}),
(\ref{eqpsidiscr}) shows
\begin{eqnarray}
\label{s4eqr}
2\sin k \ln\frac{R(x_n,k)}{R(x_{n-1},k)} & = &
-\sum_{s=x_{n-1}}^{x_n-1} V(s)\sin\psi(s,k) + O\left(
\sum_{s=x_{n-1}}^{x_n-1} V(s)^2 \right), \\
\label{s4eqpsi}
\psi(m,k) & = & \omega(m,k)+\theta(m,k),
\end{eqnarray}
where $\omega$ has been defined above, and $\theta$ satisfies
\begin{equation}
\label{s4eqtheta}
\theta(m+1,k)-\theta(m,k)= -\frac{V(m)}{\sin k}\cos\psi(m,k) + O\left(
V(m)^2\right).
\end{equation}
The constants hidden in the $O(\ldots)$ terms of
(\ref{s4eqr}), (\ref{s4eqtheta}) are uniform in $k$, provided
$k$ is restricted to a compact subset of $(0,\pi)$.
Analogues of Propositions \ref{P21}, \ref{P22} remain true.
The proofs are completely analogous and will thus be left
to the reader.
Now fix $\delta>0$.
We will show that
\begin{equation}
\label{s4sum}
\sum_{n=1}^{\infty} \left| \sum_{s=x_{n-1}}^{x_n-1}
V(s) e^{i\omega(s,k)} e^{i\theta(s,k)} \right|
<\infty
\end{equation}
for almost every $k\in (\delta,\pi-\delta)$. This will imply
the assertion by the analogue of Proposition \ref{P21}.
Summation by parts lets us rewrite the sum over $s$ from
(\ref{s4sum}) as
\begin{equation}
\label{s4a1}
e^{i\theta(x_n)} \sum_{s=x_{n-1}}^{x_n-1} V(s) e^{i\omega(s)}
+ \sum_{s=x_{n-1}}^{x_n-1}\left( e^{i\theta(s)} -
e^{i\theta(s+1)}\right) \sum_{t=x_{n-1}}^s
V(t) e^{i\omega(t)}.
\end{equation}
As in Sect.\ 2, we see from the analogue of Proposition \ref{P22}
that the first term is already summable for almost every $k$.
Namely, we have that
\begin{eqnarray*}
\int_{\delta}^{\pi-\delta} dk\, \left| \sum_{s=x_{n-1}}
^{x_n-1} V(s) e^{i\omega(s,k)}\right| & \le &
\pi^{1/2} \left( \int_{\delta}^{\pi-\delta} dk\, \left| \sum_{s=x_{n-1}}
^{x_n-1} V(s) e^{i\omega(s,k)}\right|^2 \right)^{1/2}\\
& \le & C\|V_n\|_2 \in l_1({\Bbb N}).
\end{eqnarray*}
Taylor's theorem and (\ref{s4eqtheta}) imply that
\[
e^{i\theta(s)} -
e^{i\theta(s+1)} = \frac{iV(s)}{\sin k} \cos \psi(s)e^{i\theta(s)}
+O\left( V(s)^2\right).
\]
Plug this into (\ref{s4a1}). The contribution coming from
the remainder $O(V(s)^2)$ is summable
for almost every $k$. In fact,
\[
\int_{\delta}^{\pi-\delta} dk\, \sum_{s=x_{n-1}}^{x_n-1}
V(s)^2 \left| \sum_{t=x_{n-1}}^s
V(t) e^{i\omega(t)} \right| \le C\|V_n\|_2^3 \in l_1({\Bbb N}).
\]
So, it suffices to deal with terms of the form
\[
\sum_{s=x_{n-1}}^{x_n-1} V(s) e^{i\sigma\omega(s)}
e^{im\theta(s)}
\sum_{t=x_{n-1}}^s
V(t) e^{i\omega(t)}.
\]
Now the machine based on Lemma \ref{L21} can be used to
pass to increasingly finer subdivisions of the segments
$[x_{n-1},x_n)$. As above, additional terms coming from
the $O(V^2)$ remainders are easily seen to be summable.
We keep the notations of Sect.\ 2.
In particular, we let again $N_n= \max\{1,[1/\|V_n\|_2]\}$.
So, our final task is to show that for appropriately chosen
$y_i$'s and for some $j$,
\begin{equation}
\label{s4a2}
\sum_{n=1}^{\infty} \sum_{l_1,\ldots, l_j=1}^{N_n}
\sum_{s=y_j(l_j-1)}^{y_j(l_j)-1} |V(s)|
\left| \sum_{t=y_j(l_j-1)}^s V(t) e^{i\omega(t)}\right|
<\infty
\end{equation}
for almost every $k$. The problem is that we can no
longer pick the $y_i$'s according to (\ref{n1}). This difficulty
is overcome as follows. Fix $n$; then,
we have to pick $N_n^j+1$ numbers
$y_j(n,l_1,\ldots,l_j)\in \{x_{n-1},\ldots , x_n \}$.
It will be convenient to relabel these
numbers as $y_0,y_1,\ldots, y_{N_n^j}$. The restriction
of $V$ to $\{y_{m-1},y_{m-1}+1,\ldots,y_m-1\}$ will be denoted
by $V_{nm}$. Now let $y_0=x_{n-1}$, and if $y_0,\ldots,
y_{m-1}$ have been chosen, let $y_m$ be the smallest
integer $>y_{m-1}$ such that $\|V_{nm}\|_1 > \|V_n\|_1 N_n^{-3j/4}$.
If no such number which is less than $x_n$
exists, set $y_m=x_n$. It is clear
that this final index (let us call it $M$) satisfies
$M \le N_n^{3j/4}$. Since
$N_n^j - ( N_n^{3j/4}+1) > N_n^{3j/4}$ for large enough
$n$, we can add the numbers $y_m-1\quad (m=1,\ldots,
M)$ to this collection of $y_i$'s. The remaining
$y_i$'s can now be chosen arbitrarily.
By construction,
this subdivision has the following property: Renumber
the $y_i$'s so that $x_{n-1}=y_0\le y_1 \le \ldots
\le y_{N_n^j}=x_n$. Then
$\{y_{m-1},\ldots,
y_m-1\}$ is either a single point, or $\|V_{nm}\|_1
\le \|V_n\|_1 N_n^{-3j/4}$. (In fact, some of these segments
may even be empty, in which case the second alternative holds.)
It is also clear how the numbers with the original
labeling $y_i(n,l_1,\ldots,l_i)$ are obtained from
this ``linear'' arrangement $y_0,\ldots,y_{N_n^j}$. Namely, a
little thought shows that we have to set
$y_i(n,l_1,\ldots,l_i)=y_m$, where $m=\sum_{r=1}^{i-1}
(l_r-1)N_n^{j-r} + l_iN_n^{j-i}$.
Now integrate the summands of (\ref{s4a2}) with respect
to $dk$. The result can be bounded by (a constant times)
\[
\sum_{m=1}^{N_n^j}\|V_{nm}\|_1\|V_{nm}\|_2.
\]
Consider separately the sum over those $m$ for which
$y_{m-1}=y_m-1$ and the sum over the remaining indices.
The first contribution obviously does not exceed
$\|V_n\|_2^2$, and the second one can be estimated by
\[
N_n^{-3j/4}\|V_n\|_1 \sum_{m=1}^{N_n^j} \|V_{nm}\|_2
\le N_n^{-j/4} \|V_n\|_1 \|V_n\|_2
\le C \|V_n\|_1 \|V_n\|_2^{j/4+1}.
\]
This is also summable, provided $j\ge 4N$.
$\Box$
\section{Perturbations of general Schr\"odinger operators}
Our methods also allow us to treat the following, more
general situation. Consider the Schr\"odinger operator
\[
H=-\frac{d^2}{dx^2} +U(x)+V(x)
\]
on $L_2(0,\infty)$, viewed as a perturbation of
$H_0=-d^2/dx^2+U(x)$. In order to be able to apply
\cite[Theorem 5]{Stolz}, we will need a mild assumption
on the negative part of $U$, namely
$\sup_{n\in{\Bbb N}}\int_n^{n+1}U_-(x)\, dx <\infty$.
We are interested in relations between
$\Sigma_{ac}(H_0)$ and $\Sigma_{ac}(H)$. Results of this
flavor were first proved in \cite{Kis2}.
Here, this extension relies on the generalized Pr\"ufer
transformation, as introduced in \cite{KRS}. We recall
briefly the basic properties of this transform. Let
$f(x,E)$ be a complex solution of the reference equation
\begin{equation}
\label{s5refse}
-f''(x)+U(x)f(x)=Ef(x)
\end{equation}
with the additional property
that $f$ and $\overline{f}$ are linearly independent.
Define a differentiable function $\gamma$ by writing
$f=|f|e^{i\gamma}$. Constancy of the Wronskian
$W=f\overline{f'}-\overline{f}f'$ shows that
$\gamma'(x)|f(x)|^2=c$. By replacing $f$ by $\overline{f}$
if necessary, we may assume that $c>0$. Then we also
have that $\gamma'(x,E)>0$ for all $x,E$.
Now any real-valued solution $y$ of
the full equation
\begin{equation}
\label{s5fse}
-y''(x)+(U(x)+V(x))y(x)=Ey(x)
\end{equation}
can be expressed in terms of $f$ and generalized Pr\"ufer
variables $R(x,E)>0,\psi(x,E)$ by
\[
\left( \begin{array}{c} y \\ y'\end{array} \right) =
\mbox{Im } Re^{i(\psi/2-\gamma)} \left( \begin{array}{c}
f \\ f' \end{array} \right).
\]
It is shown in \cite{KRS} that $R,\psi$ are well-defined and
obey the equations
\begin{eqnarray}
\label{s5eqr}
(\ln R)' & = & \frac{V}{2\gamma'} \sin\psi,\\
\label{s5eqpsi}
\psi(x,E)& = & \omega(x,E)+\theta(x,E),
\end{eqnarray}
where
\[
\omega(x,E)= 2\gamma(x,E)- \int_0^x \frac{V(t)}{\gamma'(t,E)}\, dt
\]
and $\theta'= (V/\gamma')\cos\psi$.
Note the complete analogy of these equations
to (\ref{eqr}), (\ref{eqpsi}).
The main new feature is that we no longer have Proposition
\ref{P22}. The property stated there now becomes an
assumption.
\begin{Theorem}
\label{T51}
Let $S\subset {\Bbb R}$ be a Borel set with the following properties:\\
1. For all $E\in S$, all solutions of (\ref{s5refse}) are bounded.\\
2. For some choice of $f(x,E)$ as above,
the integral operator $T: L_2(0,\infty)\to L_2(S)$, defined
for bounded functions $g$ of compact support by
\[
(Tg)(E)= \int_0^{\infty} \frac{e^{i\omega(x,E)}}{\gamma'(x,E)}
g(x)\, dx ,
\]
is norm bounded.
Furthermore, assume that $V$ satisfies the hypotheses of
Theorem \ref{T11}.
Then $\Sigma_{ac}(H)\supset S$. Moreover, for almost every $E\in S$,
there are solutions $y,\overline{y}$ of (\ref{s5fse})
with the asymptotic form
\[
\left( \begin{array}{c} y(x,E) \\ y'(x,E) \end{array} \right)
=\left( \left( \begin{array}{c} f(x,E) \\ f'(x,E) \end{array} \right)
+o(1) \right)
e^{i(\omega(x,E)/2-\gamma(x,E))}
\quad (x\to\infty).
\]
\end{Theorem}
{\it Sketch of the proof.} Let $S_n=\{E\in S: \sup_{x\ge 0}
(1/\gamma'(x,E)) \le n \}$. Then, by assumption, $S=\bigcup_{n\in
{\Bbb N}} S_n$, and it thus suffices to prove the assertion
for $E\in S_n$, where $n$ is fixed but arbitrary.
Now we can repeat the proofs of Theorems \ref{T11}, \ref{T331}
step by step, starting from the generalized Pr\"ufer equations
(\ref{s5eqr}), (\ref{s5eqpsi}). We cannot use Proposition
\ref{P21}, but the asymptotic formulae for $y, \overline{y}$
imply that
all solutions are bounded at these energies (compare remark
2 at the end of Sect.\ 3).
$\Box$
{\it Remarks.} 1. By \cite{Stolz} (see also \cite{Sbdd}),
assumption 1 implies
that $\Sigma_{ac}(H_0)\supset S$.
2. In \cite{Kis2}, assumption 2 is established for $U=0$ and
for periodic $U$. It would be interesting to investigate
this condition in more general contexts. Note also that we
do not really need the full force of the norm bound, but
only the special case $g=V\chi_{(s,t)}$.
3. The fact that the proof of
Theorem \ref{T51} is almost identical with the
proofs of Theorems \ref{T11}, \ref{T331} may be viewed as
an argument against this result. I prefer to think
of this as an illustration of the power of the generalized
Pr\"ufer transformation.
\section{Proof of Theorems 1.3, 1.4}
The first ingredient to the proof
is the following consequence of a classical result.
\begin{Lemma}
\label{L32}
If $|V(x)|\le Cx^{-\alpha}$ for all large $x$, then the set
\[
S_{\epsilon}=\{ k: \lim_{N\to\infty}
\int_0^N x^{\epsilon}V(x) e^{ikx}\, dx
\mbox{ does not exist }\}
\]
has Hausdorff dimension $\dim S_{\epsilon}\le 2(1+\epsilon-\alpha)$.
\end{Lemma}
{\it Proof.} A classical result on the set $S_{\epsilon}$ says that
$S_{\epsilon}$ is of $\gamma$-capacity zero for every
$\gamma>2(1+\epsilon-\alpha)$ (see, e.g., \cite[Sect.\ V]{Carl} or
\cite[Sect.\ XIII.11]{Zyg}). In the literature, this result
is usually formulated for Fourier series (rather than
integrals), but the proof extends to the continuous case
without any difficulties.
$S_{\epsilon}$ is a Borel set, so the usual connections
between capacities and Hausdorff measures (see, e.g.,
\cite[Sect.\ IV]{Carl}) imply $\dim S_{\epsilon}\le 2(1
+\epsilon-\alpha)$.
$\Box$
Now let $\epsilon_n=1-\alpha+n^{-1}$ and consider the sets
$S_n\equiv S_{\epsilon_n}\cap (0,\infty)$. The integration by parts
argument from \cite{Kis1} shows that if $k>0$ and $2k\notin S_n$,
then $\lim_{N\to\infty} \int_0^N V(t)e^{2ikt}\, dt$ exists
and, moreover, $\int_x^{\infty} V(t)e^{2ikt}\, dt =
O(x^{-\epsilon_n})$. Hence, again by \cite{Kis1} (or the
alternative version of this argument in \cite[Sect.\ 3]{KLS}),
all solutions of (\ref{se}) (with $E=k^2$) are bounded.
So, for every boundary condition $\beta$,
the singular part of the
spectral measure (restricted to $(0,\infty)$) is supported
by every $\widetilde{S}_n:=\{E=k^2/4: k\in S_n\}$
and hence also by
$\bigcap_{n\in {\Bbb N}} \widetilde{S}_n$.
By Lemma \ref{L32}, this intersection
has dimension $\le 4(1-\alpha)$.
This proves Theorem \ref{T14}. $\Box_1$
The second ingredient to the proof of Theorem
\ref{T13} is an estimate on the solutions of (\ref{se}).
To simplify the notation, we will subsequently assume that
$|V(x)|\le C/x$ for all $x\ge 1$. The general case
(i.e.\ $C=\limsup x\, |V(x)|$) is dealt with by first
increasing $C$ slightly and then considering only big enough $x$.
We will use the notation
$\|y\|_x^2=\int_0^x |y(t)|^2\, dt$ (cf.\ \cite{GP}).
The following Lemma is based on a precise analysis of
the Pr\"ufer equations (compare also \cite[Section 3.2]{EK}).
\begin{Lemma}
\label{L33}
For every $E>(2C/\pi)^2$, there exist $\alpha, \beta >0$, such that
every nontrivial solution $y$ of the Schr\"odinger equation
(\ref{se}) satisfies estimates of the form
\[
A_1 x^{\alpha} \le \|y\|^2_x \le A_2 x^{\beta}
\]
(where $A_1,A_2$ are positive constants, and, say, $x\ge 1$).
\end{Lemma}
{\it Proof.} Fix $E>(2C/\pi)^2$, and let $y$ be a solution of
(\ref{se}).
Introduce the corresponding Pr\"ufer variables
$R,\psi$ (compare Sect.\ 2).
It is easy to see that we have
\[
A\int_0^xR^2(t)\, dt \le \|y\|_x^2 \le \int_0^x R^2(t)\, dt
\quad (x\ge 1, A>0),
\]
so it suffices to analyze $R$. Let $x_n\: (n\in {\Bbb N}_0)$ satisfy
$1=x_0 0 \end{array} \right. .
\]
This definition makes sense, because $\vartheta_{n-1}$ depends
only on the values of $V(x)$ for $x
(2C/\pi)^2$, we can find $\delta
=\delta(E)>0$
so that $\|u\|_x\|v\|_x^{-\delta}\to \infty$ for any two
solutions $u,v$ of the Schr\"odinger equation (\ref{se}).
By the refined subordinacy theory \cite{JL}
(see also \cite{Remso}), this implies that there is a
$\gamma=\gamma(E)>0$ so that the $\gamma$-derivative
of the spectral measure $\rho$ vanishes at $E$:
\[
(D_{\gamma}\rho)(E)=\limsup_{\epsilon\to 0+}
\frac{\rho(E-\epsilon,E+\epsilon)}{(2\epsilon)^{\gamma}} = 0.
\]
By general properties of Hausdorff measures
\cite{Rog}, this means
that $\rho$ gives zero weight to every $S\subset ((2C/\pi)^2,\infty)$
with $\dim S=0$.
However, Theorem \ref{T14} with $\alpha=1$ says that
$\rho_{s}$ is supported on a set of zero Hausdorff
dimension. This completes the proof of Theorem \ref{T13}.
$\Box_2$
We conclude this paper with a few remarks on how to
obtain analogues of Theorems \ref{T13}, \ref{T14} for
the discrete Schr\"odinger equation (\ref{dse}). First
of all, generalizing Theorem
\ref{T14} is straightforward. In fact, the proof becomes somewhat easier.
Now we try to prove an analogue of Lemma \ref{L33}. We
use the notations of Sect.\ 4. Instead of
(\ref{aux5}), we now have to analyze the following equation:
\[
\sin k\ln R^2(x) = -\sum_{n=1}^N \sum_{s=x_{n-1}}^{x_n-1} V(s)
\sin (2ks+\vartheta_{n-1}) + o(\ln x)\quad (x\to\infty).
\]
As above, we have $\vartheta_n=\psi(x_n)-2kx_n$, and $N$ satisfies
$x_N\le x < x_{N+1}$. Also, we assumed that $x_n\in {\Bbb N}$,
$x_n\to\infty$, and $x_n/x_{n-1}\to 1$.
Proceeding as in the proof of Lemma \ref{L33}, we get
\[
|\ln R^2(x)| \le \frac{C}{\sin k} \sum_{n=1}^N
\ln\frac{x_n}{x_{n-1}}\: \frac{1}{x_n-x_{n-1}}
\sum_{s=x_{n-1}}^{x_n-1} |\sin(2ks+\vartheta_{n-1})|
+o(\ln x),
\]
provided $|V(n)|\le C/n$.
We now suppose that, in addition, $x_n-x_{n-1}\to\infty$. Then,
using the unique ergodicity of the rotation on the torus with irrational
rotation number (see, e.g., \cite{Sinai}),
we deduce that if $k/\pi\notin {\Bbb Q}$, then
\[
\frac{1}{x_n-x_{n-1}}
\sum_{s=x_{n-1}}^{x_n-1} |\sin(2ks+\vartheta_{n-1})|
= \frac{1}{\pi} \int_0^{\pi} |\sin\varphi| \, d\varphi +o(1)
\quad (n\to\infty).
\]
Hence, for these $k$ we obtain
\[
|\ln R^2(x)| \le \left(\frac{2C}{\pi\sin k} +o(1)\right) \ln x
\quad(x\to\infty).
\]
Since a countable set cannot support continuous measures, we can now
conclude the proof of the analogue of Theorem \ref{T13} as in
the continuous case. Recall also that $E=2\cos k$. We have thus
shown
\begin{Theorem}
\label{T61}
If $C:=\limsup_{n\to\infty} n\, |V(n)|< \pi/2$, then $\sigma_{sc}(H)
\cap (-E_0,E_0)=\emptyset$, where $E_0=2\sqrt{1-(2C/\pi)^2}$.
In particular, if $V(n)=o(1/n)$, then $H$ is purely absolutely
continuous on $(-2,2)$.
\end{Theorem}
The second part uses the trivial remark that $o(1/n)$ potentials
do not have embedded eigenvalues. However, the question of
sharp bounds for eigenvalues of $O(1/n)$ potentials is much
more subtle than in the continuous case
and will not be discussed here.
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\end{document}