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\begin{document}
\title{Embedded singular continuous spectrum for one-dimensional
Schr\"odinger operators}
\author{Christian Remling}
\maketitle
\begin{center}
(submitted to {\it Trans.\ Amer.\ Math.\ Soc.})
\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]
Current address (until May 31, 1997):
253-37 Department of Mathematics,
California Institute of Technology,
Pasadena, CA 91125,
U.S.A.\\[0.2cm]
1991 AMS Subject Classification: 34L40, 81Q10\\
Key words: Schr\"odinger equation, singular continuous spectrum,
subordinate solutions
\begin{abstract}
We investigate one-dimensional Schr\"odinger operators
with sparse potentials (i.e.\ the potential consists
of a sequence of bumps with rapidly growing barrier
separations).
These examples illuminate various phenomena related to
embedded singular continuous spectrum.
\end{abstract}
%\newpage
\section{Introduction}
In this paper, we will study one-dimensional Schr\"odinger equations
on the half-line
\begin{equation}
\label{se}
-y''(x)+V(x)y(x)=Ey(x),\quad x\in [0,\infty).
\end{equation}
We are interested in the spectral properties of the corresponding
operators $H_{\alpha}=-\frac{d^2}{dx^2}+V(x)$ on $L_2([0,\infty))$
with boundary
conditions $y(0)\cos\alpha+y'(0)\sin\alpha=0,\:
\alpha\in [0,\pi)$ (see e.g.\
\cite{WMLN} for the basic theory). (\ref{se}) describes the
motion of a quantum mechanical particle, and important physical
properties of this system depend directly on the spectral
characteristics of the operators $H_{\alpha}$ (for more
background information, consult e.g.\ \cite{RS}).
Here, we will analyze two classes of potentials $V$ related to the
question of the occurrence of singular continuous spectrum
which is embedded in the absolutely continuous spectrum. We will
construct potentials so that $\sigma_{ac}(H_{\alpha})=[0,\infty)$
and, for a set of boundary conditions $\alpha$ of positive measure,
we have that $\sigma_{sc}(H_{\alpha})\cap (0,\infty)\not=
\emptyset$ (see Theorem \ref{T31} below). There is also a
``complementary'' construction: Using similar ideas, we will
obtain potentials with $\sigma_{sc}(H_{\alpha})=[0,\infty)$,
$\sigma_{ac}(H_{\alpha})\cap (0,\infty)\not=\emptyset$ for all $\alpha$
(= Theorem \ref{T32}).
I do not know of any previous examples for these types
of spectral behavior.
It is more difficult to obtain embedded singular {\it
continuous} spectrum than, say, embedded point spectrum because
singular continuous spectrum is related to the {\it subordinacy}
of the generalized eigenfunction
(the notion of subordinacy
was introduced and analyzed in \cite{GP}; for
subsequent developments, see
\cite{JL,Rso}). Here, the term ``generalized eigenfunction''
simply refers to a solution of (\ref{se}) which satisfies the
boundary condition at $x=0$.
More specifically, one encounters the following
problem: The singular part of the spectral measure is supported
on the set where the generalized eigenfunction is subordinate
\cite{GP}, but even if this set is large, it does not automatically
follow that there actually is singular continuous spectrum. Of
course, as for the point spectrum, the situation is completely
different: If the generalized eigenfunction is square integrable,
then, trivially, the corresponding energy is an eigenvalue. In
fact, potentials with embedded point spectrum which is dense
in $[0,\infty)$ have been known for a relatively long time
\cite{Na,Spp}.
We will illuminate these remarks with our second class of
potentials. These examples have
purely absolutely continuous spectrum on $(0,\infty)$, but,
as we will prove, the set of energies for which the generalized
eigenfunction is subordinate has (local) Hausdorff dimension
$1$ (= Theorem \ref{T41}b)!
(Recall that, by general principles, this set always has
Lebesgue measure zero.) Again, this is, to the best of my
knowledge, the first explicit example for these phenomena.
All potentials in this paper will be sparse potentials (i.e.\
mainly $V=0$). This is no coincidence: Sparse potentials lead
to non-trivial asymptotics of the solutions of (\ref{se}), and
there are powerful methods which allow a detailed analysis
of these asymptotics. Here, we will rely mainly on further
extensions of the techniques
recently developed in \cite{KLS,R}. However, sparse
potentials were already studied
in the celebrated work \cite{P}; further papers
using sparse potentials in one way or another are
\cite{GMT,KMP,Rpp,Svii,SSt}. Although the methods of \cite{KLS}
and \cite{R} are very similar in spirit, the actual implementation
of the basic strategy differs in some respects
(actually, the results of
\cite{KLS} are stronger). It turns out that both
viewpoints are needed here. The examples with embedded singular
continuous spectrum make heavy use of the ideas of \cite{KLS},
whereas the second class of examples will be analyzed with
an extension of the techniques developed in \cite{R}.
This paper is organized as follows: In the next Section,
we fix the notations and explain the basic strategy for
an effective analysis of Schr\"odinger equations with
sparse potentials (as developed
in \cite{KLS,R}). Sections 3 and 4 investigate in detail the
examples described above.
While I was proofreading this paper, I obtained an interesting
preprint by Molchanov \cite{Mo} that also discusses sparse
potentials. Molchanov's work has some overlap with
\cite{KLS,R} and also with Section 3 of this paper.
I would like to thank U.\ Keich and T.\ Wolff for useful
discussions, the Deutsche Forschungsgemeinschaft for financial
support, and I am grateful for the hospitality of Caltech
where this work was done.
\section{Preliminaries}
The potentials we will study will have the form
\begin{equation}
\label{V}
V(x)=\sum_{n=1}^{\infty} g_nV_n(x-a_n)
\end{equation}
with $g_n>0, V_n\in L_1([-B_n,B_n])$; the intervals
$[a_n-B_n,a_n+B_n]$
are assumed to be disjoint. Let $L_n=a_n-B_n-a_{n-1}-B_{n-1}$
(with $a_0=B_0:=0$) and
\begin{equation}
\label{in}
I_n=\int_{-B_n}^{B_n} |V_n(x)|\, dx.
\end{equation}
The $a_n$'s can of course be recovered from the $L_n,B_n$, so
it suffices to specify these latter parameters.
Fix $\alpha\in [0,\pi)$, and, for $k>0$,
let $y(x,k)$ be the solution of
(\ref{se}) with $E=k^2$ and $y(0,k)=-\sin\alpha, y'(0,k)=
\cos\alpha$. Note that $y$ satisfies the boundary condition
$\alpha$ at $x=0$.
The (modified) Pr\"ufer variables $R(x,k),\varphi(x,k)$ are
defined by
\[
\left( \begin{array}{c} y \\ y' \end{array} \right)
= R \left( \begin{array}{c} \sin\varphi \\ k\cos\varphi
\end{array} \right).
\]
Here, we demand that $R>0$ and $\varphi$ be continuous in $x$.
Clearly, $R,\varphi$ also depend on
$\alpha$, but this dependence will
not be made explicit in the notation.
We write $R_n(k)=R(a_n-B_n,k), \varphi_n(k)=\varphi(a_n-B_n,k)$
(so $R_n,\varphi_n$ are the Pr\"ufer variables immediately
before the $n$th barrier).
$R,\varphi$ obey the equations
\begin{eqnarray}
\label{prf1}
(\ln R)'& = & \frac{V}{2k}\sin 2\varphi ,\\
\label{prf2}
\varphi' & = & k - \frac{V}{k}\sin^2 \varphi.
\end{eqnarray}
In particular, we have $R(x)\equiv R_n$ on $x\in [a_{n-1}+B_{n-1},
a_n-B_n]$
and
\begin{equation}
\label{phi}
\varphi_n(k)=\varphi(a_{n-1}+B_{n-1},k)+kL_n.
\end{equation}
Fix a compact interval $J=[k_1,k_2]\subset (0,\infty)$. Then,
for $k\in J$, we can integrate (\ref{prf1}), (\ref{prf2})
over the interval $[a_n-B_n,a_n+B_n]$ and use an elementary Taylor
expansion in the parameter $g_nI_n$. This routine calculation
yields (compare \cite{KLS})
\begin{eqnarray}
\label{eqR}
\ln \frac{R_{n+1}(k)}{R_n(k)} & = & \frac{g_n}{2k}\int_{-B_n}^{B_n}
dx\, V_n(x)\sin 2\theta_n(x,k) \nonumber\\
&& - \frac{g_n^2}{k^2}\int_{-B_n}^{B_n}dx\, V_n(x)\cos 2
\theta_n(x,k)\int_{-B_n}^x dt\, V_n(t)\sin^2
\theta_n(t,k)\nonumber\\
&&+ O(g_n^3I_n^3).
\end{eqnarray}
where $\theta_n(x,k):=k(x+B_n)+\varphi_n(k)$.
The remainder $O(g_n^3I_n^3)$ is bounded by $C(J)g_n^3I_n^3$
where $C$ is independent of $k\in J$ and $n\in {\Bbb N}$.
Finally, let
\[
C_n=\max_{k\in J} \left|
\frac{d\varphi(a_{n-1}+B_{n-1},k)}{dk}\right|.
\]
The crucial observation is that, because of (\ref{phi}), for
appropriate probability measures on $J$, the Pr\"ufer angles
$\varphi_n(k)$ (evaluated modulo $\pi$) are approximately independent
random variables, provided that $L_n\gg C_n$. This property
can be exploited by computing moments \cite{KLS} or by investigating
the joint distribution of the $\varphi_n$'s \cite{R}. With either
method, one can analyze (\ref{eqR}) rather accurately.
In order to get quantitative conditions on the $L_n$'s, we
need a priori control on the $C_n$'s.
\begin{Lemma}[\cite{KLS}]
\label{L21}
Assume $g_nI_n\to 0$ and $(L_n+B_n)/L_{n+1}\to 0$. Then
there is a constant $C=C(J)$ such that
$C_n\le CL_{n-1}$ and
\[
\max_{k\in J}\left| \frac{d^2\varphi(a_{n-1}+B_{n-1},k)}{dk^2}\right|
\le C\left( 1+\sum_{m=1}^{n-1} g_mI_mL_m^2 \right) .
\]
\end{Lemma}
{\it Proof.} This follows from a Gronwallization of the differential
equations satisfied by $\partial\varphi/
\partial k, \partial^2\varphi/\partial k^2$; see
\cite[Proposition 5.1]{KLS} for details. $\Box$
\section{Embedded singular continuous spectrum}
Suppose $g_nI_n\in l_3$. Then (\ref{eqR}) can be written as
\begin{eqnarray}
\label{1}
\ln R_{N+1}(k) & = & \frac{1}{2k} \im \sum_{n=1}^N X_n(k)
-\frac{1}{2k^2} \re \sum_{n=1}^N Y_n(k) \nonumber\\
&& +\frac{1}{8k^2} \re \sum_{n=1}^N Z_n(k)
+\frac{1}{8k^2} \sum_{n=1}^N g_n^2 |\widehat{V}_n(2k)|^2
+\rho_N(k)
\end{eqnarray}
where $|\rho_N(k)|\le C$ for all $N\in {\Bbb N}, k\in J$.
Here, we have set
\begin{equation}
\label{FT}
\widehat{V}_n(2k)=\int_{-B_n}^{B_n} V_n(x) e^{2ikx}\, dx
\end{equation}
and
\begin{eqnarray*}
X_n(k) & = & g_n \vk e^{2i(\varphi_n(k)+kB_n)}, \\
Y_n(k) & = & g_n^2 e^{2i(\varphi_n(k)+kB_n)} \int_{-B_n}^{B_n} dx\,
V_n(x) e^{2ikx}\int_{-B_n}^x dt\, V_n(t),\\
Z_n(k) & = & g_n^2 (\vk)^2 e^{4i(\varphi_n(k)+kB_n)}.
\end{eqnarray*}
Since $X_n,Y_n,Z_n$ contain the highly oscillatory
factors $\exp (2i\varphi_n)$, we expect that the leading term of
(\ref{1}) will be $\sum g_n^2 |\vk|^2$. Thus,
by choosing the shape of the bumps $V_n$ carefully,
we can get non-trivial $k$-dependence of the asymptotics
of $R_N(k)\:(N\to\infty)$. This idea was already used in
\cite{KLS} to construct potentials with $\sigma_{ac}=
[E_1,E_2]\subset (0,\infty)$, $\sigma_{sc}=[0,\infty)
\setminus (E_1,E_2)$ (see \cite[Theorem 6.3]{KLS}).
As we will see, it is more difficult to obtain
{\it embedded} singular continuous spectrum.
The barriers $V_n$ will have the form
\begin{equation}
\label{vv}
V_n(x)=\chi_{(-B_n,B_n)}(x)W(x)
\end{equation}
where $W$ is the Fourier transform of the characteristic function
of a Cantor type set $F$. So, construct $F$ as follows:
Let $\delta_n>0$ be sufficiently small prescribed numbers.
Fix $F_0=[a,b]\subset (0,\infty)$ and let $F_1=F_0\setminus
(c_1^{(0)}-\delta_0,c_1^{(0)}+\delta_0)$ where
$c_1^{(0)}=(a+b)/2$ is the center of $F_0$. In general, if
$F_n$ is a disjoint union of $2^n$ closed intervals with centers
$c_m^{(n)}\: (m=1,\ldots, 2^n)$, set
$F_{n+1}=F_n\setminus \bigcup_{m=1}^{2^n} (c_m^{(n)}-
\delta_n,c_m^{(n)}+\delta_n)$. The set $F=\bigcap F_n$ is closed,
nowhere dense and has Lebesgue measure $|F|=b-a-\sum_{n=0}^{\infty}
2^{n+1}\delta_n$. We assume that $|F|>0$, and we define
\begin{equation}
\label{vvv}
W(x)=\int_F \cos 2kx \, dk.
\end{equation}
\begin{Lemma}
\label{L31}
Let $F,W(x)$ be as above.
Suppose $ \sup_{n\in {\Bbb N}}\delta_n 2^{\gamma n}<\infty$ for
some $\gamma>1$. Then $W(x)=O((1+|x|)^{-1+1/\gamma})$.
\end{Lemma}
{\it Proof.} Let $f_n(x)=\int_{F_n}
\cos 2kx\, dk$. Obviously, $|f_n(x)-f(x)|\le |F_n\setminus F|\to 0$.
Furthermore, by construction of the $F_n$,
\begin{eqnarray*}
f_{n+1}(x)-f_n(x) & = & -\int_{F_n\setminus F_{n+1}}
\cos 2kx \, dk = -\sum_{m=1}^{2^n}\int_{c_m^{(n)}-\delta_n}
^{c_m^{(n)}+\delta_n} \cos 2kx\, dk \\
& = & -\frac{1}{2x} \sum_{m=1}^{2^n} \left(
\sin 2x(c_m^{(n)}+\delta_n) - \sin 2x(c_m^{(n)}-\delta_n) \right)\\
& = & -\frac{1}{x}\sin 2\delta_nx \sum_{m=1}^{2^n} \cos 2c_m^{(n)}x.
\end{eqnarray*}
In particular, the infinite sum
$\sum (f_{n+1}(x)-f_n(x))$ is absolutely convergent, and thus
\begin{eqnarray}
\label{3}
|f(x)| &=& \left| f_0(x) +\sum_{n=0}^{\infty} (f_{n+1}(x)-f_n(x))
\right|\nonumber\\
&\le & \frac{1}{|x|}\left( 1+ \sum_{n=0}^{\infty} 2^n |\sin
2\delta_nx|\right).
\end{eqnarray}
For $|x|>2^{-\gamma}$, define $N(x)\in {\Bbb N}_0$ by
demanding $2^{-\gamma}<2^{-\gamma N(x)}|x|\le 1$. Now the assertion
is obtained by considering separately the sums $\sum_{n6$. Let $g_n=n^{-1/2},B_n=n^{\beta}$
with $(2-4/\gamma)^{-1}<\beta<\gamma/8$, and assume that
$n^{\beta/(2\gamma)}L_{n-1}/L_n\to 0$.
Then the half-line Schr\"odinger operators $H_{\alpha}$
with potential $V$ given by (\ref{V}), (\ref{vv}), (\ref{vvv}) satisfy
$\sigma_{ac}(H_{\alpha})=\sigma_{ess}(H_{\alpha})=
[0,\infty)$, $\sigma_p(H_{\alpha})\cap (0,\infty) =\emptyset$
and, for a set of boundary conditions $\alpha$
of positive measure, $\sigma_{sc}(H_{\alpha})\cap
(0,\infty)\not= \emptyset$.
\end{Theorem}
{\it Remarks.} 1. The following proof also works
under more general assumptions on $\gamma,g_n,B_n,L_n$.
However, these general conditions are very clumsy, and
it seems pointless to make them explicit.
2. The proof will show that, more precisely,
$\sigma_{sc}(H_{\alpha})\cap
{\Bbb R}\setminus F^2 =\emptyset$ for almost all $\alpha$, and
$\sigma_{sc}(H_{\alpha})\cap F^2 \not= \emptyset$ for a set of
$\alpha$'s of positive measure. Here,
$F^2=\{ k^2:k\in F\}$. Note that the method of \cite{KLS}
(which would establish that the spectrum is purely absolutely
continuous on $(0,\infty)\setminus F^2$ for {\it every}
boundary condition) does not work here because
of the slow decay of $W(x)$.
{\it Proof.} The assertion on $\sigma_{ess}$ follows from the fact
that $V(x)\to 0$. Furthermore, a standard Gronwall estimate shows
that because of the rapid growth of the barrier separations $L_n$,
(\ref{se}) has no $L_2$-solutions if $E>0$.
Moving on to the non-trivial parts of the proof, we
fix $f\in C_0^{\infty}(0,\infty)$ with $f\ge 0$,
$\mbox{supp }f\cap F=\emptyset$ and $\int f(k)\,dk=1$. In the
first part of the proof, we will
show with the aid of Proposition \ref{P31} that almost surely
with respect to the probability measure $dP(k)=f(k)\,dk$, the
right-hand side of (\ref{1}) remains bounded as $N$ goes to
infinity. Using (again) a Gronwall estimate, one can easily
extend this statement to $x\in\bigcup_n [a_n-B_n,a_n+B_n]$,
i.e., $R(x,k)$ is bounded on all of $x\in [0,\infty)$.
Running through this argument for two different
initial angles $\varphi(0,k)$ then shows that {\it all} solutions
are bounded for almost every $k\in \mbox{supp }f$. Since $f$ is
arbitrary and $F$ is nowhere dense, this will establish the claim
on $\sigma_{ac}$ (by \cite[Theorem 5]{St}).
Let
\[
\epsilon_n=\left( \int_{|x|>B_n} |W(x)|^2\,dx\right)^{1/2}.
\]
We will need
\begin{Lemma}
\label{L32}
a) Let $f(k)$ be a bounded function.
Then there is a constant $C=C(f)$, such that
\[
\int \left| \vk-\pi\chi_{F\cup -F}(k)/2\right|^2 f(k)\,dk
\le C\epsilon_n^2,\\.
\]
b) $\lim_{n\to\infty}\vk = \pi\chi_{F \cup -F}(k)/2$ for almost
every $k$.
\end{Lemma}
{\it Proof.} a) A calculation using (\ref{FT}), (\ref{vv}),
(\ref{vvv}), and the Plancherel formula
\cite[Theorem 9.13(b)]{Rud}
yields
\[
\int \left| \vk -\pi \chi_{F\cup -F}(k)/2\right|^2\, dk =
\pi\epsilon_n^2.
\]
This obviously implies the assertion.
b) Since $\gamma>6>2$, Lemma \ref{L31} shows that
$W\in L_p$ for some $p<2$. Thus the
assertion follows from Zygmund's Theorem on the pointwise
convergence of Fourier transforms \cite{Z} and the fact
that $B_n\to\infty$. $\Box$
We are now ready to compute the moments of $X_n,Y_n,Z_n$. Note that
Lemma \ref{L31} implies that ($I_n$ was defined in eq.\ (\ref{in}))
\begin{equation}
\label{est}
I_n\le Cn^{\beta/\gamma},\quad\quad \epsilon_n\le Cn^{\beta/
\gamma-\beta/2}.
\end{equation}
In particular, we have that $\sum (g_nI_n)^3< \infty$; thus
we may indeed use (\ref{1}).
Let us first verify that $X_n$ satisfies the assumptions of
Proposition \ref{P31}. Since we are interested in asymptotic
properties (namely, the boundedness of $\sum X_n$ almost
everywhere), we may restrict $n$ to large values $n\ge n_0$.
In the sequel, we will do so (if necessary) without explicit
mention.
An integration by parts shows that (for $m0$ so that $L_s^2/L_{n-1}^2\le Ce^{-\alpha(n-s)}$
for all $s\le n-1$. Hence
\begin{eqnarray*}
\sum_{s=1}^{n-1} g_sI_sL_s^2 &\le & C L_{n-1}^2 e^{-\alpha n}
\sum_{s=1}^{n-1} s^{-1/2+\beta/\gamma} e^{\alpha s}\\
& \le & CL_{n-1}^2 e^{-\alpha n} \int_1^n s^{-1/2+\beta/\gamma}
e^{\alpha s}\, ds
\le C n^{-1/2+\beta/\gamma}L_{n-1}^2.
\end{eqnarray*}
(Recall that $C$ does not necessarily have the same value
in every formula.) Here, we have estimated the integral by an
integration by parts.
Since $\mbox{supp }f\cap F=\emptyset$, we get from Lemma \ref{L32}a)
\begin{eqnarray*}
\int |\widehat{V}_m\widehat{V}_n|f &\le & \left( \int
|\widehat{V}_m|^2 f
\right)^{1/2}\left(\int |\widehat{V}_n|^2 f \right)^{1/2}\\
& = & \left( \int\left| \widehat{V}_m -
\pi\chi_{F\cup -F}/2\right|^2 f
\int\left| \widehat{V}_n -\pi\chi_{F\cup -F}/2\right|^2
f\right)^{1/2}\\
&\le & C\epsilon_m\epsilon_n.
\end{eqnarray*}
Thus we can bound (\ref{30}) by
\[
Cm^{-1/2+\beta/\gamma-\beta/2}n^{-1+2\beta/\gamma-\beta/2}
\frac{L_{n-1}^2}{L_n^2}.
\]
The other contributions from (\ref{20}) (where $d/dk$ acts on
the numerator) are much easier to deal with.
We use the obvious estimates $|\widehat{V}_i|\le
I_i$, $|\widehat{V}_i'|\le 2B_iI_i$.
It follows that these terms can be bounded
by $Cn^c/L_n$ for some $c$.
In conclusion, we see
that $EX_m\overline{X}_n$ satisfies an estimate
of the form
\[
|EX_m\overline{X}_n| \le
C \sum_{m=1}^{n-1}\left( \frac{n^c}{L_n}+
m^{-1/2+\beta/\gamma-\beta/2}n^{-1+2\beta/\gamma-\beta/2}
\frac{L_{n-1}^2}{L_n^2}\right) .
\]
We assume that the exponent of $m$ is larger than
$-1$; the
proof in the other case is completely analogous.
Then summing over $m$ yields
\begin{equation}
\label{cor}
\sum_{m=1}^n |EX_m\overline{X}_n|
\le C\left( \frac{n^{c+1}}{L_n} +
n^{-1/2+3\beta/\gamma-\beta}
\frac{L_{n-1}^2}{L_n^2}\right).
\end{equation}
Note that, as promised, the first term on the right-hand side is
completely harmless because of the rapid growth of the $L_n$
(faster than exponential).
In order to control $E|X_n|^2$, we use again the fact that
$\mbox{supp }f\cap F=\emptyset$ and apply Lemma \ref{L32}a):
\begin{eqnarray*}
E|X_n|^2&=&g_n^2\int |\vk|^2 f(k)\, dk\\
&=& g_n^2 \int \left| \vk-\pi\chi_{F\cup -F}(k)/
2\right|^2 f(k)\, dk\\
&\le & Cg_n^2\epsilon_n^2\le Cn^{-1+2\beta/\gamma-\beta}.
\end{eqnarray*}
Now it is straightforward to verify that this estimate and
(\ref{cor}) indeed ensure that Proposition \ref{P31}, with
\[
\rho_n= C\left( n^{-1+2\beta/\gamma-\beta}
+ \frac{n^{c+1}}{L_n} +
n^{-1/2+3\beta/\gamma-\beta}
\frac{L_{n-1}^2}{L_n^2}\right),
\]
applies to $\sum X_n$.
In the remainder of this Section, it will be convenient
to simplify the
notation by using the following convention: We will not write out
terms which {\it obviously} decay so rapidly that they do
not pose any difficulties. An example would be the contribution
$n^{c+1}/L_n$ in (\ref{cor}); note, however, that this term
could even be the dominant one in (\ref{cor}), namely, if the
$L_n$ grow unreasonably fast. In any event, we will alert
the reader by writing $\lesssim$ instead of $\le$ whenever
this convention has been applied.
The above strategy can also be used to control $\sum Y_n, \sum Z_n$.
For instance, the leading term in the estimate on
$|EY_m\overline{Y}_n|\:(m0$.
In particular, we can find a subsequence $N_i\to\infty$ so
that $\sum_{i=1}^{\infty} E|\sum_{n=1}^{N_i} X_n|^2 \ln^{-2}N_i
<\infty$, and now the Borel-Cantelli Lemma says that
\[
P\left( \left|\sum_{n=1}^{N_i} X_n\right|\ge \delta \ln N_i
\mbox{ for infinitely many }i\right)=0.
\]
Of course, analogous statements hold if $X_n$ is replaced with
$Y_n$ or $Z_n$.
On the other hand, Lemma \ref{L32}b) implies that
$\lim_{n\to\infty}\vk=\pi/2$ for almost every $k\in F$.
If $\delta$ from above is small enough, we thus see from
(\ref{1}) that
\[
\lim_{i\to\infty}R_{N_i}(k)\ge \lim_{i\to\infty} \left(
c_1\sum_{n=1}^{N_i} g_n^2 - c_2\delta \ln N_i\right)=
\infty
\]
for almost every $k\in F$. Hence the absolutely continuous
part of the spectral measure $\rho_{\alpha}$ gives zero weight
to $F^2=\{ k^2:k\in F\}$ \cite[Theorem 1.2]{LS}.
As noted at the beginning of the
proof, this also holds for the point part of $\rho_{\alpha}$.
Moreover, $\alpha$ was arbitrary, so
the spectral averaging formula (see e.g.\
\cite{dRSS}) becomes
\[
0<|F^2|=\int_0^{\pi} \rho_{\alpha}(F^2)\, d\alpha
=\int_0^{\pi} \rho_{\alpha}^{(sc)}(F^2)\, d\alpha.
\]
This forces $\rho_{\alpha}^{(sc)}(F^2)>0$ for a set of $\alpha$'s
of positive measure, as desired.
It remains to prove (\ref{21}). To this end, let
$S_N=\sum_{n=1}^N X_n$. Then
\begin{equation}
\label{ind}
E|S_N|^2\le E|S_{N-1}|^2 + E|X_N|^2 + 2|ES_{N-1}\overline{X}_N|.
\end{equation}
Clearly, $E|X_N|^2 \le Cg_N^2=CN^{-1}$. In order to estimate
$ES_{N-1}\overline{X}_N$, we use the integration by parts
argument from the first part of this proof. This gives
\[
|ES_{N-1}\overline{X}_N|\le Cg_N \int \left| \frac{d}{dk}
\left( \frac{f(k)\overline{\widehat{V}_N(2k)} S_{N-1}(k)}
{\varphi'_N(k)+B_N}\right)\right|\, dk.
\]
This time, there are two potentially dangerous terms: the first one
coming from the derivative of the denominator and the second
one involving $dS_{N-1}/dk$. The first contribution is treated
as above to obtain the bound
\[
Cg_N \sum_{s=1}^{N-1} g_sI_s \frac{L_s^2}{L_N^2}
\int |f\widehat{V}_N S_{N-1}|\, dk
\le CN^{-1+\beta/\gamma}\frac{L_{N-1}^2}{L_N^2}
\left( E|S_{N-1}|^2\right)^{1/2}.
\]
The last estimate follows by the Cauchy-Schwarz inequality,
Lemma \ref{L32}a), and the usual bound on $\sum_{s=1}^{N-1} g_sI_s
L_s^2$.
As for the second term, we note that the leading term of
\[
\frac{dS_{N-1}}{dk} = \sum_{n=1}^{N-1} g_n \frac{d}{dk}
\vk e^{2i(\varphi_n(k)+kB_n)}
\]
comes from differentiating $e^{2i\varphi_n}$. Now the usual
techniques show that the corresponding contribution
to $ES_{N-1}\overline{X}_N$ can be bounded by $Cg_N^2 L_{N-1}/
L_N$.
As before, we need not worry about the remaining terms which
can be bounded by an expression of the form $CN^c/L_N$, and
the rapid growth of $L_N$ guarantees that these terms are
unimportant. So, if we put everything together, (\ref{ind})
becomes
\[
E|S_N|^2\lesssim E|S_{N-1}|^2 + CN^{-1} + CN^{-1+\beta/\gamma}\frac
{L_{N-1}^2}{L_N^2} \left( E|S_{N-1}|^2\right)^{1/2}.
\]
By an inductive argument, one can now prove that
\[
\left( E|S_N|^2\right)^{1/2}\lesssim
C\sum_{n=1}^N n^{-1+\beta/\gamma}
\frac{L_{n-1}^2}{L_n^2} + C \left( \sum_{n=1}^N n^{-1}\right)^{1/2}.
\]
In fact, the statement needed here is exactly \cite[Lemma 6.2]{KLS};
it would be pointless to repeat that proof here. In any event,
using the assumptions of Theorem \ref{T31},
we see that we have the required bound $E|S_N|^2=o(\ln^2 N)$.
The proof of (\ref{21}) for $Y_n$ and $Z_n$ is
similar. Again, we sketch the argument for $Y_n$ and leave the
proof for $Z_n$ to the reader.
An elementary estimate yields $E|Y_N|^2\le CN^{-2+4\beta/\gamma}$.
As usual, $ES_{N-1}\overline{Y}_N$ (here, $S_n=\sum_{t=1}^n Y_t$,
of course) is treated with an integration
by parts, and the most serious attention has to be paid to
the contribution involving $(d/dk)(1/(\varphi'_N+B_N))$. This
term can be bounded by
\[
C(g_NI_N)^2\sum_{s=1}^{N-1} g_sI_s\frac{L_s^2}{L_N^2}
\int f|S_{N-1}|\, dk \le
CN^{-3/2+3\beta/\gamma}\frac{L_{N-1}^2}{L_N^2}
\left( E|S_{N-1}|^2\right)^{1/2}.
\]
If we collect all terms and use the inductive argument from
above, we finally get the following estimate
\[
\left( E|S_N|^2\right)^{1/2} \lesssim C\sum_{n=1}^N
n^{-3/2+3\beta/\gamma}\frac{L_{n-1}^2}{L_n^2}
+ C\left(\sum_{n=1}^N n^{-2+4\beta/\gamma}\right)^{1/2}.
\]
Again, a routine verification shows that this latter expression
is of order $o(\ln N)$ (in fact, it is even bounded), as desired.
$\Box$
One can interchange the roles of $F$ and $(0,\infty)\setminus
F$ to obtain potentials with embedded {\it absolutely}
continuous spectrum. More precisely, proceed as follows:
Pick an even function $g\in {\cal S}({\Bbb R})$
(the Schwartz class) with
$g>0$ and $g(k)=1$ if $k\in F$. Define
\begin{equation}
\label{w1}
W(x)=\int_0^{\infty} \left( g(k)-\chi_F(k)\right) \cos 2kx\, dk.
\end{equation}
\begin{Theorem}
\label{T32}
Assume that $F, \gamma, g_n, B_n, L_n$ satisfy the
assumptions of Theorem \ref{T31}. Then, for all $\alpha$,
the half-line Schr\"odinger operator $H_{\alpha}$
with potential $V$ given by (\ref{V}), (\ref{vv}), (\ref{w1}) satisfies
$\sigma_{sc}(H_{\alpha})=\sigma_{ess}(H_{\alpha})=
[0,\infty)$, $\sigma_p(H_{\alpha})\cap (0,\infty) =\emptyset$ and
$\sigma_{ac}(H_{\alpha})=F^2$.
\end{Theorem}
Large parts of this {\it proof} are similar to the
corresponding arguments of the proof of Theorem \ref{T31}. Therefore,
these parts of the proof will only be sketched.
As before,
it is easy to verify the assertions on $\sigma_{ess},\sigma_{p}$.
We also have an analogue of Lemma \ref{L32} where the function
$\pi\chi_{F\cup -F}(k)/2$ now is replaced with
$\widehat{W}(k):=\pi(g(k)
-\chi_{F\cup -F}(k))/2$. Note that
$\widehat{W}(k)=0$ if $k\in F$. Moreover, if $I\subset (0,\infty)$
is a compact set with $I\cap F=\emptyset$, then $\inf_{k\in I}
|\widehat{W}(k)|>0$. Finally, we still have the estimates (\ref{est}),
because $g\in {\cal S}({\Bbb R})$ implies that
the part $\int g(k)\cos 2kx\, dk$ of $W$ decays faster than any
power of $x$.
To prove the assertion on $\sigma_{sc}$, fix $f\in C_0^{\infty}
(0,\infty)$ with $f\ge 0$, $\mbox{supp }f\cap F=\emptyset$, and
$\int f\, dk=1$. Now we can repeat the arguments
from the last part of the proof of Theorem \ref{T31}.
In this way, we see that $\rho_{ac}^{(\alpha)}((0,\infty)
\setminus F^2)=0$ for all $\alpha$, hence, since
$F$ is nowhere dense and $\sigma_{ess}=[0,\infty), \sigma_p
\cap (0,\infty)=\emptyset$, we must have $\sigma_{sc}(H_{\alpha})
=[0,\infty)$, as claimed.
Now let $dP(k)=|F|^{-1}\chi_F(k)\, dk$. We want to estimate
$EX_m\overline{X}_n\:(m\le n)$, where the expectation
is computed with this probability
measure. To this end, we first observe
that there exist functions $f_N(k)
\in C_0^{\infty}(0,\infty)\:(N\in {\Bbb N})$ with
$0\le f_N \le 1$ and
\begin{equation}
\label{fN}
\int\left| \chi_F(k)-f_N(k)\right|\, dk\le C2^{(1-\gamma)N},
\quad \int \left| f_N'(k)\right|\, dk = 2^{N+1}
\end{equation}
($\gamma$ is from Lemma \ref{L31}). To see this, simply
approximate $\chi_{F_N}$ by an appropriate smooth function.
Here, $F_N$ is the set obtained in the $N$th step of the
construction of $F$ (see the discussion preceding Lemma
\ref{L31}).
Using these approximations of $\chi_F$, we get (for $m0$ also
satisfies $\rho_{ac}^{(\alpha)}(F'^2)>0$ \cite[Theorem 5]{St}.
It is easy to see from the properties of $F$ that this forces
$\sigma_{ac}\supset F^2$. On the other hand, we know already
that $\sigma_{ac}\subset F^2$, hence $\sigma_{ac}=F^2$, as claimed.
$\Box$
\section{Subordinate solutions}
To begin with, recall the results from \cite{GP}.
Write $\|y\|_x:=(\int_0^x|y(t)|^2\,dt)^{1/2}$.
A solution $y\not\equiv 0$ of (\ref{se}) is called subordinate if
$\lim_{x\to\infty} \|y\|_x/\|w\|_x=0$ for every linearly
independent solution $w$ of the same equation.
The generalized eigenfunction $v_{\alpha}(x,E)$ is, by definition,
the solution of (\ref{se}) with the initial values
$v_{\alpha}(0,E)=-\sin\alpha, v'_{\alpha}(0,E)=\cos\alpha$.
Note that $v_{\alpha}$ satisfies the boundary condition described
by $\alpha$.
The following basic result relates the notion
of subordinacy to the boundary behavior of the $m$-function
(for more information on the $m$-function, see e.g.\ \cite{CL}).
\begin{Theorem}[\cite{GP}]
\label{Tgp}
$v_{\alpha}(\cdot,E)$ is subordinate if and only if
$\lim_{\epsilon\to 0+}|m_{\alpha}(E+i\epsilon)|=\infty$.
\end{Theorem}
Motivated by Theorem \ref{Tgp}, we consider the sets
\[
S_{\alpha} =\{ E\in {\Bbb R}:
v_{\alpha}(\cdot,E) \mbox{ is subordinate}\}.
\]
Then, by Theorem \ref{Tgp} and basic facts on the $m$-function,
$S_{\alpha}$ supports the singular part of the spectral
measure $\rho_{\alpha}$ (see \cite{GP}). Note that
each $S_{\alpha}$ has
Lebesgue measure zero because the $m$-function
has a finite limit almost everywhere. Therefore, the
result below is of interest. It shows that $S_{\alpha}$ can be large
without there being any singular continuous spectrum (namely,
if $g_n\in l_2\setminus l_1$ in Theorem \ref{T41}).
These phenomena can {\it not} occur for point or absolutely
continuous spectra. That is to say, the existence (or
non-existence) of point and absolutely continuous spectrum,
respectively, can always be read off from the size of the
corresponding minimal supports. This is due to the fact
that point and absolutely continuous measures are
equivalent to counting and Lebesgue measure, respectively,
restricted to these supports. Singular continuous measures
do not, in general, have such a universal property. See also
\cite{dRJLS,L} for a discussion of related issues in the
context of Schr\"odinger operators.
\begin{Theorem}
\label{T41}
Assume that $B_n=B$, $V_n(x)=W(x)$ (where $W\in L_1([-B,B])$,
$W\not\equiv 0$), $g_n\to 0$,
and $L_{n-1}/L_n\to 0$. Consider the Schr\"odinger operators
$H_{\alpha}$ with potential $V$ given by (\ref{V}). Then
for all boundary conditions $\alpha$, we have:\\
a) \cite{KLS}
If $g_n\in l_2$, then
the spectrum of $H_{\alpha}$ is purely absolutely
continuous on $(0,\infty)$.\\
b) If $g_n\notin l_1$, then
for any open set $U\subset (0,\infty)$, the set
$S_{\alpha}\cap U$ has Hausdorff dimension $1$.
\end{Theorem}
{\it Remarks.} 1. We could also treat $n$-dependent barriers
$V_n$ with our methods,
but in this case one needs additional assumptions
on the Fourier transforms $\widehat{V}_n$.
2. If $g_n\notin l_2$, the spectrum is purely singular
continuous on $(0,\infty)$ by \cite[Theorem 1.6(2)]{KLS}.
Note that part b) of the Theorem continues to hold. However,
in this case this statement follows quite easily from the
results of \cite{JL,Rso}, and the following rather involved
proof would clearly be inappropriate here.
{\it Proof.} a) This is \cite[Theorem 1.6(1)]{KLS}.
b) Here is the strategy of the proof. Fix $\alpha$, and assume,
without loss of generality, that $U$ is contained in a
compact subset of $(0,\infty)$. Also, as above,
it will be convenient
to work with wavenumbers $k=\sqrt{E}$ instead of energies
$E$. Since $I_n$ is constant, (\ref{eqR}) yields
\begin{equation}
\label{abc}
\ln R_{N+1}(k)=\frac{1}{2k}\im \widehat{W}(2k) \sum_{n=1}^N g_n
e^{2i\varphi_n(k)} + \rho_N(k)
\end{equation}
where $|\rho_N(k)|\le C(U)\sum_{n=1}^N g_n^2$ for
all $k\in U$ and
\[
\widehat{W}(2k)=\int_{-B}^B W(x) e^{2ik(x+B)}\, dx.
\]
We will construct a probability measure $P$ on $U$ so that the
sum on the right-hand side of (\ref{abc}) goes to $-\infty$ almost
surely with respect to $P$. Moreover, $P$ will give zero weight
to sets of Hausdorff dimension less than $1$. In this way, we
will obtain ``small'' solutions on a set of Hausdorff dimension $1$.
To carry out this program,
pick numbers $N_n\in {\Bbb N}$, such that $N_n\to\infty$ and
$N_nN_{n-1}L_{n-1}/L_n\to 0$ as $n\to \infty$. Since
$\widehat{W}$ is analytic, its zeros are isolated. Thus it
clearly suffices to prove the claim for the case when
$U$ is an open interval with $\inf_{k\in U} |\widehat{W}(2k)|>0$
(and, as above, $\inf U>0$, $\sup U<\infty$).
For $n\ge n_0$, we let
\begin{equation}
\label{ll}
l_n:=\frac{\pi}{N_nL_n}\left( 1 +\frac{\theta_n}{N_n}\right),
\end{equation}
with as yet unspecified $n_0\in {\Bbb N}$ and
$\theta_n\in [0,1]$.
We also set $l_{n_0-1}=|U|$. Now a straightforward computation
shows that it is possible to choose first $n_0$ sufficiently
large and then inductively the
$\theta_n\: (n\ge n_0)$, such that
\[
r_n:= \frac{l_{n-1}}{l_nN_n} \in {\Bbb N}\quad \forall n\ge n_0.
\]
For instance, we have that for $n>n_0$ and for
fixed $\theta_{n-1}$ the
quantity
\[
\frac{l_{n-1}}{l_nN_n}=\frac{L_n}{N_{n-1}L_{n-1}} \frac
{1+\theta_{n-1}/N_{n-1}}{1+\theta_n/N_n}
\]
runs over an interval of size $\ge L_n/(L_{n-1}N_{n-1}(N_n+1))$
as $\theta_n$ runs over $[0,1]$. Since this latter expression
tends to infinity, the corresponding interval must contain
an integer, provided $n$ is large enough. A similar argument
works in the case $n=n_0$.
In order to simplify the notation,
we will assume that $n_0=1$. The reader can verify easily that
our arguments are valid in the general case as well.
By definition of $r_1$ and $l_0$,
the original interval $U=I^{(0)}$ can be
divided into $r_1N_1$ subintervals $I^{(1)}_i$ of equal length
$l_1$. Each of these subintervals $I^{(1)}_i$, in turn,
can be divided into $r_2N_2$ sub-subintervals $I^{(2)}_j$ of equal
length $l_2$ etc. So we obtain a sequence of
partitions $U=\bigcup_j I^{(n)}_j$ which become increasingly finer.
Every interval $I^{(n)}_j$ belonging to the $n$th partition has
length $l_n$ and contains exactly $r_{n+1}N_{n+1}$ elements of
the $(n+1)$st partition.
Now we define correspondingly discrete approximations $\psi_n$
of the Pr\"ufer angles $\varphi_n$. The variable $\psi_n$
will take the values $t\pi/N_n\:(t=0,1,\ldots,N_n-1)$, and
$\psi_n$ will be constant on every $I^{(n)}_j$. Let
$n\in {\Bbb N}$, fix one of the intervals $I^{(n-1)}_i$, and
consider the corresponding subintervals $I^{(n)}_j\subset
I^{(n-1)}_i$. By construction, there are $r_nN_n$ such intervals
$I^{(n)}_j$. To fix the notation, let us assume that we obtain
these intervals if $j$ runs from $1$ to $r_nN_n$. We further
assume that this labeling is the natural one in the sense that
if $j0$, $\sup U<\infty$,
and $\inf_{k\in U} |\widehat{W}(2k)|
>0$. We further demand that the phase of $\widehat{W}$ does not
vary much if $k$ runs over $U$. More precisely, write
$\widehat{W}(2k)=|\widehat{W}(2k)|e^{i\beta(2k)}$; we suppose that
\begin{equation}
\label{beta}
\sup_{k,k' \in U} |\beta(2k)-\beta(2k')|\le \pi/8.
\end{equation}
This, of course, can be ensured by taking $U$ sufficiently small.
Next, we want to use Lemma \ref{L41} to construct appropriate
probability measures on $U$. Thus we need to specify
the $p_n(t)$ from (\ref{star}). To this end,
first choose $t_0(n)\in \{ 0,1,\ldots, N_n-1\}$ such that
\begin{equation}
\label{32}
\left|\frac{2t_0(n)\pi}{N_n}+\beta(2k)+\pi \right|
\le \frac{\pi}{4}
\end{equation}
for all $k\in U$ (the left-hand side of (\ref{32}) is to be
evaluated modulo $2\pi$). This is
possible by (\ref{beta}), at least if we assume, without loss of
generality, that $N_n\ge 8$.
We choose the $N_n$ as even numbers and set
\[
p_n(t_0(n))=\ldots=p_n(t_0(n)-1+N_n/2)= 2/N_n,
\]
and $p_n(t)=0$ otherwise. (In this definition, the argument
of $p_n$ has to be evaluated modulo $N_n$.) Now, Lemma \ref{L41}
yields a probability measure $P$, such that the $\psi_n$ are
independent with distributions determined by the $p_n(t)$,
according to Lemma \ref{L41}b).
We compute
\begin{eqnarray*}
E(e^{2i\psi_n}) & = & \frac{2}{N_n} \sum_{t=t_0(n)}^{t_0(n)-1+N_n/2}
e^{2\pi it/N_n} = -\frac{4}{N_n}
\frac{e^{2\pi it_0(n)/N_n}}{e^{2\pi i/N_n}-1}\\
& = & \frac{2i}{\pi} e^{2\pi it_0(n)/N_n} + O(N_n^{-1}).
\end{eqnarray*}
Note that by (\ref{32}), we have $\im ie^{2\pi it_0(n)/N_n}
e^{i\beta(2k)} \le -1/\sqrt{2}$ for all $n\in {\Bbb N}, k\in U$. Thus
by combining Lemma \ref{L41}c) and Lemma \ref{L42}
we see that on a $k$-set of
Hausdorff dimension $1$, the Pr\"ufer radius $R$ satisfies
\begin{equation}
\label{fin}
\ln R_{N+1}(k) \le -c \sum_{n=1}^N g_n
\end{equation}
for all large enough $N\ge N_0=N_0(k)$. The constant $c>0$ depends
only on $U$. If we consider the
corresponding set of energies $E=k^2$, we still have a set
of Hausdorff dimension $1$ because the map $k\to E=k^2$ is
bi-Lipschitz on every compact subset of $(0,\infty)$.
Recall that $R$ is given by $R(x,k)^2=v_{\alpha}(x,k)^2
+v'_{\alpha}(x,k)^2/k^2$ (cf.\ Section 2)
where $v_{\alpha}$ is the generalized
eigenfunction introduced at the beginning of this Section.
Now let $y$ be any solution of the Schr\"odinger equation
(\ref{se}) which is linearly independent of $v_{\alpha}(\cdot,k)$,
and let $\tilde{R}^2=y^2+y'^2/k^2$ and $\tilde{\varphi}$
be the corresponding Pr\"ufer
variables. Constancy of the Wronskian $W(v_{\alpha},y)=v_{\alpha}
y'-v'_{\alpha}y$ yields $R\tilde{R}\sin (\varphi-\tilde{\varphi})
=w\not= 0$, hence $\tilde{R}\ge |w|/R$. It is easy to see that
for the potentials under consideration, one has inequalities
of the type $\|y\|_x^2 \ge c \int_0^x \tilde{R}^2$ (with
$c>0$, of course)
for all sufficiently large $x$. Hence (\ref{fin}) together
with the estimate $\tilde{R}\ge |w|/R$ guarantee
that the solution $v_{\alpha}$ is subordinate. $\Box$
The methods of this proof clearly extend to more general
Hausdorff measures. More specifically, let $h(t)$ be an increasing,
right continuous function on $[0,\infty)$ with $h(0)=0$ and
$h(t)>0$ for $t>0$. Then one can define a (generalized)
Hausdorff measure $\mu_h$ (see \cite[Section 2.1]{Ro}; the
details are not of interest here). The usual $\gamma$-dimensional
Hausdorff measures
are obtained with the choice $h(t)=t^{\gamma}$. Now only a
minor modification in the proof of Lemma \ref{L41}c) is
needed to prove the following result: Let $h$ be as
above with $\lim_{t\to 0+} h(t)/t=\infty$.
If the $L_n$ grow sufficiently rapidly (this condition can be
put in a more quantitative version, of course), then
$\mu_h(S_{\alpha}\cap U)=\infty$ for all $\alpha$ and all
open sets $U\subset (0,\infty)$. In fact, we even have that
$S_{\alpha}\cap U$ is not $\sigma$-finite with respect to
$\mu_h$.
I do not think that this extension of Theorem \ref{T41}
gives much additional insight,
but it does provide explicit examples
where the set $S_{\alpha}$ is arbitrarily large in the
measure theoretic sense (given the restriction that $|S_{\alpha}|
=0$).
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