\input amstex
\loadbold
\documentstyle{amsppt}
\magnification=1200
\baselineskip=15 pt
%\NoBlackBoxes
\TagsOnRight
\def\gap{\vskip 0.1in\noindent}
\def\ref#1#2#3#4#5#6{#1, {\it #2,} #3 {\bf #4} (#5), #6.}
%References
\def\aron {1} % Aronszajn
\def\avs {2} % Avron-Simon IDS
\def\brfs {3} % Brenner-Fishman
\def\car {4} % Carmona
\def\cala {5} % Carmona-Lacroix
\def\cfs {6} % Cornfeld-Fomin-Sinai
\def\cyc {7} % Cycon et al
\def\dasi {8} % Davies-Simon (CMP)
\def\dasii {9} % Davies-Simon {JFA)
\def\deis {10} % Deift-Simon
\def\del {11} % Delyon
\def\dss {12} % Delyon-Simon-Souillard
\def\dono {13} % Donoghue
\def\furs {14} % Furstenberg
\def\gp {15} % Gilbert-Pearson
\def\goso {16} % Goldsheid-Sorets
\def\gor {17} % Gordon SP
\def\grfs {18} % Griniasty-Fishman
\def\ght {19} % Guillement-Helffer-Treton
\def\hesj {20} % Helffer-Sjostrand
\def\ish {21} % Ishii
\def\jito {22} %Jitomirskaya - to be published
\def\jlprl {23} % Jitomirskaya-Last PRL
\def\jli {24} % Jitomirskaya-Last PSI
\def\js {25} % Jitomirskaya-Simon
\def\kap {26} % Kahn-Pearson
\def\katz {27} % Katznelson
\def\kis {28} % Kiselev
\def\kls {29} % Kiselev-Last-Simon
\def\kot {30} % Kotani
\def\kots {31} % Kotani (support)
\def\kotfmv {32} % Kotani FMV
\def\lastz {33} % Last (zero measure)
\def\lath {34} % Last (thesis)
\def\lapaii {35} % Last (Per-Approx II)
\def\lsii {36} % Last-Simon
\def\lesa {37} % Levitan-Sargsjan
\def\pas {38} % Pastur
\def\pcmp {39} % Pearson (CMP)
\def\pea {40} % Pearson
\def\rue {41} % Ruelle
\def\sch {42} % Schnol
\def\ssg {43} % Simon (semigroups)
\def\scmp {44} % Simon (CMP 1982)
\def\svan {45} % Simon (Vancouver school)
\def\ssp {46} % Simon-Spencer
\def\sst {47} % Simon-Stolz
\def\thou {48} % Thouless
\topmatter
\title Eigenfunctions, Transfer Matrices, and Absolutely
Continuous Spectrum of One-dimensional Schr\"odinger
Operators
\endtitle
\rightheadtext{Eigenfunctions, Transfer Matrices, and A.C.~Spectrum}
\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
\thanks To be submitted to {\it{Annals of Math.}}
\endthanks
\date November 18, 1996
\enddate
\endtopmatter
\document
\vskip 0.3in
\flushpar{\bf \S 1. Introduction}
\vskip 0.1in
In this paper, we will primarily discuss one-dimensional discrete Schr\"odinger
operators
$$
(hu)(n)=u(n+1)+u(n-1)+V(n)u(n) \tag 1.1D
$$
on $\ell^2(\Bbb Z)$ (and the half-line problem, $h_+$, on $\ell^2
(\{n\in\Bbb Z\mid n >0\}) \equiv \ell^2 ({\Bbb Z}^+)$) with $u(0)=0$ boundary
conditions. We will also discuss the continuum analog
$$
(Hu)(x)=-u''(x)+V(x)u(x) \tag 1.1C
$$
on $L^2 (\Bbb R)$ (and its half-line problem, $H_+$, on $L^2 (0,\infty)$ with
$u(0)=0$ boundary conditions).
We will focus on a new approach to the absolutely continuous spectrum
$\sigma_{\text{\rom{ac}}}(h)$ and, more generally, $\Sigma_{\text{\rom{ac}}}
(h)$, the essential support of the a.c.~part of the spectral measures.
What is new in our approach is that it relies on estimates on the transfer matrix, that
is, the $2\times 2$ matrix $T_E (n,m)$ which takes $\binom{u(m+1)}{u(m)}$ to $\binom{u(n+1)}{u(n)}$ for solutions $u$ of $hu=Eu$ (in the continuum case use
$\binom{u'(x)}{u(x)}$ instead of $\binom{u(x+1)}{u(x)}$). We let $T_E (n)\equiv
T_E (n,0)$. For example, we will prove the following:
\proclaim{Theorem 1.1} Let $h_+$ be the operator {\rom{(1.1D)}} on $\ell^2
(\{n\in\Bbb Z \mid n>0\})$ with $u(0)=0$ boundary conditions. Let
$$
S=\biggl\{ E\biggm| \varliminf_{L\to\infty} \frac{1}{L}\, \sum^L_{n=1}
\| T_E (n)\|^{2} < \infty\biggr\}.
$$
Then $S$ is an essential support of the a.c.~part of the spectral measure for $h_+$
\rom(i.e., $S=\Sigma_{\text{\rom{ac}}}(h)$\rom) and $S$ has zero measure with
respect to the singular part of the spectral measure.
\endproclaim
The behavior of the transfer matrix is a reflection of the behavior of eigenfunctions
since $T$ is built out of eigenfunctions. Indeed, if $u$ and $w$ are any two linearly
independent solutions of $hu=Eu$ normalized at $0$, then $\frac{1}{L}\sum^L_{n=1}
\|T(n)\|^2$ and $\frac{1}{L}\sum^{L+1}_{n=1} [|u(n)|^2 + |w(n)|^2]$ are comparable
and so Theorem~1.1 relates the a.c.~spectrum to the behavior of eigenfunctions.
That there is a connection between eigenfunctions and a.c.~spectrum is not new.
Gilbert-Pearson [\gp] related a.c.~spectrum to subordinate solutions. Typical is
the following (actually due to [\kap]; see also [\jlprl,\jli]): Call a solution $u$ of
$hu=Eu$ subordinate if and only if for any linearly independent solution $w$,
$$
\sum^L_{n=1} |u(n)|^2 \bigg/ \sum^L_{n=1} |w(n)|^2 \to 0 \tag 1.2
$$
as $L\to \infty$. Let
$$
S_0 =\{E\mid \text{there is no subordinate solution}\}.
$$
Then $S_0$ is an essential support of the a.c.~part of the spectral measure for $h_+$
and $S_0$ has zero measure with respect to the singular part of the spectral measure.
The Gilbert-Pearson theory provides one-half of the proof of Theorem~1.1. Indeed, we
will show that $S\subset S_0$. The other direction is intimately related to some new
eigenfunction estimates which we discuss in Section~2. Its relation to the theory of
Browder, Berezinski, Garding, Gel'fand, and Kac is discussed in the appendix.
Related to Theorem~1.1 is the following, which also relies on the eigenfunction estimate
of Section~2:
\proclaim{Theorem 1.2} Let $h_+$ be as in Theorem~{\rom{1.1}}. Let $m_j, k_j$
be arbitrary sequences in $\{n\in\Bbb Z\mid n>0\}$ and let
$$
S_1 =\biggl\{E\biggm| \varliminf_{j\to\infty} \|T_E (m_j, k_j)\| <\infty\biggr\}.
$$
Then $S_1$ supports the a.c.~part of the spectral measure for $h_+$ in that
$\rho_{\text{\rom{ac}}}(\Bbb R\backslash S_1)=0$.
\endproclaim
These two theorems allow us to recover virtually all the major abstract results proven
in the past fifteen years on the a.c.~spectrum for ergodic Schr\"odinger operators with
the exception of Kotani's results [\kot,\kotfmv] on $\{E\mid\gamma (E)=0\}$. More
significantly, they establish new results and settle an important open problem. Among
the results recovered via a new proof are the Ishii-Pastur theorem [\ish,\pas], Kotani's
support theorem [\kots], and the results of Simon-Spencer [\ssp].
In a companion paper with A.~Kiselev [\kls], we will use Theorems~1.1, 1.2 and
Theorem~1.3 below to analyze, recover, and extend results on decaying random
potentials [\scmp,\dss,\del], sparse potentials [\pcmp,\pea], and $n^{-\alpha}
(1>\alpha >\frac{3}{4})$ potentials [\kis].
Theorem~1.1 and Fatou's lemma immediately imply that if $Q$ is any subset of
$\Bbb R$ and
$$
\sup_{n} \int_Q \|T_E (n)\|^2 \, dE <\infty, \tag 1.3
$$
then $Q$ lies in the essential support of $d\rho_{\text{\rom{ac}}}$ (for Fatou's
lemma and (1.3) show for a.e.~$E\in Q$ we have that $\varliminf \frac{1}{L}
\sum^L_{n=1} \|T_E (n)\|^2 <\infty$) but (1.3) does not seem to eliminate the
possibility of singular spectrum on $Q$ (on the set of Lebesgue measure zero where
Fatou does not apply). In this regard, the following result, which is an extension of
ideas of Carmona [\car], is of interest:
\proclaim{Theorem 1.3} Suppose that
$$
\varliminf_{n\to\infty} \int^b_a \|T_E (n)\|^p\,dE <\infty
$$
for some $p>2$. Then the spectrum is purely absolutely continuous on $(a,b)$.
\endproclaim
It is interesting to compare Theorems~1.2 and 1.3. A priori, one might think there
could be potentials so there exist $n_1 0\}$.
We say that $W$ is a right limit of $V$ if and only if there exist $n_j \to\infty$ so
that $V(n-n_j) \to W(n)$ as $j\to\infty$ for each fixed $n>0$.
\enddefinition
Then we will prove from Theorem~1.1 and the eigenfuction expansion results of
Section~2 that
\proclaim{Theorem 1.4} If \, $W$ is a right limit of $V$ and $\tilde h_+, h_+$ are
the half-line Schr\"odinger operators associated to $W,V$ respectively, then
$\Sigma_{\text{\rom{ac}}} (h_+)\subset \Sigma_{\text{\rom{ac}}}(\tilde h_+)$.
\endproclaim
\remark{Remark} This result is particularly interesting because it is easy to see that
$\sigma_{\text{\rom{ess}}} (\tilde h_+) \subset\sigma_{\text{\rom{ess}}} (h_+)$
with the inclusion in the opposite direction.
\endremark
Our proof of Theorem~1.4 depends on the shift to transfer matrices rather than
eigenfunctions.
This theorem will have an important corollary:
\proclaim{Theorem 1.5} Let $W$ be an almost periodic function on $\Bbb Z$
\rom(resp.~$\Bbb R$\rom). Let $h$ \rom(resp.~$H$\rom) be the full-line operator
given by {\rom{(1.1)}}. For each $W_\omega$ in the hull of $W$\!, let $h_\omega$
\rom(resp.~$H_\omega$\rom) be the corresponding operator. Then the a.c.~spectrum,
indeed the essential support of the a.c.~spectrum, of $h_\omega$ is independent of
$\omega$.
\endproclaim
\remark{Remarks} 1. The result holds more generally than almost periodic potentials.
It suffices that the underlying process be minimally ergodic.
2. We will also recover the Deift-Simon [\deis] result that the multiplicity of the
a.c.~spectrum is $2$.
3. Following Pastur [\pas] and others (see [\cala,\cyc]), it is known that the spectrum
and its components are a.e.~constant on the hull. In 1982, Avron-Simon [\avs] proved
that the spectrum is everywhere constant rather than a.e.~constant in the almost
periodic case. Theorem~1.5 has been believed for a long time, but this is its first
proof. It is known (see Jitomirskaya-Simon [\js]) that the s.c.~ and p.p.~components
need {\it{not}} be everywhere constant.
\endremark
In this paper, we will also obtain rigorous spectral results on the operator $h_+$ where
$V(n)=\lambda\cos(n^\beta)$, and $1<\beta$ is not an integer.
Theorem~1.5 is reminiscent of the invariance of the a.c.~spectrum under rank one
perturbations for all couplings. This is no coincidence. In our development of
Theorems~1.1--1.2, what distinguishes a.c..~spectrum from non-a.c.~spectrum is its
invariance under boundary conditions.
While the main focus of this paper is on the a.c.~spectrum and transfer matrices,
we will say something about point spectrum also. In this introduction, we will focus
on the discrete case with $V$ bounded. In [\sst], using constancy of the Wronskian,
Simon-Stolz proved
\proclaim{Theorem 1.6 ([\sst])} If $\sum^\infty_{n=1} \|T_E (n)\|^{-2}=\infty$, then
$hu=Eu$ has no solution which is $L^2$ at infinity.
\endproclaim
As we will see in Section~8, it can happen that $\sum^\infty_{n=1} \|T_E (n)\|^{-2}
<\infty$ without there being a solution $L^2$ at infinity; indeed, without there even
being a bounded solution, but $\sum^\infty_{n=1} \|T_E (n)\|^{-2} <\infty$ has one
important consequence. Call a solution $u$ of $hu=Eu$ strongly subordinate if for
any linearly independent solution $w$ we have that
$$
[u(n)^2 +u(n+1)^2] \big/ [w(n)^2 + w(n+1)^2] \to 0
$$
as $n\to\infty$. It is easy to see that any strongly subordinate solution is subordinate.
We will prove that
\proclaim{Theorem 1.7} If $V$ is bounded and $\sum^\infty_{n=1} \|T_E (n)\|^{-2}
<\infty$, then there is a strongly subordinate solution of $hu=Eu$. This solution,
$u_\infty$, obeys the estimate
$$
\|u_\infty (n)\|^2 \leq \| T_E(n)\|^{-2} + \frac{\pi^2}{4} \, \| T_E(n)\|^2
\biggl(\, \sum^\infty_{m=n} \frac{1}{\|T_E(m)\|^2}\biggr)^2.
$$
In particular, if
$$
\sum ^\infty_{n=1} \biggl\{\|T_E (n)\|^2 \biggl(\,\sum^\infty_{m=n}
\|T_E (m)\|^{-2} \biggr)^2 \biggr\} <\infty,
$$
then $hu=Eu$ has an $L^2$ solution.
\endproclaim
Theorem~1.7 is essentially an abstraction of a well-known argument of Ruelle [\rue].
We will use it in [\kls,\lsii] to prove point spectrum in certain models, including new
and simplified proofs of the results of Simon [\scmp] and some of the results of
Gordon [\gor].
The plan of this paper is as follows. In Section~2 we develop eigenfunction estimates.
Their relation to the BGK eigenfunction expansions is discussed in the appendix which
includes higher-dimensional results. In Section~3 we use the results of Section~2 and
the Gilbert-Pearson theory to prove Theorems~1.1 and 1.2 and we will use Carmona's
formula to prove Theorem~1.3. In Section~4 we recover and extend the Simon-Spencer
[\ssp] results. In Section~5 we prove Theorem~1.4 and in Section~6 we prove
Theorem~1.5 and some other consequences of Theorem~1.4, including the Kotani
support theorem. In Section~7 we discuss $\lambda\cos (n^\beta)$. In Section~8
we prove Theorems~1.7 and 1.8.
We would like to thank Bert Hof and Svetlana Jitomirskaya for useful discussions.
B.S.~would like to thank M.~Ben-Artzi for the hospitality of the Hebrew University
where some of this work was done.
\vskip 0.3in
\flushpar{\bf \S 2. Eigenfunction Estimates}
\vskip 0.1in
We consider half-line problems in this section. In the discrete case for fixed $V(n)$ and
$z\in\Bbb C$, define $u_D (n), u_N (n)$ to be the solution of $hu=zu$ ($h$ given by
(1.1D)) with boundary conditions
$$\align
u_D (0)=0 & \qquad u_D (1) = 1 \\
u_N (0) = 1 & \qquad u_N (1) = 0.
\endalign
$$
We will use $X$ to denote $D$ or $N$ in formulas where either is valid, and $Y$ for
the opposite condition.
In the continuum case, $u_D, u_N$ obey $Hu=zu$ ($H$ given by (1.1C)) with boundary
conditions
$$\align
u_D (0) = 0 & \qquad u'_D(0) = 1 \\
u_N (0)=1 & \qquad u'_N (0) = 0.
\endalign
$$
Of course, $u$ is $z$-dependent and we will sometimes use $u(\,\cdot \, ; z)$. It is
standard that $u(n;z)$, $u(x;z)$, and $u'(x;z)$ are entire functions of $z$ for real
$x,n$.
The solutions $u$ are related to the transfer matrix by
$$
T_E (n) = \pmatrix u_N (n+1) & u_D (n+1) \\
u_N (n) & u_D (n)
\endpmatrix \tag 2.1D
$$
in the discrete case and
$$
T_E (x) =\pmatrix u'_N (x) & u'_D (x) \\
u_N (x) & u_D (x)
\endpmatrix \tag 2.1C
$$
in the continuum case.
For $z\in\Bbb C_+ =\{z\mid\text{Im } z >0\}$, there is a unique solution $L^2$ at
$+\infty$ (for arbitrary $V$ in the discrete case and for $V$ which is limit point at
infinity in the continuum case). Both it and its derivative (in the continuum case) are
everywhere non-vanishing. In the continuum, we denote the solution by $\varphi^D_+
(x;z)$ if normalized by $\varphi^D_+ (0;z)=1$ and $\varphi^N_+ (x;z)$ if normalized
by $(\varphi^N_+)'(0;z)=-1$, and in the discrete case $\varphi^D_+ (0;z)=1$,
$\varphi^N_+ (1;z)=-1$. This normalization is chosen so that the Wronskian of
$\varphi^X_+$ and $u_X$ is $+1$.
The $m$-functions are defined by
$$
\varphi^X_+ (\,\cdot\,; z)=\pm u_Y (\,\cdot\,; z) + m_X (z) u_X (\,\cdot\,;z) \tag 2.2
$$
where we take the plus sign in case $X=D$ and minus in case $X=N$. (Noting that the
Wronskian of $\varphi^X_+$ and $\varphi^Y_+$ is zero, we see that $m_X (z)m_Y (z)
= -1$.)
It is well known (see, e.g., [\cala,\katz]) that the $m$-functions are Herglotz (i.e., analytic
with $\text{Im }m>0$ on $\Bbb C_+$) and that the measures
$$
d\rho^X (E) =\lim_{\epsilon\downarrow 0}\,\frac{1}{\pi}\, \text{Im }m_X
(E + i\epsilon)\, dE \tag 2.3
$$
are spectral measures for the operator $H_X$ ($h$ or $H$ with appropriate boundary
conditions; i.e., in the continuum case $H_{D,N}$ are defined on $L^2 (0,\infty)$ with
$u(0)$ or $u'(0)$ boundary conditions, and in the discrete case $H_D$ (resp.~$H_N$)
is defined on $\ell^2 (\Bbb Z_+)$ (resp. $\ell^2 (\{2,3,\dots,\})$) with $u(0)=0$
(resp.~ $u(1)=0$) boundary conditions). That is, $H_X$ is unitarily equivalent to
multiplication by $E$ on $L^2 (\Bbb R, d\rho^X (E))$. Note that in (2.3) (and similarly
(2.8) below), the limit is intended in the weak sense, that is, holds when smeared in $E$
with continuous functions of compact support.
In the discrete case and in the continuum case with $X=N$, we have
$$
\int \frac{d\rho^X (E)}{|E|+1} < \infty \tag 2.4a
$$
and
$$
m_X (z) = \int \frac{d\rho^X (E)}{E - z}. \tag 2.4b
$$
In the continuum case with $X=D$, we only have
$$
\int \frac{d\rho^D (E)}{E^2 + 1} <\infty \tag 2.5a
$$
and a Herglotz representation
$$
m_D (z)=a_0 + \int \biggl(\frac{1}{E - z} - \frac{E}{1+E^2}\biggr)\,
d\rho^D (E) \tag 2.5b
$$
for a suitable real constant $a_0$.
We are heading toward a proof of the following theorems:
\proclaim{Theorem 2.1D} In the discrete case, for any $V$ and $n$,
$$
\int |u_X (n;E)|^2 \,d\rho^X (E) = 1. \tag 2.6D
$$
\endproclaim
\proclaim{Theorem 2.1C} In the continuum case for any $V\geq 0$ and all $x$,
$$
\int \frac{|u_X (x;E)|^2}{E+1} \, d\rho^X (E) \leq \frac{1}{2} \,
(1\mp e^{-2|x|}) \tag 2.6C(a)
$$
where $\mp$ correspond to $X=D/N$. Moreover, for a universal constant $C$, we
have that for all $x$
$$
\int \frac{\bigl[\int^{x+1}_{x-1} |u'_X (y;E)|^2 \,dy\bigr]}{(E+1)^2} \,
d\rho^X (E) \leq C . \tag 2.6C(b)
$$
\endproclaim
\remark{Remarks} 1. In (2.6C(b)), if $x<1$, interpret $x-1$ as $0$.
2. Obviously, $V\geq 0$ can be replaced by $V\geq c$ for any $c$ if $(E +1)^{-1}$
in (2.6C) is replaced by $(E -c+1)^{-1}$. The proof shows that as long as $-V \leq
\alpha (-\Delta) + \beta$ for some $\alpha < 1$, estimates similar to (2.6C) hold (with
$(E +1)^{-1}$ replaced by $(E + |\beta| + (1-\alpha)^{-1}$) and the $\frac{1}{2}$
(resp.~$1$) in the inequality replaced by $\frac{1}{2} (1-\alpha)^{-1}$
(resp.~$(1-\alpha)^{-1}$). Thus, the result allows any $V$ whose negative part is
uniformly locally $L^1$.
\endremark
\smallskip
As a preliminary we note that
\proclaim{Lemma 2.2} {\rom{(a)}} $\epsilon^2 |m(E + i\epsilon)|\to 0$ as $\epsilon
\downarrow 0$ uniformly for $E$ in compact subsets of $\Bbb R$.
{\rom{(b)}} $\epsilon |\text{\rom{Re }} m(E +i\epsilon)|\to 0$ as $\epsilon \downarrow
0$ and is uniformly bounded for $E$ in compacts.
\endproclaim
\demo{Proof} (a) is a direct consequence of (2.4/2.5). (b) follows from those formulas
and the dominated convergence theorem. \qed
\enddemo
The resolvent, $(H_X -z)^{-1}$, of the operator $H_X$ has a continuous integral
kernel (in the continuum case). In general, this kernel $G_X (x,y;z)$ has the form
$$
G_X (x,y;z)=u_X (x_< ;z) \varphi^X_+ (x_>; z) \tag 2.7
$$
where $x_< =\min (x,y)$, $x_> =\max (x,y)$. This formula is easy to verify and shows
that $G$ is continuous.
\proclaim{Theorem 2.3}
$$
\lim\limits_{\epsilon\downarrow 0}\,\frac{1}{\pi}\, \text{\rom{Im}}\, G_X
(x,x; E+i\epsilon)\, dE = |u_X (x,E)|^2 \, d\rho^X (E). \tag 2.8
$$
\endproclaim
\demo{Proof} By (2.7) and (2.2),
$$
G_X (x,x; E + i\epsilon)=\pm u_Y (x, E +i\epsilon) u_X (x; E +i\epsilon)
+m_X (E +i\epsilon) u_X (x; E +i\epsilon)^2.
$$
Since $u_{X,Y}$ are entire and real for $z$ real, we have that
$\lim _{\epsilon\downarrow 0} \,\text{Im}\, u_Y (x , E +i\epsilon) u_X
(x, E +i\epsilon) =0$. Similarly, $u_X (x; E + i \epsilon)^2 = u_X (x; E)^2 +
i\epsilon a(x; E) + O(\epsilon^2)$ where $u^2_X$ and $a(x)$ are real. Thus,
$$
\text{Im }[m_X (E +i\epsilon) u_X (x; E +i\epsilon)^2] = \boxed{1} +
\boxed{2} + \boxed{3}
$$
with
$$
\boxed{1} = u_X (x;E)^2 \text{ Im }m_X (E +i\epsilon)\to
\pi |u_X (x,E)|^2 \,d\rho^X (E)
$$
by (2.3) and
$$
\boxed{2}=\epsilon a(x; E)\text{ Re }m(E +i\epsilon)\to 0
$$
by Lemma~2.2(b) and
$$
\boxed{3} =\text{Im}[O(\epsilon^2)m(E +i\epsilon)]\to 0
$$
by Lemma~2.2(a). Thus, (2.8) is proven. \qed
\enddemo
\remark{Remarks} 1. (2.8) is essentially a version of the spectral theorem. We will
discuss this further in the appendix.
2. The same method shows more generally that
$$
\lim\limits_{\epsilon\downarrow 0} \frac{1}{\pi}\, G_X (x,y; E +i\epsilon) =
u_X (x,E) u_X (y,E)\, d\rho^X (E). \tag 2.8$^\prime$
$$
3. (2.8/2.8$^\prime$) are not new; they are implicit, for example, in Section~II.3 of
Levitan-Sargsjan [\lesa].
\endremark
\demo{Proof of Theorem {\rom{2.1D}}} (2.8) says that $|u|^2 \,d\rho$ is the spectral
measure for $H_X$ with vector $\delta_n$. Thus, $\int |u(n;E)|^2\,d\rho^X (E)=
(\delta_n, \delta_n) =1$. \qed
\enddemo
\demo{Proof of Theorem {\rom{2.1C}}} $G_D (x,x;z)$ is analytic in $\Bbb C\backslash
[0,\infty)$ and goes to zero as $|z| \to \infty$. It follows that
$$
G_X (x,x; -1)=\int \frac{|u(x,E)|^2\, d\rho^X (E)}{E +1}\,.
$$
But since $V\geq 0$, $(H_X +1)^{-1}\leq (H^{(0)}_X +1)^{-1}$ where $H^{(0)}_X$
is the operator when $V=0$. Thus,
$$
G_X (x,x;-1) \leq G^{(0)}_X (x,x;-1)=\frac{1}{2}\, (1 \pm e^{-2|x|})
$$
by the method of images formulas for $G^{(0)}$. This proves (2.6C(a)).
To prove (2.6C(b)) where $x\geq 2$, pick $g$ a $C^\infty$ function with $0\leq g\leq
1$, $g$ supported on $[-2,2]$, and $g\equiv 1$ on $[-1, 1]$. Let $f(y)=g(y-x)$. Then
$$\align
\int^{x+1}_{x-1} (u')^2\, dy & \leq \int f^2 (u')^2\, dy \\
& = -\int f^2 u''u \, dy -\frac{1}{2}\int (u^2)' (f^2)' \, dy \\
& = \int f(E -V) u^2\, dy + \frac{1}{2} \int (f^2)'' u^2\, dy \\
& \leq C(1+|E|) \int^{x+2}_{x-2} u^2\, dy.
\endalign
$$
Thus, (2.6(b)) for $x\geq 2$ follows from (2.6(a)).
A similar calculation works for $x=1$. Explicitly, pick $f$ which is supported on
$[0,3)$ and $f\equiv 1$ on $[0,2]$. Because $u(0)u'(0)=0$, the above calculations still
show that
$$\align
\int^2_0 |u(y)'|^2\, dy & \leq \int f(E-V) u^2\, dy + \frac{1}{2} \int (f^2)'' u^2 \, dy \\
& \leq C(1+|E|) \int^3_0 u^2 \, dy.
\endalign
$$
(2.6(b)) for $x=1$ and $x\geq 2$ imply the result for all $x$. \qed
\enddemo
\remark{Remark} If $V$ is uniformly locally $L^2$, one can show that (2.6C(b)) holds
without the need for integrating over $y$.
\endremark
\vskip 0.3in
\flushpar{\bf \S 3. Criteria for A.C.~Spectrum}
\vskip 0.1in
Our main goal in this section is to prove Theorems~1.1 and 1.2 as well as a continuum
analog of Theorem~1.2. We begin with an estimate based on the Gilbert-Pearson theory
and then apply the bounds of Section~2. We will then provide a new proof of the
Pastur-Ishii theorem. Finally, we present a condition for purely a.c.~spectrum.
Fix $V$ and $E$. For each $\theta\in [0,\pi)$, let $\Phi_\theta$ be the vector formed
from the solution with $(\sin\theta, \cos\theta)$ boundary conditions at $0$, that is,
$$
\Phi_\theta (\,\cdot\,)=T_E (\,\cdot\,) \binom{\sin\theta}{\cos\theta} \tag 3.1a
$$
and let $\Psi_\theta$ be $\Phi_{\pi/2 +\theta}$, that is,
$$
\Psi_\theta (\,\cdot\,)=T_E (\,\cdot\,) \binom{\cos\theta}{-\sin\theta}. \tag 3.1b
$$
Define $u_\theta, v_\theta$ by $\Phi_\theta (n) = \binom{u_\theta (n+1)}{u_\theta(n)}$,
$\Psi_\theta(n) = \binom{v_\theta (n+1)}{v_\theta (n)}$.
The Wronskian of $u$ and $v$ is constant, that is, $\langle \Phi, J\Psi\rangle =1$ with
$J=\left(\smallmatrix 0&1 \\ -1&0\endsmallmatrix\right)$. It follows by the
Cauchy-Schwarz inequality that
$$
\| \Phi_\theta (n)\| \, \|\Psi_\theta (n)\| \geq 1. \tag 3.2
$$
Clearly, $\| \Psi_\theta (\,\cdot\,)\| \leq \|T_E (\,\cdot\,)\|$ by (3.1b). Let us use the
symbol $\frac{1}{L} \int^L_0 \cdot \, dx$ for the integral in the continuum case and
for the sum $\frac{1}{L} \sum^L_{n=1}\cdot\,$ in the discrete case. Then
$$
\frac{1}{L} \int^L_0 \|\Psi_\theta (x)\|^2\, dx \leq \frac{1}{L} \int^L_0
\|T_E (x) \|^2 \, dx. \tag 3.3
$$
By (3.2),
$$\align
1 &\leq \biggl( \frac{1}{L}\int^L_0 \|\Phi_\theta (x)\| \, \|\Psi_\theta (x)\| \,
dx\biggr)^2 \\
& \leq\biggl( \frac{1}{L} \int^L_0 \|\Phi_\theta (x)\|^2 \biggr) \biggl( \frac{1}{L}
\int^L_0 \|\Psi_\theta (x)\|^2 \, dx\biggr). \tag 3.4
\endalign
$$
(3.3) and (3.4) immediately imply
\proclaim{Lemma 3.1}
$$
\frac{\int^L_0 \| \Psi_\theta (x)\|^2 \, dx}{\int^L_0 \| \Phi_\theta (x)\|^2 \, dx} \leq
\biggl( \frac{1}{L} \int^L_0 \| T_E (x)\|^2 \, dx \biggr)^2 \tag 3.5
$$
\endproclaim
Recall the definitions of Gilbert-Pearson. A solution $u_\theta$ is called subordinate if
and only if
$$
\lim\limits_{x\to\infty} \,
\frac{\int^L_0 | u_\theta (x)|^2\, dx}{\int^L_0 | v_\theta (x) |^2 \, dx} = 0. \tag 3.6
$$
To use (3.6), we must deal with the fact that $\Phi, \Psi$ are not quite the same as
$u,v$. In the discrete case, we have that
$$
\sum^{L+1}_{n=1} | v_\theta (n)|^2 \leq \sum^L_{n=1} \| \Psi_\theta (n)\|^2
$$
while
$$\align
\sum^{L+1}_{n=1} |u_\theta (n) |^2 &\geq \frac12 \sum^L_{n=1} (|u_\theta (n)|^2 +
|u_\theta (n+1)|^2) \\
&\geq \frac12 \sum^L_{n=1} \| \Phi_\theta (n)\|^2
\endalign
$$
so returning to $\int^L_0 \,\cdot\, dx$ notation for the sum
$$
\frac{\int^{L+1}_0 |v_\theta (x)|^2 \, dx}{\int^{L+1}_0 |u_\theta (x)|^2 \, dx}
\leq \frac{2 \int^L_0 \| \Psi_\theta (x)\|^2\, dx}{\int^L_0 \| \Phi_\theta (x)\|^2 \, dx}
$$
so
$$
\frac{\int^{L+1}_0 |v_\theta (x)|^2\, dx}{\int^{L+1}_0 |u_\theta (x)|^2 \, dx} \leq
2 \biggl( \frac{1}{L} \int^L_0 \|T_E (x) \|^2\, dx \biggr)^2. \tag 3.7D
$$
In the continuum case, one can mimic the proof of Theorem~2.1C to see that if $V
\geq 0$, then
$$
\int^L_0 (u_\theta (x)^2 + u'_\theta (x)^2)\, dx \leq C (1+|E|) \int^{L+1}_0
u_\theta (x)^2\, dx.
$$
Thus,
$$\align
\frac{\int^{L+1}_0 v^2_\theta (x)\, dx}{\int^{L+1}_0 u^2_\theta (x)\, dx} & \leq
C(1+|E|) \, \frac{\int^{L+1}_0 \| \Psi_\theta (x)\|^2 \, dx}{\int^L_0 \| \Phi_\theta
(x)\|^2 \, dx} \\
&\leq C (1+|E|) \biggl( \frac{1}{L} \int^L_0 \| \Psi_\theta (x) \|^2 \, dx \biggr)
\biggl( \frac{1}{L} \int^{L+1}_0 \| \Psi_\theta (x)\|^2 \, dx \biggr) \\
&\leq \biggl( \frac{(L+1)}{L}\biggr)^2 C(1+|E|) \biggl( \frac{1}{L+1}
\int^{L+1}_0 \|T_E (x) \|^2\, dx \biggr)^2. \tag 3.7C
\endalign
$$
(3.6) and (3.7) imply that
\proclaim{Theorem 3.2} If $H$ has a subordinate solution at energy $E$, then
$$
\lim\limits_{L\to \infty} \, \frac{1}{L} \int ^L_0 \| T_E (x)\|^2\, dx
= \infty. \tag 3.8
$$
\endproclaim
Let $Q = \{E\mid H \text{ has a subordinate solution at energy } E\}$ and let
$S_0 = \Bbb R \backslash Q$. Recall $S$, the set of Theorem~1.1, is given by
$$
S= \biggl\{E \biggm| \varliminf\limits_{L\to\infty} \frac{1}{L} \int ^L_0
\| T_E (x)\|^2 \, dx <\infty \biggr\}.
$$
Theorem~3.2 says that $Q\subset \Bbb R \backslash S$ so $S\subset S_0$.
Gilbert-Pearson have shown that $S_0$ is the essential support of the a.c.~part, $\mu_{\text{\rom{ac}}}$, of the spectral measure of $H_+$. Thus, $S\subset S_0$
implies that if $A\subset S$ and $|A| > 0$, then $\mu_{\text{\rom{ac}}}(A) >0$.
Theorem~1.1 thus follows from
\proclaim{Proposition 3.3} For a.e.~$E$ w.r.t.~ $\mu_{\text{\rom{ac}}}$, we
have that $E\in S$.
\endproclaim
\demo{Proof} In terms of the measures $d\rho^X$ of Section~2, let $d\mu(E) = \min
(d\rho^D, d\rho^N)$ in the discrete case and $d\mu =(1+E^2)^{-1} \min (d\rho^D,
d\rho^N)$ in the continuum case. Since the singular parts of $d\rho^D$ and
$d\rho^N$ are disjoint and the a.c.~parts are mutually equivalent (see, e.g., [\svan]),
$d\mu$ is equivalent to the a.c.~part of the spectral measure for $H_+$. By (2.1)
and (2.6), we have that for each $n$,
$$
\int d\mu (E) \| T_E (n)\|^2 \leq 4 \tag 3.9D
$$
in the discrete case and for each $x_0 \geq 1$,
$$
\int d\mu (E) \int ^{x_0 +1}_{x_0 -1} \| T_E (x)\|^2 \leq C \tag 3.9C
$$
in the continuum case. Here $C$ is a universal constant. It follows that
$$
\int d\mu (E) G_L (E) \leq C, \tag 3.10
$$
where
$$
G_L (E) =\frac{1}{L} \sum^L_{n=1} \|T_E (n)\|^2
$$
in the discrete case and
$$
G_L (E) =\frac{1}{Q(L)}\, \int ^L_0 \| T_E (x)\|^2\, dx,
$$
where $Q(L)$ is the smallest even integer less than $L$ (so $L/Q(L) \to 1$ as
$L\to\infty$).
By (3.10) and Fatou's lemma, $\int d\mu (E) \varliminf G_L (E) <\infty$, so
$\varliminf G_L (E) <\infty$ a.e. w.r.t. $d\mu$, that is, $E\in S$ for a.e.~$E$
w.r.t.~to $d\mu$. \qed
\enddemo
\remark{Remark} An immediate consequence of Theorem~1.1 is that if $V_\omega$
is an ergodic family of potentials and the Lyapunov exponent $\gamma (E)>0$ on a
Borel set $T\subset\Bbb R$, then for a.e.~$\omega$, $\Sigma_{\text{\rom{ac}}}
(H_\omega) \cap T =\emptyset$. For by Fubini's theorem for a.e.~$\omega$, for
a.e.~$E\in T$, we have $\lim \frac{1} {n} \ln \|T(n)\| >0$ so that a fortiori, $\lim \frac{1}{L}\sum^L_{n=1}\|T(n)\|^2 = \infty$ and thus, for a.e.~$\omega$, $S\cap T$
has zero Lebesgue measure. This result is the celebrated Ishii-Pastur theorem
[\ish,\pas,\cala,\cyc]. Note that our proof is more direct than the one that goes through
the construction of exponentially decaying eigenfunctions.
\endremark
To prove Theorem~1.2 (and also Theorem~1.4), we need to extend (3.9) from $T(n)$
to $T(n,m)$. As in that equation, $d\mu$ is the $\min$ of $d\rho^N$ and $d\rho^D$
which is an a.c.~measure for $h_+$:
\proclaim{Theorem 3.4D} For any $n,m$, $\int \|T_E (n,m)\| d\mu (E) \leq 4$.
\endproclaim
\demo{Proof} We have that $\| T(n,m)\|\leq \|T(n,0)\| \, \|T(0,m)\| = \| T(n,0)\| \,
\|T(m,0)\|$, so by the Schwarz inequality,
$$
\int \| T_E (n,m)\| \, d\mu \leq \biggl( \int \| T_E (n,0)\|^2 \, d\mu\biggr)^{1/2}
\biggl(\int \| T_E (m,0)\|^2 \, d\mu\biggr)^{1/2} \leq 4
$$
by (3.9). \qed
\enddemo
An immediate consequence of this theorem and Fatou's lemma is
\proclaim{Theorem 3.5D ($\equiv$ Theorem 1.2)} Let $m_j, k_j$ be arbitrary
sequences in $\{n\in\Bbb Z\mid n >0\}$. Then for a.e.~$E$ in the a.c.~part of the
spectral measure for $h_+$, we have that $\varliminf _{j\to\infty} \|T_E (m_j, k_j)\|
< \infty$.
\endproclaim
The continuum versions of these results are straightforward analogs following the
above proof using (3.9C). Here $d\mu (E)=(1+E^2)^{-1} \min (d\rho^D (E),
d\rho^N (E))$.
\proclaim{Theorem 3.4C} For each $x_0, y_0$ and a universal constant $C$,
$$
\int d\mu(E) \biggl[ \, \int ^{x_0 +1}_{x_0 -1} dx
\int ^{y_0 +1}_{y_0 -1} dy \, \| T_E (x,y)\| \biggr] < C.
$$
\endproclaim
\proclaim{Theorem 3.5C} Let $x_j, y_j$ be arbitrary sequences in $\{x\in\Bbb R
\mid x > 0\}$. Then for a.e.~$E$ in the a.c.~part of the spectral measure for $H_+$,
we have that
$$
\varliminf\limits_{j\to\infty} \int ^{x_j +1}_{x_j -1} dx \int^{y_j +1}_{y_j -1}
dy \, \| T_E (x,y)\| < \infty.
$$
\endproclaim
We will need the following variant of these ideas in Section~5:
\proclaim{Theorem 3.6D} In the discrete case,
$$
\int d\mu(E) \biggl( \frac{1}{L} \sum^{n+L}_{m=n+1} \|T(m,n)\|^2 \biggr)^{1/2}
\leq 4. \tag 3.11
$$
\endproclaim
\demo{Proof} Since $\|T(m,n)\| \leq \|T(m)\| \, \|T(n)\|$, we have that $(\frac{1}{L}
\sum^{n+L}_{m=n+1} \|T(m,n)\|^2) \leq \mathbreak \|T(n)\| (\frac{1}{L}
\sum^{n+L}_{m=n+1} \|T(m)\|^2 )^{1/2}$, so (3.11) follows from (3.9D) and the
Schwarz inequality. \qed
\enddemo
In the same way, we get
\proclaim{Theorem 3.6C} In the continuum case for a universal constant $C$,
$$
\int d\mu (E) \biggl( \, \int ^{x_0 +1}_{x_0 -1} dx \, \frac{1}{Q(L)}
\int ^{x+L}_{x} \| T_E (x,y)\|^2 \, dy \biggr)^{1/2} \leq C.
$$
\endproclaim
\vskip 0.1in
\centerline{* \qquad * \qquad *}
\vskip 0.1in
\proclaim{Theorem 3.7 ($=$ Theorem 1.3)} Suppose that for some $x_n \to \infty$,
$$
\lim_{n\to\infty} \int ^b_a \|T_E (x_n)\|^p\, dE <\infty
$$
for some $p>2$. Then for any boundary condition at zero, the spectral measure is
purely absolutely continuous on $(a,b)$. More generally, if $W$ is an arbitrary
function on $(-\infty, \infty)$ so that
\roster
\item"\rom{(i)}" $W=V$ on $(0,\infty)$
\item"\rom{(ii)}" $W$ is limit point at both $-\infty$ and $\infty$.
\endroster
Then $H= -\frac{d^2}{dx^2}+W$ has purely a.c.~spectrum on $(a,b)$.
\endproclaim
\demo{Proof} Fix a boundary condition $\theta$ at zero and let $u_\theta =(\cos
(\theta), \sin(\theta))$. For any $x$, let
$$
d\mu^\theta_x (E) =\pi^{-1} dE \big/ \|T_E (x) u_\theta \|^2.
\tag 3.12
$$
Then Carmona [\car] proves that as $x\to\infty$,
$$
d\mu^\theta_x \to d\mu^\theta, \tag 3.13
$$
the spectral measure for boundary condition $\theta$ (the convergence in (3.13) is in
the vague sense, i.e., it holds after smearing with continuous functions in $E$). Since
$T$ is unimodular, $\|T^{-1}\| = \|T\|$ so $\| Tu_\theta \| \geq \| T\|^{-1} \|u_\theta \|$
and thus, $d\mu^\theta_x (E) = F^\theta_x (E)\, dE$ with
$$
|F_x (E)| \leq \|T_E (x)\|^2. \tag 3.14
$$
For the whole-line problem, Carmona proves a result similar to (3.12/3.13), but in
(3.12) $\|T_E (x) u_\theta \|$ is replaced by $\| T_E u_{\theta (E)}\|$ with $\theta (E)$
dependent on $E$ (and $x$) but (3.14) still holds. The result now follows from the next
lemma. \qed
\enddemo
\proclaim{Lemma 3.8} Let $f_n (\lambda)$ be a sequence of functions on $(a,b)\subset
\Bbb R$ so that for some $q>1$,
$$
\int f_n (\lambda)^q \, d\lambda \leq C
$$
uniformly in $n$. Suppose that $f_n (\lambda) \, d\lambda $ converge to a measure
$d\mu (\lambda)$ weakly. Then $d\mu$ is purely absolutely continuous.
\endproclaim
\demo{Proof} The ball of radius $C$ in $L^q$ is compact in the weak-* topology, so
there exists a subsequence $f_{n(i)}$ and $f_\infty \in L^p$ so that $\int f_{n(i)}
(\lambda) g(\lambda) \, d\lambda \to \int f_\infty (\lambda) g(\lambda) \, d\mu(\lambda)$
for all $g\in L^{q'}$ with $q'$ dual to $p$. Thus, $d\mu =f_\infty \, d\lambda $ is
absolutely continuous. \qed
\enddemo
\vskip 0.1in
\centerline{* \qquad * \qquad *}
\vskip 0.1in
We end this section with two remarks that shed some light on the earlier theorems in
this section. The first concerns an explicit relationship between the $m$-function and
the basic average $\frac{1}{L} \int^L_0 \|T_E (x)\|^2 \, dx$ which is connected with
Lemma~3.1:
\proclaim{Proposition 3.9} We have for any $\theta$ that
$$
\text{\rom{Im }} m_\theta \biggl( E+i \,\frac{1}{L}\biggr) \leq
\left(5 + \sqrt{24}\right) \biggl[ \frac{1}{L} \sum^{L+1}_{n=0}
\|T_E (n)\|^2 \biggr]
$$
where $\|T_E (0)\|$ is short for $1$.
\endproclaim
\demo{Proof} Let $u_1$ be the solution with $\theta$ boundary conditions normalized
at $n=1$ and $u_2$ the solution with complementary $(\frac{\pi}{2} -\theta)$ boundary
conditions. Then Jitomirskaya-Last [\jlprl,\jli] prove that if $\|f\|_L = (\sum^L_{n=1}
f(n)|^2)^{1/2}$ and $\epsilon (L)$ is defined by
$$
\|u_1\|_L \, \|u_2\|_L =(2\epsilon)^{-1}, \tag 3.15
$$
then
$$
|m(E+i\epsilon)| \leq \left(5+\sqrt{24}\, \right) \, \frac{\|u_2\|_L}{\|u_1\|_L}\, .
$$
If $L$ is odd, let $f=(u_1 (1), u_1 (2), \dots, u_1 (L-1))$ and let $g=(u_2 (2),
-u_2 (1), u_2 (4), -u_2 (3), \mathbreak \dots, -u_2 (L-1))$. Then constancy of the
Wronskian implies that $\langle f,g\rangle =\frac{L}{2}$, so by the Schwarz inequality,
$$
\frac{L-1}{2} \leq \|u_1\|_L \, \|u_2 \|_L = \frac{1}{2\epsilon (L)}\, . \tag 3.16
$$
For $L$ even, the inequality holds with $\frac{L}{2}$, so a fortiori, (3.16) holds. Thus,
$\frac{1}{L}\geq \epsilon(L+1)$. Since $\text{Im }m (E+i\epsilon)/ \epsilon$ is
monotone increasing as $\epsilon$ decreases,
$$\align
\frac{\text{Im }m(E+iL^{-1})}{L^{-1}} &\leq \frac{\text{Im } m(E+i\epsilon(L+1))}
{\epsilon(L+1)} \\
&\leq \frac{|m (E+i\epsilon)|}{\epsilon} \leq \frac{5+\sqrt{24}}{\epsilon} \,
\frac{\|u_2\|_{L+1}}{\|u_1\|_{L+1}}\, .
\endalign
$$
By (3.15), $(\epsilon \|u_1\|_{L+1})^{-1} \leq 2\|u_2\|_{L+1}$, so
$$
\frac{\text{Im }m(E+iL^{-1})}{L^{-1}} \leq 2 \left( 5+\sqrt{24} \right)
\|u_2\|^2_{L+1}\,.
$$
Now $|u_2 (n)|^2 + |u_2 (n+1)|^2 \leq \|T(n)\|^2$, so
$$
\sum^{L+1}_{n=0} \|T(n)\|^2 \geq |u(0)|^2 + |u(L+2)|^2 + 2 \|u_2\|^2_{L+1},
$$
proving that $2\|u_2\|^2_{L+1} \leq \sum^{L+1}_{n=0} \|T(n)\|^2$ and the claimed
inequality. \qed
\enddemo
The second result concerns the fact that $\varliminf \frac{1}{L} \sum^L_{n=1}
\|T(n)\|^2 <\infty$ says nothing about upper bounds. We claim that this sum cannot
grow too fast, at least for a.e.~$E$ w.r.t.~$d\mu_{\text{\rom{ac}}}$.
\proclaim{Theorem 3.10} Fix $\delta > 0$. For a.e.~$E$
w.r.t.~$d\mu_{\text{\rom{ac}}}$, we have that for any $L\geq 2$,
$$
\biggl( \frac{1}{L} \sum^L_{n=1} \|T_E (n)\|^2 \biggr) \leq C_E
(\log L)^{1+\delta}.
$$
\endproclaim
\remark{Remarks} 1. $(\log L)^{1+\delta}$ can be replaced by any increasing
function $f(n)$ with $\sum f(2^n)^{-1} <\infty$, for example, $(\log L)(\log
(\log L))^{1+\delta}$.
2. If we replace $\| T_E (n)\|$ by $\| u(n; E)\|$, this result holds for $d\mu (E)$ rather
than just for $d\mu_{\text{\rom{ac}}}(E)$.
\endremark
\demo{Proof} Let $g_k (E) =2^{-k} \sum^{2^k}_{n=1} \|T_E (n)\|^2$. Then by
(3.9D), $\int g_k (E)\, d\mu_{\text{\rom{ac}}} (E) \leq 4$ so \linebreak
$\sum^\infty_{k=1} k^{-1-\delta} g_k (E) \in L^1 (d\mu_{\text{\rom{ac}}})$,
which, in particular, implies that
$$
g_k (E) \leq C_E k^{1+\delta} \tag 3.17
$$
for a.e.~$E$ w.r.t.~$d\mu_{\text{\rom{ac}}}$.
Let $2^{k-1} \leq L\leq 2^k$. Then
$$
L^{-1} \sum^L_{n=1} \|T_E (n)\|^2 \leq 2^{-k-1} \sum^{2^k}_{n=1}
\|T_E (n) \|^2 \leq 2 g_k (E),
$$
so (3.17) completes the proof. \qed
\enddemo
\vskip 0.3in
\flushpar {\bf {\S 4. Barriers and A.C.~Spectrum}}
\vskip 0.1in
Theorem~1.2, which we proved in Section~3, is ideal for showing that barriers can
prevent a.c.~spectrum, an idea originally developed by Simon-Spencer [\ssp]. In this
section, we will explain how to recover their results using Theorem~1.2. Our techniques
here allow one to go further since they can handle the case where $V$ goes to zero. We
will illustrate this at the end of this section. A more thorough analysis of this case will
be made in a forthcoming paper [\lsii]. As the simplest example of the strategy, we
recover
\proclaim{Theorem 4.1 ([\ssp])} Let $h_+$ be a Jacobi matrix on $\ell^2
({\Bbb Z}^+)$. Suppose $\varlimsup |V(n)| = \infty$. Then $h_+$ has no
a.c.~spectrum.
\endproclaim
\demo{Proof} Pick $n_j$ so $|V(n_j)|\to\infty$. Then
$$
T_E (n_j, n_j -1) = \pmatrix E - V(n_j) & -1 \\ 1 & 0 \endpmatrix,
$$
so for all $E$, $\|T_E (n_j, n_j -1)\|\to\infty$ as $j\to \infty$. By Theorem~1.2, the
a.c.~spectrum must be empty. \qed
\enddemo
To recover some of the other results of [\ssp], we need bounds that show if $E$ is in
the middle of a gap of size $2\delta$, then the transfer matrix over a length $L$ has
an a priori bound that grows as $L\to\infty$ in a way independent of the potential. We
could obtain this using Combes-Thomas estimates with explicit constants (as in [\ssp])
or using the periodic potential methods of [\lath,\lapaii], but we will instead use the idea
of approximate eigenfunctions. Our simple estimates can be viewed as a quantitative
version of an idea of Sch'nol [\sch]. Basically, we will see that any solution is found to
grow exponentially at a pre-assigned rate in some direction.
\proclaim{Theorem 4.2} Suppose $h$ is a one-dimensional operator of the form
{\rom{(1.1D)}} on a subset $D$ of $\Bbb Z$ with $D_n \equiv \{-n, - n+1, \dots,
n-1, n\}\subset D$. Suppose that there is an operator $B$ on $\ell^2 (D')$ for some
$D'\subset \Bbb Z$ with $D_n \subset D'$ so that
\roster
\item"\rom{(i)}" $\text{\rom{spec}}(B) \cap (E-\delta, E+\delta) = \emptyset$.
\item"\rom{(ii)}" $Bu = hu$ if $u$ vanishes outside $D_n$.
\endroster
Then
\roster
\item"\rom{(a)}" Any solution of $hu=Eu$ obeys
$$
|u(\ell)|^2 + |u(-\ell)^2| \geq \delta^2 (1+\delta^2)^{\ell-1} |u(0)|^2 \quad
\text{for }\ell=1,2, \dots, n+1 \tag 4.1
$$
and
$$
|u(\ell)|^2 + |u(-\ell)|^2 \geq \delta^2 (1+\delta^2)^{\ell-2} (|u(0)|^2 + |u(1)|^2 +
|u(-1)|^2) \quad \text{for }\ell = 2,3,\dots, n+1. \tag 4.2
$$
\item"\rom{(b)}" For any vector $\varphi\in \Bbb R^2$,
$$
\| T(\ell,0) \varphi\|^2 + \| T(-\ell, 0)\varphi\|^2 \geq \delta^2 (1+\delta^2)^{\ell-1}
\| \varphi\|^2 \quad \text{for } \ell=1,2,\dots, n. \tag 4.3
$$
\item"\rom{(c)}" We have that
$$
\| T(-n, n)\| \geq \tfrac12 \delta^2 (1+\delta^2)^{n-1}. \tag 4.4
$$
\endroster
\endproclaim
\remark{Remarks} 1. One remarkable aspect of these estimates is that they (and their
multidimensional case continuum analog) are independent of $V$\!.
2. The point, of course, is that since $\delta >0$, $(1+\delta^2)^\ell$ grows to infinity
as $\ell\to\infty$. We show it is exponentially fast, but that is not needed.
3. While the estimates are elegant and explicit, it is likely the exponent is not optimal.
For $\delta$ small, $(1+\delta^2)^n \simeq \exp(n \log (1+\delta^2))\sim \exp(n\delta^2)$.
One would expect that $\| T(n, -n)\| \sim \exp(2\delta n)$ for $\delta$ small (and fixed)
and $n$ large.
\endremark
\demo{Proof} Let $\chi_j$ be the characteristic function of $\{-j, \dots, j\}$. Then
$$
((h-E)(\chi_j u)) (\ell) = -\delta_{j,\ell+1} u(\ell+1) - \delta_{j, -\ell-1} u(-\ell-1)
$$
so if we define
$$
a_j \equiv |u(j)|^2 + |u(-j)|^2 \quad \text{for } j=1,2,\dots
$$
and
$$
a_0 \equiv |u(0)|^2
$$
we have that
$$
\| (H-E) (\chi_j u) \|^2 = a_{j+1}. \tag 4.5
$$
Clearly,
$$
\| \chi_j u\|^2 = \sum^j_{k=0} a_k. \tag 4.6
$$
But by hypothesis (i), (ii), if $j=0,1,\dots, n$,
$$
\| (H-E)\chi_j u\|^2 = \| (B-E)\chi_j u \|^2 \geq \delta^2 \| \chi_j u \|^2. \tag 4.7
$$
(4.5), (4.6), and (4.7) imply that
$$
\delta^2 \biggl( \, \sum^j_{k=0} a_k \biggr) \leq a_{j+1} \tag 4.8
$$
for $j=0,1,2,\dots, n$.
It follows inductively that for $\ell=1,2,\dots$
$$
a_\ell \geq \delta^2 (1+\delta^2)^{\ell-1} a_0 \tag 4.9
$$
for (4.9) holds for $\ell=1$ (by 4.8), and if (4.9) holds for $a_1, \dots, a_j$, then
by (4.8),
$$
a_{j+1} \geq \delta^2 \biggl( 1+\sum^j_{k=1} \delta^2 (1+\delta^2)^{k-1} \biggr)
a_0 = \delta^2 (1+\delta^2)^j a_0.
$$
(4.9) is precisely (4.1).
A virtually identical inductive argument proves (4.2). (4.3) follows from (4.1) and its
translate:
$$
|u(\ell + 1)|^2 + |u(-\ell+1)|^2 \geq \delta^2 (1+\delta^2)^{\ell-1} |u(1)|^2; \qquad
\ell=1,\dots, n.
$$
To prove (4.4), let $\alpha = \frac12 \delta^2 (1+\delta^2)^{n-1}$ so that (4.3) becomes
$$
\| T(n, 0) \varphi\|^2 + \|T(-n,0)\varphi\|^2 \geq 2\alpha \| \varphi\|^2. \tag 4.10
$$
If $\alpha \leq 1$, (4.4) is trivial so suppose that $\alpha >1$. Picking any unit vector
$\varphi$, we conclude that
$$
\| T(n,0)\|^2 \geq \alpha \qquad \text{or} \qquad \| T(-n,0)\|^2 \geq \alpha.
$$
Suppose the former. Since $T(n,0)$ is uninodular, we can find a unit vector $\varphi_0$
so that $\| T(n,0) \varphi_0\| = \|T(n,0)\|^{-1}$. Thus,
$$
\| T(n,0) \varphi_0 \|^2 \leq \frac{1}{\alpha} \, \| \varphi_0 \|^2 \leq \alpha \|\varphi_0 \|^2
$$
because we are supposing that $\alpha >1$. Thus, by (4.10),
$$
\| T(-n, 0) \varphi_0 \|^2 \geq \alpha \| \varphi_0 \|^2 \geq \alpha^2
\| T(n,0)\varphi_0 \|^2.
$$
It follows that
$$
\| T(n, -n)\|^2 = \| T(-n, n)\|^2 \geq \alpha^2
$$
which is (4.4). \qed
\enddemo
Once we have Theorem~4.2, we immediately conclude by Theorem~1.2 that
\proclaim{Theorem 4.3} Suppose that $h$ has the form {\rom{(1.1D)}} on ${\Bbb Z}^
+$ so that there exist $x_n \geq n$ and $W_n$ on $\tilde D_n \supset \{ x_n - n, \dots,
x_n + n\}$ so that
\roster
\item"\rom{(i)}" $(\alpha,\beta) \cap \text{\rom{spec}} (-\frac{d^2}{dx^2}+ W_n )
= \emptyset$ for some boundary conditions on $\tilde D_n$.
\item"\rom{(ii)}" $W_n (j) = V(j)$ for $j\in \{x_n -n, \dots, x_n +n\}$.
\endroster
Then $(\alpha, \beta)$ is disjoint from the a.c.~spectrum of $h$.
\endproclaim
\demo{Proof} Fix $E\in (\alpha, \beta)$. Then by Theorem~4.2,
$$
\lim_{n\to\infty} \| T_E (x_n -n, x_n + n)\| = \infty.
$$
It follows by Theorem~1.2 that $(\alpha, \beta)$ is disjoint from the a.c.~spectrum.
\qed
\enddemo
With this result, one can recover the theorems in [\ssp] that depend on gaps in the
spectrum.
Before leaving the subject of Theorem~4.2, we note that (4.1) has a continuum,
higher-dimensional analog.
\proclaim{Theorem 4.4} For any $K>0$ and dimension $\nu$, there exists a universal
constant $C_\nu (K)$ depending only on $\nu$ and $K$ so that if $V$ is in the local
Kato class and there exists an operator $B$ on $L^2 (\Bbb R^\nu)$ so that
\roster
\item"\rom{(i)}" $Bu = (-\Delta + V)u$, all $u\in C^\infty_0 (D_n)$ where $D_n =
\{ x\mid |x| \leq n+1 \}$
\item"\rom{(ii)}" $\sigma (B) \cap (E-\delta, E+\delta)=\emptyset$
\item"\rom{(iii)}" $\| V\chi_{\{x\mid \, |x| \leq n+1\}} \| \leq K$ where the norm is the
$K_\nu$ Kato class norm {\rom{[\cyc,\ssg]}}
\item"\rom{(iv)}" $|E| \leq K$
\endroster
then any $L^2_{\text{\rom{loc}}}$ distributional solution of $(-\Delta + V)u = Eu$
in $D_n$ obeys
$$
\int_{j\leq |x|\leq j+1} |u(x)|^2\, dx \geq C_\nu (K)\delta^2 (1+C_\nu (K) \delta^2)^{j-2}
\int_{|x| \leq 1} |u(x)|^2 \, dx \tag 4.11
$$
for $j=1,2,\dots, n$.
\endproclaim
\demo{Proof} Let $\chi_j$ be the characteristic function of $\{ x\mid |x| \leq j\}$. It is
fairly easy to see one can construct a sequence, $f_j$, of $C^\infty$ functions on
$\Bbb R^\nu$ so that
$$\align
f_j \chi_j &= \chi_j \\
f_j \chi_{j+1} &= f_j
\endalign
$$
and
$$
\sup_j \| D^\alpha f_j \| \equiv d_\alpha < \infty \tag 4.12
$$
for each multi-index $\alpha$.
We claim that with $H=-\Delta + V$\!,
$$
\| (H-E) f_j u \|^2 \leq C_\nu (K)^{-1} \| (\chi_{j+1} - \chi_j) u\|^2. \tag 4.13
$$
Accepting this for a moment, we will prove (4.8). We have for $j\leq n-1$,
$$
\| (H-E) f_j u \|^2 = \| (B-E) f_j u\|^2 \geq \delta^2 \| f_j u\|^2
\geq \delta^2 \| \chi_j u \|^2.
$$
Thus with $a_j = \| (\chi_{j+1} - \chi_j)u \|^2$, we see that
$$
C_\nu \delta^2 \biggl(\, \sum^j_1 a_\ell \biggr) \leq a_{j+1}
$$
so that as in the proof of Theorem~4.2,
$$
a_j \geq C_\nu \delta^2 (1+C_\nu \delta^2)^{j-2} a_1
$$
which is (4.11).
To prove (4.13), notice that
$$
(H-E) f_j u = (-\Delta f_j) u + 2(\nabla f_j) \, \cdot \, \nabla u
$$
so
$$
\| (H-E) f_j u \|^2 \leq 2 \| (-\Delta f_j) u\|^2 + 8 \| (\nabla f_j)
\, \cdot \, \nabla u \|^2. \tag 4.14
$$
By Theorem~C.2.2 of [\ssg], we can bound $\| \nabla f \, \cdot\, \nabla u\|^2$ by a
constant $C_1$ (depending on $K$) times $\| (\chi_{j-1} - \chi_j) u\|^2$ so by (4.14),
we have the estimate (4.13). \qed
\enddemo
\proclaim{Theorem 4.5} Fix $\alpha < \frac12$ and let $\{a_n\}^\infty_{n=1}$ be
identically independently distributed random variables with distribution $\frac12
\chi_{[-1,1]} (x)\, dx$. Then there exists $N_1 < N_2 <\cdots $ so that for any $m_1,
\dots, m_n, \dots \geq 0$ and a.e.~$\{a_n\}$ the potential on $\Bbb Z^+$:
$$
V(n) = \cases 0 & n\leq m_1 \\
(n-m_1)^{-\alpha} a_n & m_1 < n \leq m_1 + N_1 \\
0 & m_1 + N_1 < n \leq m_1 + N_1 + m_2 \\
\vdots & {} \\
(n-m_1 - N_1 - \cdots - m_j)^{-\alpha} a_n & m_1 + \cdots + N_{j-1} +
m_j < n \leq m_1 +\cdots + N_j \\
0 & m_1 + \cdots + N_j < n \leq m_1 + \cdots + N_j + m_j
\endcases
$$
has no a.c.~sprectrum.
\endproclaim
\remark{Remark} The choice can be made so that by Theorem~1.6, there is no
point spectrum, that is, so the spectrum is purely singular continuous.
\endremark
\demo{Proof} Let $\tilde T_E (0,n)$ be the transfer matrix for the power-decaying
potential $n^{-\alpha} a_n$. By [\scmp], for a.e.~$\{a_n\}$ and a.e.~$E\in [-2,2]$,
$$
\lim\limits_{n\to\infty} \|\tilde T_E (0,n)\| = \infty. \tag 4.15
$$
Let $A^{(1)}_\ell = \{ E\mid \inf_{n\geq \ell} \| \tilde T_E (0,n)\| \geq 1 \}$. By (4.15)
$|[-2,2]\backslash A_\ell | \downarrow 0$ as $\ell \to \infty$ so we can pick $N_1$ so
that $[-2,2] \backslash|A^{(1)}_{N_1}| \leq 2 ^{-1}$. Now inductively pick $N_j$
given $N_1, \dots, N_{j-1}$ so if
$$
A^{(j)}_\ell = \biggl\{ E \biggm| \inf\limits_{n\geq\ell} \| \tilde T_E (N_1 +
\cdots + N_{j-1}, N_1 + \cdots + N_{j-1} + n)\| \geq n\biggr\}
$$
then $[-2,2]\backslash |A^{(j)}_{N_j}| \leq 2^{-j}$.
For this choice of $N_j$'s, the theorem holds since for a.e.~$E$, $E\in A^{(j)}_{N_j}$
for all large $j$ and thus for such $E$, $\| T_E (m_1 + \cdots + N_{j-1}+m_j, m_1 +
\cdots + m_j + N_j) \| \geq j$. Theorem~3.5 implies $\sigma_{\text{\rom{ac}}} =
\emptyset$. \qed
\enddemo
[\kls] will have a much more effective analysis of this type of example.
\vskip 0.3in
\flushpar {\bf {\S 5. Semicontinuity of the A.C.~Spectrum}}
\vskip 0.1in
In this section, we will prove Theorem~1.4. Consider first the discrete case. Pick $n_j$
so $V(n-n_j) \to W(n)$ as $j\to\infty$ for each $n$. Let $T_V$ (resp.~$T_W$) denote
the transfer matrix for the Jacobi matrix with $V$ (resp.~$W$) along the diagonal. By
Theorem~3.6D,
$$
\int d\mu_V (E) \biggl( \frac{1}{L} \sum^{n_j + L}_{m=n_j + 1} \| T_V (m, n_j)\|^2
\biggr)^{1/2} \leq 4 \tag 5.1
$$
where $d\mu_V (E)$ is a measure equivalent to the a.c.~part of the spectral measure
for $V$\!. Since $V(n-n_j)\to W(n)$ as $j\to\infty$, we have that
$$
T_V (n_j + m, n_j) \to T_W (m,0)
$$
so (5.1) implies that
$$
\int d\mu_V (E) \biggl( \frac{1}{L} \sum^L_{m=1} \| T_W (m,0)\|^2 \biggr)^{1/2}
\leq 4. \tag 5.2
$$
It follows by Fatou's lemma that for a.e.~$E$ with respect to $d\mu_V (E)$, we have
$$
\varliminf \, \frac{1}{L} \sum^L_{m=1} \|T_W (m,0)\|^2 < \infty.
$$
Such $E$ are thus a.e.~in $\Sigma_{\text{\rom{ac}}} (h_0 +W)$, that is,
$\Sigma_{\text{\rom{ac}}} (h_0 + V) \subset \Sigma_{\text{\rom{ac}}} (h_0 + W)$
as claimed. \qed
\smallskip
The proof in the continuum case is similar, except that we use Theorem~3.6C in
place of Theorem~3.6D.
We note that the notion of right/left limits, which enters in Theorem~1.4, is in the
spirit of the notion of limit class introduced by Davies-Simon [\dasii].
\vskip 0.3in
\flushpar {\bf {\S 6. Consequences of Semicontinuity of the A.C.~Spectrum}}
\vskip 0.1in
Let $(\Omega, T, \mu)$ be a metric ergodic process, that is, $T$ is a continuous
invertible bijection from $\Omega \to \Omega$ with $\Omega$ a compact metric space
(recall that any separable compact space is metrizable) and $\mu$ a probability measure
with support $\mu = \Omega$.
\definition{Definition} A point $\omega_0 \in \Omega$ is called right prototypical if and
only if $\{T^n \omega_0 \mid n \geq 0\}$ is dense in $\Omega$, and left prototypical if
and only if $\{T^n \omega_0 \mid n\leq 0\}$ is dense in $\Omega$. If $\omega_0$ is both
left and right prototypical, we say it is prototypical.
\enddefinition
The ergodic theorem implies that a.e.~$\omega_0 \in \Omega$ is prototypical. Fix a
continuous function $f \: \Omega \to \Bbb R$ and let $h_\omega$ on $\ell^2 (\Bbb Z)$
be defined by $(h_\omega u)(n) = u(n+1) + u(n-1) + f(T^n \omega)u(n)$.
\proclaim{Theorem 6.1} The essential support of the a.c.~spectrum of $h_\omega$ is the
same for all prototypical points and is of multiplicity $2$. Moreover, for any prototypical
$\omega_0$ and any $\omega\in \Omega$, we have $\Sigma_{\text{\rom{ac}}}
(h_{\omega_0}) \subset \Sigma_{\text{\rom{ac}}} (h_\omega)$.
\endproclaim
\demo{Proof} Let $h^\pm_\omega$ be the operators on $\ell^2 (\pm n \geq 1)$ with $u(0)
=0$ boundary conditions. By general principles (see, e.g., Davies-Simon [\dasi]), the
restriction of $h_\omega$ to its a.c.~subspace is unitarily equivalent to the restriction of
$h^+_\omega \oplus h^-_\omega$ to its a.c.~subspace. Thus, the theorem follows from
\roster
\item"\rom{(i)}" if $\omega_0$ is prototypical and is $\omega$ arbitrary, then
$\Sigma_{\text{\rom{ac}}} (h^\pm_{\omega_0}) \subset \Sigma_{\text{\rom{ac}}}
(h^\pm_\omega )$
\item"\rom{(ii)}" for prototypical $\omega_0$, $\Sigma_{\text{\rom{ac}}}
(h^+_{\omega_0}) = \Sigma_{\text{\rom{ac}}} (h^-_{\omega_0})$
\endroster
for (i) implies equality if both $\omega_0$ and $\omega$ are prototypical and (ii) implies
multiplicity 2.
To prove (i), pick $n_j \to \infty$ so $T^{-n_j} \omega_0 \to \omega$. Then, since
$V_{T^{-m}\omega_0} (n) = f (T^{n-m} \omega_0) = V_{\omega_0} (n-m)$, we have
that $V_{\omega_0} ( \, \cdot\, - n_j) \to V_\omega (\, \cdot \,)$ so (i) follows from
Theorem~1.4.
To prove (ii), let $\omega_0$ be prototypical and $\omega\neq \omega_0$, also
prototypical. Pick $n_j \to \infty$ so $T^{-n_j} \omega_0 \to \omega$. Fix $L$ and use
the fact that $\| T(n,m)\| = \| T(m,n)\|$ (since $T$ is unimodular) to note that
$$\align
\frac{1}{L} \sum^L_{m=1} \| T_\omega (-1, -m) \|^2 &= \frac{1}{L}
\sum^L_{m=1} \| T_\omega (-m, -1) \|^2 \\
&= \lim \frac{1}{L} \sum^L_{m=1} \| T_{\omega_0} (n_j - m, n_j -1)\|^2
\endalign
$$
so by Theorem~3.6D,
$$
\int d\mu_{\omega_0} (E) \biggl(\frac{1}{L} \sum^L_{m=1}
\| T_\omega (-1, -m)\|^2 \biggr) < 4
$$
where $d\mu_{\omega_0} (E)$is an a.c.~measure for $h^+_{\omega_0}$. Thus, as in
the last section, $\Sigma_{\text{\rom{ac}}} (h^-_\omega) \supset
\Sigma_{\text{\rom{ac}}} (h^+_{\omega_0})$. By symmetry, (ii) holds. \qed
\enddemo
\remark{Remark} That the typical a.c.~spectrum is of multiplicity 2 is a result of
Deift-Simon [\deis] proven using Kotani theory. Our proof is different.
If $V$ is almost periodic, then every $\omega \in \Omega$ is prototypical. Thus,
Theorem~6.1 implies Theorem~1.5. More generally, if $(T, \Omega, \mu)$ is minimal
(or if it is strictly ergodic which implies minimal), then every $\omega \in \Omega$ is
prototypical, and we see that the a.c.~spectrum is constant (rather than just a.e.~constant)
on $\Omega$.
\endremark
\example{Example} Consider the sequence $V _1, V_2, V_3, \dots $ given by
$$
0,1,0,0,0,1,1,0,1,1,0,0,0,0,0,1,0,1,0, \dots
$$
defined as follows. For two finite sequences of $0$'s and $1$'s of length $n$, say $w_1,
\dots, w_n$, and $\tilde w_1, \dots \tilde w_n$, say $w < \tilde w$, if and only if
$$
w_1 = \tilde w_1, \dots, w_j = \tilde w_j, \qquad w_{j+1} < \tilde w_{j+1}.
$$
With this order, the sequences of length $n$ are well-ordered, for example,
$$
(0)<(1), \quad (0,0) < (0,1) < (1,0) < (1,1), \quad (0,0,0) < (0,0,1) < (0,1,0) < \cdots.
$$
$V$ is obtained by placing the two sequences of length 1 in order, then the four
sequences of length 2, etc. Clearly, $V$ is prototypical for a Bernoulli model. By
Furstenberg's theorem, that model has no a.c.~spectrum, so $V$ is an explicit sequence
for which we know that $\sigma_{\text{\rom{ac}}} (h_0 + V) = \emptyset$.
\endexample
Another consequence of Theorem~6.1 is a new proof of the Kotani support theorem:
\proclaim{Theorem 6.2} Let $\Omega$ be the compact metric space of sequences
$V_n$ with $|V_n|\leq a$ with the product topology. Let $f \: \Omega \to \Bbb R$ by
$f(V) = V_0$ and $T \: \Omega \to \Bbb R$ by $(TV)_n = V_{n+1}$. Let $\mu_1,
\mu_2$ be two measures on $\Omega$ under which $T$ is ergodic. Let $\Sigma_i$
be the essential support of the a.c.~spectrum of the prototypical $h_\omega$ for the
process $(\text{\rom{supp}} (\mu_i), T, \mu_i)$. If $\text{\rom{supp}}(\mu_1)
\subset \text{\rom{supp}}(\mu_2)$, then $\Sigma_1 \supset \Sigma_2$.
\endproclaim
\demo{Proof} Let $\omega_i \in \text{supp}(\mu_i)$ be $\mu_i$-prototypical. Since
$\omega_1 \in \text{supp}(\mu_2)$, Theorem 6.1 implies $\Sigma (h_{\omega_2})
\subset \Sigma (h_{\omega_1})$. \qed
\enddemo
\remark{Remark} In a sense, Theorem~1.4 is a deterministic version of the Kotani
support theorem, so it is not surprising that it implies the Kotani theorem.
\endremark
While we have stated these theorems in this section for the discrete case, they all extend
easily to the continuum case.
\vskip 0.3in
\flushpar {\bf {\S 7. The Potential $\lambda \cos (n^\beta), \; \beta > 1$}}
\vskip 0.1in
Jacobi matrices with potentials of the form $V (n) \equiv \lambda \cos (n^\beta)$, where
$\lambda,\beta$ are real parameters with $\beta > 1$, had been studied numerically and
heuristically by Griniasty-Fishman [\grfs] and Brenner-Fishman [\brfs]. The particular
case $1<\beta < 2$ had been studied in more detail by Thouless [\thou]. The numerical
evidence indicates that for $\beta \geq 2$, such potentials exhibit ``localization'' with
the same Lyapunov exponents as those of random potentials (Anderson model) with the
same coupling. The case $1<\beta < 2$ is different, and far less conclusive. One still
expects ``localization'' away from $E=0$ (the center of the spectrum) and for large
$\lambda$, but the Lyapunov exponents are smaller and seem to vanish for $E=0$ and
small $\lambda$. Mathematical results exist for the case where $\beta$ is an integer
and $\lambda$ is large (larger than 2, to be precise), in which case it is known
[\goso,\jito] that there is no absolutely continuous spectrum. More precisely, for every
polynomial $p(n)$ with a leading coefficient that is an irrational multiple of $\pi$, it is
known that $\lambda \cos (p(n))$ can be obtained as a realization (an element) of an
ergodic family of potentials coming from a suitable ergodic transformation on the
$d$-dimensional torus [\cfs] (with $d$ the degree of $p(n)$); and that the
corresponding ergodic families have only positive Lyapunov exponents as long as
$\lambda > 2$ [\goso,\jito]. Further, the corresponding ergodic families are minimal
[\furs], and so it follows from our Theorem~6.1 that {\it every} realization of such a
family has no absolutely continuous spectrum. We note that this is also true for the case
$\beta = 1$, where the absence of a.c.~spectrum follows from earlier results [\avs]. Our
purpose in this section is to extend these results to cases where $\beta$ is not an integer.
We discuss half-line problems here, and denote by $h_0^+$ the free Laplacian on
$\ell^2 ({\Bbb Z}^+)$. The results are also valid for full-line problems if we replace
$n$ by $|n|$. We shall prove the following:
\proclaim{Theorem 7.1}
For any $\lambda >2$ and $\beta > 1$, $\Sigma_{\text{\rom{ac}}}
(h_0^+ + \lambda \cos (n^\beta)) = \emptyset$.
\endproclaim
\remark{Remarks} 1. $\cos(\,\cdot\,)$ in the above theorem can be replaced by any real
analytic function $f(\,\cdot\,)$ of period $2\pi$, in which case the theorem would hold
for $\lambda$ ``large enough.'' This follows from the argument below combined with the
results of Goldsheid-Sorets [\goso]. The explicit $\lambda > 2$ for the $\cos(\,\cdot\,)$
case is due to Jitomirskaya [\jito].
2. The result is actually also more general in the sense that one can replace $n^\beta$
by, for example, $\sum_{j=1}^k a_j n^{\beta_j}$, where $k$ is any positive integer,
$\beta_j > 0$ for each $j$, and the $a_j$'s are some real numbers (except if the largest
$\beta_j$ is an integer, in which case we would need some further condition, such as that
the corresponding $a_j$ would be an irrational multiple of $\pi$).
3. For $1 < \beta < 2$, the result also follows for $\lambda = 2$, by results
of Helffer-Sj\"ostrand [\hesj] and Last [\lastz].
\endremark
\demo{Proof} Let $\lambda > 2$. We only need to consider the case where $\beta$ is
not an integer. Fix $k < \beta < k+1$, where $k$ is an integer, and consider
$(n+m)^\beta$ where $n$ is large and $m\ll n$. By writing $(n+m)^\beta = n^\beta
(1+n/m)^\beta$ and expanding $(1+n/m)^\beta$ as a Taylor series, we obtain:
$$
(n+m)^\beta = \sum_{\ell=0}^\infty a_\ell n^{\beta -\ell} m^\ell =
\sum_{\ell=0}^k a_\ell n^{\beta -\ell} m^\ell + O(n^{\beta-k-1}m^{k+1}), \tag 7.1
$$
where $a_0 = 1$, $a_\ell = (1/\ell !)\prod_{j=0}^{\ell-1}(\beta-j)$ for $\ell\geq 1$.
Let $b_{\ell,n} = \langle a_\ell n^{\beta -\ell}/2\pi\rangle$ where $\langle\,\cdot\,
\rangle$ denotes fractional part (i.e., $\langle x\rangle = x-[x]$). Since
$(n+1)^{\beta -k}-n^{\beta -k} \to 0$, $\{b_{k,n}\}_{n=1}^\infty$ is clearly dense in
$[0,1]$ and we can pick a convergent subsequence $\{n_j\}$ so that $b_{k,n_j}\to b_{k,\infty}\equiv b_k$, where $b_k$ is an irrational. Moreover, by compactness, we
can find a subsequence of that for which $b_{\ell,n_j}\to b_\ell$ for all $\ell\leq k$,
where the $b_\ell$'s (for $\ell < k$) are some numbers in $[0,1]$. For the resulting
polynomial $p(n)\equiv\sum_{\ell=0}^k b_\ell n^\ell$ we see from (7.1) that $\langle p(n)/2\pi\rangle$ is a pointwise limit of translations of $\langle n^\beta/2\pi\rangle$.
Thus, the potential $\lambda\cos (p(n))$ is a right limit of $\lambda \cos (n^\beta)$, and
by Theorem~1.4, we have $\Sigma_{\text{\rom{ac}}} (h_0^+ + \lambda \cos (n^\beta))
\subset \Sigma_{\text{\rom{ac}}} (h_0^+ + \lambda \cos (p(n))) = \emptyset$ (where
the last equality follows from the discussion above). \qed
\enddemo
In the above proof we only needed to show that some fixed realization of a suitable
ergodic process is obtained as a limit of translations of $\langle n^\beta/2\pi\rangle$.
However, since the underlying ergodic systems are minimal, it follows that translations
of that realization are themselves dense in the ergodic family. Thus, one sees that we can
actually obtain {\it every} realization as such a limit. We can combine this with Kotani's
result [\kotfmv] --- that ergodic potentials taking finitely many values have no absolutely
continuous spectrum (unless they are periodic) --- to show that if $f(\,\cdot\,)$ is any real
periodic piecewise constant function on the line (with only finitely many discontinuities
per period), then for any $\beta > 1$ that is not an integer, $\Sigma_{\text{\rom{ac}}}
(h_0^+ + f(n^\beta)) = \emptyset$. The proof here is very similar to that of Theorem~7.1,
except that we need to choose a realization that does not take values in any of the points
where $f$ is discontinuous.
Finally, we would like to discuss the special case $1 < \beta < 2$ and to explain how
one could prove $\Sigma_{\text{\rom{ac}}} (h_0^+ + \lambda \cos (n^\beta)) =
\emptyset$ also for $\lambda < 2$, if one could prove that ``Hofstadter's butterfly has
wings.'' Noting that we could choose the largest order coefficient $b_k$ in the proof of
Theorem~7.1 at will, we obtain for $1 < \beta < 2$:
\proclaim{Proposition 7.2} $\Sigma_{\text{\rom{ac}}} (h_0^+ + \lambda \cos
(n^\beta)) \subset \cap_{\alpha\in F} \Sigma_{\text{\rom{ac}}} (h_0 + \lambda
\cos (\pi\alpha n))$ for any countable set $F$ of irrational $\alpha$'s.
\endproclaim
$\Sigma_{\text{\rom{ac}}} (h_0^+ + \lambda \cos (n^\beta )) = \emptyset$ would
thus follow, if we could prove the following:
\example{Conjecture} Fix $\lambda \neq 0$ and $E_0$ real. Then there exists $\delta >
0$ and irrational $\alpha$ with $\sigma (h_0 + \lambda \cos (\pi\alpha n)) \cap (E_0 -
\delta, E_0 + \delta) = \emptyset$.
\endexample
Intuitively, this conjecture comes from the fact that for $\lambda$ small, $h_0 +
\lambda \cos (\pi\alpha n)$ should have a gap about the energy $E =2\cos (\pi\alpha)$.
In numerical drawings of Hofstadter-like butterflies for various values of $\lambda$
[\ght], one indeed sees those ``stripes'' appear for small $\lambda$, and then broaden
and get more structure as $\lambda$ increases, up to the critical point $\lambda = 2$
where they form the wings of {\it the} famous Hofstadter butterfly. Unfortunately,
we do not know how to prove that these wings exist.
\vskip 0.3in
\flushpar {\bf {\S 8. Transfer Matrices and Bound States}}
\vskip 0.1in
In this section, we will prove Theorem~1.7. As noted already, Theorem~1.6 is motivation
for considering $\sum^\infty_{n=1} \| T_E (n) \|^{-2}$ as an indicator of bound states. If
it is infinite, $hu=Eu$ has no solution $L^2$ at infinity.
\example{Example 1} Take $V =0$ and $E=2$. Then $hu = Eu$ has the solutions
$$
u(n) = c_1 + c_2 n,
$$
none of which are $\ell^2$. But
$$
T_E (n) = \pmatrix n+1 & - n \\ n & 1-n \endpmatrix
$$
has $\|T_E (n)\| = \sqrt2 n + 0(1)$ and thus $\sum^\infty_{n+1} \|T_E (n) \|^{-2} <
\infty$. We see that $\sum^\infty_{n=1} \|T_E (n) \|^{-2} \mathbreak < \infty$ does not
imply that there is an $\ell^2$ solution.
\endexample
\example{Example 2} Let $V(n) = c_0 n^{-2}$, $n= 1,2,\dots$ with $c_0 < \frac14$
and $E=2$. Then standard arguments (variation of parameters) show that there are two
solutions $u_\pm (n)$ with
$$
u_\pm (n) \sim n^{\alpha_\pm}
$$
with $\alpha_\pm$ the roots of $\alpha (\alpha - 1) + c_0 =0$, that is,
$$
\alpha_\pm = \frac12 \pm \sqrt{\frac14 - c_0} \, .
$$
We see that $\|T(n)\| \sim C n^{\alpha_+}$, so since $\alpha_+ > \frac 12$, $\sum_n
\|T(n)\|^{-2} < \infty$. But if $c_0 > 0$, there is no bounded solution. Thus, $\sum_n
\|T(n)\|^{-2} < \infty$ need not even imply that there is a bounded solution!
\endexample
The following (note (8.5) and (8.7)) includes the first part of Theorem~1.7 as a special
case. Its proof just abstracts Ruelle [\rue]:
\proclaim{Theorem 8.1} Let $A_1, A_2, \dots$ be unimodular $2\times 2$ real matrices
and let $T(n) = A_n A_{n-1} \mathbreak \dots A_1$. Suppose that
$$
\sum^\infty_{n=1} \frac{\| A_{n+1}\|^2}{\| T(n)\|^2} < \infty. \tag 8.1
$$
Then there is a unit vector $u\in \Bbb R^2$ so that for any other unit vector $v\in
\Bbb R^2$, we have
$$
\frac{\|T(n) u\|}{\|T(n) v\|} \to 0. \tag 8.2
$$
\endproclaim
\demo{Proof} Let $t(n) = \| T(n)\|$ and $a(n) = \| A_n \|$. Since $|T(n)|$ is self-adjoint
and unimodular, it has eigenvalues $t(n)$ and $t(n)^{-1}$. Thus, taking $u_\theta =
\binom{\cos (\theta)}{\sin (\theta)}$ we see there exists $\theta_n$ so that
$$
\| T(n) u_\theta \|^2 = t(n)^2 \sin^2 (\theta - \theta_n ) + t(n)^{-2} \cos^2
(\theta - \theta_n) \tag 8.3
$$
for pick $\theta_n$ so that $|T(n)| u_{\theta_n} = t(n)^{-1} u_{\theta_n}$. Now by
(8.3) for $n+1$,
$$\align
t(n+1)^2 \sin^2 (\theta_n - \theta_{n+1} ) &\leq \|T(n+1) u_{\theta_n} \|^2 \\
& \leq a (n+1)^2 \|T(n) u_{\theta_n}\|^2 \\
&= a(n+1)^2 t(n)^{-2}.
\endalign
$$
Since $A_{n+1}$ is unimodular, $t(n)=\| T(n) \| \leq \|T(n+1)\| \, \|A^{-1}_{n+1}\| =
t(n+1) a(n+1)$ so
$$
t(n)^2 \sin^2 (\theta_n - \theta_{n+1}) \leq a(n+1)^4 t(n)^{-2}.
$$
Since $\sin^2 (x) \geq (\frac{2x}{\pi})^2$, we see that
$$
|\theta_n - \theta_{n+1}| \leq \frac{\pi}{2} \, \frac{a(n+1)^2}{t(n)^2}\, . \tag 8.4
$$
Thus, (8.1) implies
$$
\sum_n |\theta_n - \theta_{n+1}| < \infty.
$$
So if (8.1) holds, $\theta_n$ has a limit $\theta_\infty$ and
$$
|\theta_n - \theta_\infty | \leq \frac{\pi}{2} \sum^\infty_{m=n} \,
\biggl[\frac{a(m+1)^2}{t(m)^2} \biggr]. \tag 8.5
$$
Let $u_\infty = u_{\theta_\infty}$ and $v_\infty = u_{\pi/2 + \theta_\infty}$. Since
$\theta_n - \theta_\infty \to 0$, for $n$ large enough, we have by (8.3) that
$$
\| T(n) v_\infty \|^2 \geq \frac12 \, t(n)^2. \tag 8.6
$$
On the other hand, by (8.3) again,
$$
\| T(n) u_\infty \|^2 \leq t(n)^2 (\theta_n - \theta_\infty)^2 + t(n)^{-2}. \tag 8.7
$$
(8.6) and (8.7) imply that
$$
\frac{\|T(n) u_\infty \|^2}{\|T(n) v_\infty\|^2} \leq 2 (\theta_n - \theta_\infty )^2 +
2 t(n)^{-4} \to 0
$$
since $a(n+1)\geq 1$ and (8.1) imply that $t(n)\to\infty$. From this, (8.2) follows. \qed
\enddemo
The following includes the second part of Theorem~1.7 as a special case (where $a(n)$
is bounded):
\proclaim{Theorem 8.2} Under the hypothesis of Theorem~{\rom{8.1}}, suppose that
we also have that
$$
\sum^\infty_{m=1} \|T(m)\|^2 \biggl( \, \sum^\infty_{n=m}
\frac{\| A (n+1)\|^2}{\|T(n)\|^2} \biggr)^2 <\infty. \tag 8.8
$$
Then there is a unit vector $u_\infty$ with
$$
\sum^\infty_{n=1} \|T(n) u_\infty \|^2 < \infty.
$$
\endproclaim
\demo{Proof} Using (8.7) and (8.5), we see that $\sum^\infty_{n=1} \| T(n)
u_\infty \|^2 <\infty$ if (8.8) holds and if $\sum_n t(n)^{-2} < \infty$. But since $a(n)
\geq 1$, (8.1) implies that $\sum_n t(n)^{-2}<\infty$. \qed
\enddemo
\example{Example 3} Suppose that $\| A(n)\|$ is bounded and $t(n) \sim n^\gamma$
in the sense that $C_- n^\gamma \leq t(n) \leq C_+ n^\gamma$. Then (8.1) requires
$\gamma > \frac12$ while (8.4) requires $\gamma > \frac 32$. Notice in Example~2,
$\gamma = \alpha_+$ while an $\ell^2$ solution requires $\alpha_- < -\frac12$. Since
$\alpha_+ = 1 - \alpha_-$, Example~2 provides an example with $\gamma = \frac32$
where there is no $\ell^2$ solution (namely, take $c_0 = -\frac34$). Thus, $\gamma >
\frac32$ is best possible!
\endexample
If one has control over the limit of $\ln \|T(n)\|$, one can say more:
\proclaim{Theorem 8.3} Suppose that the hypotheses of Theorem~{\rom{8.1}} hold
and that
$$
\lim\limits_{n\to\infty}\, \frac{\ln \|T(n)\|}{f(n)} = 1
$$
and
$$
\lim\limits_{n\to\infty}\, \frac{\ln \|A(n)\|}{f(n)} = 0,
$$
where $f(n)$, a monotone increasing function, is such that
$$
\sum e^{-\epsilon f(n)} < \infty
$$
for any $\epsilon >0$. Then the $u_\infty$ of Theorem~{\rom{8.2}} obeys
$$
\lim\limits_{n\to\infty}\, \frac{\ln \|T(n) u_\infty\|}{f(n)} = -1.
$$
\endproclaim
\demo{Proof} By (8.5) and (8.7), for any $\epsilon >0$, for $n$ large
$$\align
\|T(n) u_\infty\|^2 &\leq e^{-2(1-\epsilon)f(n)} + \biggl(\frac{\pi}{2}\biggr)^2
\biggl( \, \sum^\infty_{m=n} \frac{e^{2\epsilon f(m)}}{e^{2(1-\epsilon)f(m)}}
\biggr)^2 e^{2(1+\epsilon)f(n)} \\
&\leq e^{-2(1-\epsilon)f(n)} + \biggl(\frac{\pi}{2}\biggr)^2 e^{-(2-11\epsilon)f(n)}
\biggl[ \, \sum^\infty_{m=n} e^{-\epsilon f(m)} \biggr]
\endalign
$$
so
$$
\varlimsup \, \frac{\|T(n) u_\infty \|}{f(n)} \leq -1.
$$
On the other hand, $\| T(n) u\|_\infty \geq \|T(n) \|^{-1}$ implies
$$
\varliminf \, \frac{\|T(n) u_\infty \|}{f(n)} \geq -1. \qed
$$
\enddemo
Typical cases of this theorem are $f(n) = n^\alpha$; $f(n)=n$ is Ruelle's theorem. For
the case $f(n)=\ln (n)$ where (8.9) fails, we have
\proclaim{Theorem 8.4} Suppose the hypotheses of Theorem~{\rom{8.1}} hold and
that
$$\align
\lim\limits_{n\to\infty}\, \frac{\ln \|T(n)\|}{\ln n} &= \gamma \\
\lim\limits_{n\to\infty}\, \frac{\ln \|A(n)\|}{\ln n} &= 0
\endalign
$$
where $\gamma > \frac12$. Then
$$
\varlimsup\limits_{n\to\infty}\, \frac{\ln \|T(n) u_\infty\|}{\ln n} \leq 1-\gamma \tag 8.10
$$
while
$$
\varliminf\limits_{n\to\infty}\, \frac{\ln \|T(n) u_\infty\|}{\ln n} \geq -\gamma. \tag 8.11
$$
\endproclaim
\demo{Proof} As in the last theorem, (8.11) is a consequence of $\|T(n) u_\infty \| \geq
\| T(n)\|^{-1}$. To get (8.10), we use (8.5), (8.7) to see that for any $\epsilon > 0$,
$$\align
\| T(n) u_\infty \|^2 &\leq n^{-2\gamma +\epsilon} + \biggl( \, \sum^\infty_{m=n}
m^{-2\gamma + \epsilon} \biggr)^2 n^{2\gamma + \epsilon} \\
&\leq n^{-2\gamma + \epsilon} + Cn^{2-2\gamma + 2\epsilon}. \qed
\endalign
$$
\enddemo
Example~2 shows there are cases where the limit is $1-\gamma$. [\kls] has examples
where the limit is $-\gamma$.
The ideas of this section can be applied to certain continuum problems by sampling the
wave function at a discrete set of points.
\vskip 0.3in
\flushpar {\bf {Appendix: BGK Eigenfunction Expansions}}
\vskip 0.1in
The proofs of the estimates in Section~2 are one-dimensional, relying on the relation
between Green's functions and $m$-functions. Our goal in this appendix is to discuss an
alternate proof which extends to higher dimensions. The applications of these estimates
in Section~3 are intrinsically one-dimensional, so those results do not extend to higher
dimensions. Nevertheless, we believe these general estimates may be of use elsewhere.
We recall the abstract eigenfunction expansion dubbed BGK expansions in [\ssg] after
work of Berezinski, Browden, Garding, Gel'fand, and Kac (see [\ssg] for references).
In the discrete case, they take the following form. Let $V$ be an arbitrary function on
$\Bbb Z^\nu$. Let $(Hu)(n) =\sum_{|\ell| = 1} u(n + \ell) + V(n) u(n)$. Then there exist
measures $\{d\rho_k (E)\}^\infty_{k=1}$ on $\Bbb R$ and a measure $d\rho_\infty (E)$
so that the $d\rho$'s are mutually singular [\aron,\dono,\svan]. Moreover, for a.e.~$E$
w.r.t.~$d\rho_k (E)$, there exist $k$ linearly independent functions $\{ u_{j,k}
(n; E)\}^k_{j=1}$ on $\Bbb Z^\nu$ so that
\roster
\item"{(i)}" $\sum_{|\ell| =1} u_j (n+\ell) + (V(n)-E) u_j(n) = 0$.
\item"{(ii)}" For any $f$ on $\Bbb Z^\nu$ of finite support, define $a_{j,k} (f)(E) =
\sum_n \overline{u_{j,k} (n; E)}\, f(n)$. Then
$$
a_{j,k} (Hf)(E) = E a_{j,k}(f)(E)
$$
and for any $f,g$ of finite support,
$$
\langle f,g\rangle = \sum^{\text{``$\infty$"}}_{k=1} \, \sum^k_{j=1} \int
\overline{a_{j,k} (f)(E)} \, a_{j,k} (g)(E) \, d\rho_k (E) \tag A.1
$$
with an explicit $\rho_\infty$-term intended in $\sum^{\text{``$\infty$"}}_{k=1}$.
\endroster
Pick $f=g=\delta_n$, a Kronecker delta function at $n$. Then (A.1) becomes
$$
\sum^{\text{``$\infty$"}}_{k=1}\, \sum^k_{j=1} \int |u_{j,k} (n;E)|^2 \,
d\rho_k (E) =1.\tag A.2
$$
This is essentially (2.6D) except in arbitrary dimension. In the one-dimensional case,
$d\rho_k =0$ for $k\neq 1$. If we define $d\tilde\rho (E) = |u(1; E)|^2\, d\rho_1$ and
$\tilde u (n; E) = u(n; E) u(1;E)^{-1}$, then (A.2) is exactly (2.6D).
In the continuum case, the situation is similar. One needs some minimal local regularity
on $V$ (see [\ssg]). Using the fact that $e^{-tH} (x,x) = 0(t^{-1/2})$ at $t\downarrow
0$, one can show that as $f\to\delta_x$, a $\delta$-function at $x$, $(f, (H+c)^{-\ell}f)$
stays finite and bounded in $x$ so long as $2\ell > \nu$. Thus, (A.2) in the continuum
case becomes
$$
\sum^{\text{``$\infty$"}}_{k=1} \, \sum^k_{j=1} \int |u_{j,k} (x; E)|^2
\frac{d\rho_k (E)}{(1+|E|)^\ell} \leq C
$$
uniformly in $x$ where $\ell > \frac{\nu}2$.
As in Section~3, from these bounds and Fatou's lemma, we get bounds like before for
a.e.~$E$ w.r.t.~$\sum_k d\rho_k (E)$,
$$
\varliminf\limits_{L\to\infty} \, \frac{1}{(2L+1)^\nu} \sum_{|n| \leq L} |u(n; E)|^2
< \infty.
$$
\vskip 0.3in
\Refs
\endRefs
\vskip 0.1in
\item{\aron.}\ref{N.~Aronszajn}{On a problem of Weyl in the theory of
Sturm-Liouville equations}{Am. J.~Math.}{79}{1957}{597--610}
\gap
\item{\avs.}\ref{J.~Avron and B.~Simon}{Almost periodic Schr\"odinger operators,
II. The integrated density of states}{Duke Math.~ J.}{50}{1983}{369--391}
\gap
\item{\brfs.}\ref{N.~Brenner and S.~Fishman}{Pseudo-randomness and localization}
{Nonlinearity}{4}{1992}{211--235}
\gap
\item{\car.}\ref{R.~Carmona}{One-dimensional Schr\"odinger operators with random
or deterministic potentials, New spectral types}{J.~Funct.~Anal.}{51}{1983}
{229--258}
\gap
\item{\cala.} R.~Carmona and J.~Lacroix, {\it{Spectral Theory of Random
Schr\"odinger Operators}}, Birk-h\"auser, Boston, 1990.
\gap
\item{\cfs.} I.P~Cornfeld, S.V.~Fomin, and Ya.G.~Sinai, {\it{Ergodic Theory}},
Springer, New York-Heidel-\linebreak berg-Berlin, 1982.
\gap
\item{\cyc.} H.L.~Cycon, R.G.~Froese, W.~Kirsch, and B.~Simon,
{\it{Schr\"odinger Operators}}, Springer, Berlin-Heidelberg-New York, 1987.
\gap
\item{\dasi.}\ref{E.B.~Davies and B.~Simon}{Scattering theory for systems with
different spatial asymptotics on the left and right}{Commun.~Math.~Phys.}{63}
{1978}{277--301}
\gap
\item{\dasii.}\ref{E.B.~Davies and B.~Simon}{$L^1$-properties of intrinsic
Schr\"odinger semigroups}{J.~Funct. Anal.}{65}{1986}{126--146}
\gap
\item{\deis.}\ref{P.~Deift and B.~Simon}{Almost periodic Schro\"odinger
operators, III. The absolutely continuous spectrum in one dimension}
{Commun.~Math.~Phys.}{90}{1983}{389--411}
\gap
\item{\del.}\ref{F.~Delyon}{Apparition of purely singular continuous spectrum in a
class of random Schr\"odinger operators}{J.~Statist.~Phys.}{40}{1985}{621--630}
\gap
\item{\dss.}\ref{F.~Delyon, B.~Simon, and B.~Souillard}{From power pure point to
continuous spectrum in disordered systems}{Ann.~Inst.~H.~Poincar\'e}{42}{1985}
{283--309}
\gap
\item{\dono.}\ref{W.~Donoghue}{On the perturbation of the spectra}{Commun.~Pure
Appl.~Math.}{18}{1965}{559--579}
\gap
\item{\furs.}\ref{H.~Furstenberg}{Strict ergodicity and transformations of the torus}
{Am.~ J.~Math.}{83}{1961}{573--601}
\gap
\item{\gp.}\ref{D.J.~Gilbert and D.~Pearson}{On subordinacy and analysis of the
spectrum of one-dimensional Schr\"odinger operators}{J.~Math.~Anal.}{128}
{1987}{30--56}
\gap
\item{\goso.}\ref{I.~Goldsheid and E.~Sorets}{Lyapunov exponents of the
Schr\"odinger equation with certain classes of ergodic potentials}
{Amer.~Math.~Soc.~Trans. (2)}{171}{1996}{73--80}
\gap
\item{\gor.}\ref{A.Ya.~Gordon}{Deterministic potential with a pure point spectrum}
{Math.~Notes}{48}{1990}{1197--1203}
\gap
\item{\grfs.}\ref{M.~Griniasty and S.~Fishman}{Localization by pseudorandom
potentials in one dimension}{Phys.~Rev.~Lett.}{60}{1988}{1334--1337}
\gap
\item{\ght.}\ref{J.P.~Guillement, B.~Helffer, and P.~ Treton}{Walk inside Hofstadter's
butterfly}{J.~Phys. France}{50}{1989}{2019--2058}
\gap
\item{\hesj.}\ref{B.~Helffer and J.~Sj\"ostrand}{Semi-classical analysis for Harper's
equation, III. Cantor structure of the spectrum}{M\'em.~Soc.~Math.~France (N.S.)}
{39}{1989}{1--139}
\gap
\item{\ish.}\ref{K.~Ishii}{Localization of eigenstates and transport phenomena in
one-dimensional disordered systems}{Suppl.~Prog.~Theor.~Phys.}{53}{1973}
{77--138}
\gap
\item{\jito.} S.~Jitomirskaya, to be published.
\gap
\item{\jlprl.}\ref{S.~Jitomirskaya and Y.~Last}{Dimensional Hausdorff properties
of singular continuous spectra}{Phys.~Rev.~Lett.}{76}{1996}{1765--1769}
\gap
\item{\jli.} S.~Jitomirskaya and Y.~Last, {\it{Power law subordinacy and singular
spectra, I. Half-line operators}}, in preparation.
\gap
\item{\js.}\ref{S.~Jitomirskaya and B.~Simon}{Operators with singular continuous
spectrum, III. Almost periodic Schr\"odinger operators}{Commun.~Math.~Phys.} {165}{1994}{201--205}
\gap
\item{\kap.}\ref{S.~Kahn and D.B.~Pearson}{Subordinacy and spectral theory for
infinite matrices}{Helv. Phys.~Acta}{65}{1992}{505--527}
\gap
\item{\katz.} Y.~Katznelson, {\it{An Introduction to Harmonic Analysis}},
Dover, New York, 1976.
\gap
\item{\kis.}\ref{A.A.~Kiselev}{Absolutely continuous spectrum of one-dimensional
Schr\"odinger operators and Jacobi matrices with slowly decreasing potentials}
{Commun.~Math.~Phys.}{179}{1996}{377--400}
\gap
\item{\kls.} A.~Kiselev, Y.~Last, and B.~Simon, {\it{Modifed Pr\"ufer and EFGP
transforms and the spectral analysis of one-dimensional Schr\"odinger operators}},
preprint.
\gap
\item{\kot.} S.~Kotani, {\it{Ljapunov indices determine absolutely continuous spectra
of stationary random one-dimensional Schr\"odinger operators}}, Stochastic Analysis
(K.~Ito, ed.), pp.~225--248, North Holland, Amsterdam, 1984.
\gap
\item{\kots.}\ref{S.~Kotani}{Support theorems for random Schr\"odinger operators}
{Commun.~Math.~Phys.}{97}{1985}{443--452}
\gap
\item{\kotfmv.}\ref{S.~Kotani}{Jacobi matrices with random potential taking finitely
many values}{Rev.~Math. Phys.}{1}{1989}{129--133}
\gap
\item{\lastz.}\ref{Y.~Last}{Zero measure spectrum for the almost Mathieu operator} {Commun.~Math.~Phys.}{164}{1994}{421--432}
\gap
\item{\lath.} Y.~Last, {\it{Conductance and Spectral Properties}}, D.Sc.~thesis,
Technion, 1994.
\gap
\item{\lapaii.} Y.~Last, {\it{Periodic approximants of Jacobi matrices, II. Complete
determination of the absolutely continuous spectrum}}, in preparation.
\gap
\item{\lsii.} Y.~Last and B.~Simon, {\it{Modified Pr\"ufer and EFGP transforms and
deterministic models with dense point spectrum}}, preprint.
\gap
\item{\lesa.} B.M.~Levitan and I.S.~Sargsjan {\it{Introduction to Spectral Theory}},
Trans.~Math.~Monographs, {\bf 39}, Amer.~Math.~Soc., Providence, RI, 1976.
\gap
\item{\pas.}\ref{L.~Pastur}{Spectral properties of disordered systems in the one-body
approximation}{Commun.~Math.~Phys.}{75}{1980}{167--196}
\gap
\item{\pcmp.}\ref{D.~Pearson}{Singular continuous measures in scattering theory}
{Commun.~Math.~Phys.}{60}{1978}{13--36}
\gap
\item{\pea.} D.~Pearson, {\it{Pathological spectral properties}}, Mathematical Problems
in Theoretical Physics, pp.~49--51, Lecture Notes in Physics No.~80, Springer,
Berlin-New York, 1979.
\gap
\item{\rue.}\ref{D.~Ruelle}{Ergodic theory of differentiable dynamical systems}
{Publ.~Math.~IHES}{50}{1979}{275--306}
\gap
\item{\sch.} I.~Sch'nol, {\it{On the behavior of the Schr\"odinger equation}},
Mat.~Sb. {\bf 42} (1957), 273--286 [in Russian].
\gap
\item{\ssg.}\ref{B.~Simon}{Schr\"odinger semigroups}{Bull.~Amer.~Math.~Soc.}
{7}{1982}{447--526}
\gap
\item{\scmp.}\ref{B.~Simon}{Some Jacobi matrices with decaying potentials and
dense point spectrum}{Commun.~Math.~Phys.}{87}{1982}{253--258}
\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{\ssp.} \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{\sst.} \ref{B.~Simon and G.~Stolz}{Operators with singular continuous spectrum,
V. Sparse potentials}{Proc.~Amer.~Math.~Soc.}{124}{1996}{2073--2080}
\gap
\item{\thou.}\ref{D.J.~Thouless}{Localization by a potential with slowly varying period}
{Phys.~Rev.~Lett.}{61}{1988}{2141--2143}
\gap
\enddocument