\input amstex
\documentstyle{amsppt}
\TagsOnRight
\topmatter
\title Operators with Singular Continuous Spectrum:\\ II. Rank One 
Operators
\endtitle
\rightheadtext{Singular Continuous Spectrum: Rank One Operators}
\author R. del Rio$^{1,3}$,
\thanks $^{1}$ Permanent address: IIMAS-UNAM,
Apdo. Postal 20-726, Admon. No. 20, 01000 Mexico D.F., Mexico. 
Research partially supported by DGAPA-UNAM and CONACYT.
\endthanks
N. Makarov$^2$
\thanks $^2$ This material is based upon work
supported by the National Science Foundation under Grant No. DMS-9207071.
The Government has certain rights in this material.
\endthanks
and B. Simon$^3$
\thanks $^3$ This material is based upon work
supported by the National Science Foundation under Grant No. 
DMS-9101715. The Government has certain rights in this material.
\endthanks
\leftheadtext{R. del Rio, N. Makarov and B. Simon}
\endauthor
\thanks To appear in {\it Commun.~Math.~Phys.}
\endthanks
\affil Division of Physics, Astronomy and Mathematics\\California 
Institute of Technology, 253-37\\Pasadena, CA 91125
\endaffil
\abstract For an operator, $A$, with cyclic vector $\varphi$, we study 
$A+\lambda P$ where $P$ is the rank one projection onto multiples of 
$\varphi$.  If $[\alpha,\beta] \subset \text{spec} (A)$ and $A$ has no 
a.c. spectrum, we prove that $A+\lambda P$ has purely singular 
continuous spectrum on $(\alpha,\beta)$ for a dense $G_\delta$ of 
$\lambda$\!'s.
\endabstract
\endtopmatter
\document
\flushpar{\bf \S 1. Introduction}
\smallpagebreak

The subject of rank one perturbations of self-adjoint operators and 
the closely related issue of the boundary condition dependence of 
Sturm-Liouville operators on $[0,\infty)$ has a long history.  We're 
interested here in the connection with Borel-Stieltjes transforms of 
measures $(\text{Im}\ z>0)$:
$$F(z)=\int \frac {d \rho (x)}{x-z}\tag1.1$$
where $\rho$ is a measure with
$$\int (|x|+1)^{-1} d \rho(x)<\infty .\tag1.2$$

In two fundamental papers Aronszajn [1] and Donoghue [5] related $F$ to 
spectral theory with important later input by Simon-Wolff [13].  In all 
three works, as in ours, the function ($y$ real)
$$G(y)=\int \frac {d \rho (x)}{(x-y)^2}$$
plays an important role.  Note we define $G$ to be $+\infty$ if the 
integral diverges.  Note too if $G(y)<\infty$, then the integral 
defining $F$ is finite at $z=y$ and so we can and will talk about 
$F(y)$.

Donoghue studied the situation
$$A_{\lambda}=A_{0}+\lambda P$$
where $P \psi =(\varphi, \psi)\varphi$ with $\varphi$ a unit vector 
cyclic for $A$.  $d \rho$ is then taken to be spectral measure for 
$\varphi$, that is,
$$(\varphi, e^{isA_{0}}\varphi)=\int e^{isx}d \rho(x).$$
Aronszajn studied the situation
$$H_{\text{formal}}=-\frac {d^2}{dx^2} +V(x)$$
on $[0,\infty)$ where $V$ is such that the operator is limit point at 
$\infty$.  Then, there is a one-parameter family of operators, 
$H_\theta$, with boundary condition
$$u(0) \cos\theta +u'(0) \sin\theta =0.$$
$\rho$ is the conventional Weyl-Titchmarsh-Kodaira spectral measure
for a fixed boundary condition, $\theta_0$.

An important result of the Aronszajn-Donoghue theory is
\proclaim{Theorem 1} $E$ is an eigenvalue of $A_\lambda$ \rom(resp. 
$H_\theta$\rom) if and only if
\roster
\item"\rom{(i)}" $G(E)<\infty$
\item"\rom{(ii)}" $F(E)=-\lambda^{-1}$\qquad \rom(resp. $\cot(\theta -
\theta_0)$\rom).
\endroster
\endproclaim

Our goal here is to the prove the following pair of theorems:
\proclaim{Theorem 2} $\{E|G(E)=\infty\}$ is a dense $G_\delta$ in
\rom{spec}$(A_0)$ \rom(resp. $H_{\theta_0}$\rom).
\endproclaim
\proclaim{Theorem 3} $\{\lambda |A_\lambda$ has no eigenvalues in 
\rom{spec}$(A_0)\}$ \rom(resp. $\{\theta|H_\theta$ has no eigenvalues in 
\rom{spec}$(H_{\theta_0}\}$ is a dense $G_\delta$ in $\Bbb R$ \rom(resp. 
$[0,2\pi]$\rom {))}.
\endproclaim
\bigpagebreak

While not stated precisely in those terms, Thm.~2 is a generalization 
of del Rio [4].  Gordon [8,9] has independently obtained these results 
by different methods.

Thm.~2 is quite easy and appears in \S2.  Thm.~3 is deeper and depends 
on some subtle estimates of $F$ found in \S3 and applied in \S4 to 
prove Thm.~3.

The interesting applications of Thm.~3 found in \S5 concern singular 
continuous spectrum.  For example, suppose $A_0$ has spectrum $[0,1]$ 
but has no a.c.~spectrum.  By general principles, $A_\lambda$ has no 
a.c.~spectrum either.  Then, Thm.~3 says that for a dense $G_\delta$ 
of $\lambda$, $A_\lambda$ has purely singular continuous spectrum.  
This is especially interesting because there are examples where the 
Simon-Wolff theory implies that for Lebesgue a.e.~$\lambda$, $A_\lambda$ has 
only point spectrum.  However, this is not always the case.  There 
exist $A_0$ and $P$ so that spec$(A_{0}+\lambda P)$ is purely singular 
continuous on $(0, 1)$ for {\it all} $\lambda$.  However, Thm.~3 
implies that it {\it cannot} happen that the spectrum is {\it always} 
pure point.
\bigpagebreak

\flushpar {\bf \S 2. Forbidden Energies}
\smallpagebreak

In this section we'll essentially prove Thm.~2.

\proclaim{Theorem 2.1}  Let $d \rho$ be a measure obeying \rom{(1.2)}.  
Let
$$G(y)=\int \frac{d \rho (x)}{(x-y)^2}.$$
Then, $\{ y|G(y)=\infty \}$ is a dense $G_\delta$ in 
\rom{supp}\rom($d\rho$\rom), the support of $d\rho$.
\endproclaim
\remark{Remarks \rom{1}}  By the Thm.~1, only $E$'s with 
$G(E)<\infty$ can be eigenvalues of $A_\lambda$ or $H_\theta$, so 
$E$'s with $G(y)=\infty$ are ``forbidden energies,'' that is, energies 
which cannot be eigenvalues.

2.  If supp$(d \rho)$ ($=\text{spec}(A)$) is perfect (no isolated 
points), then the theorem says the forbidden energies are locally 
uncountable in supp$(d \rho)$.

3.  Obviously, $\{y|G(y)=\infty\}\subset\text{supp}(d \rho)$.

4.  This says that $\{y\in\text{supp}(d \rho)|G(y)<\infty\}$ has  
interior empty in supp$(d \rho)$.  Even more so is the interior empty in 
$\Bbb R$.  The theorem is a stronger result than interior empty in 
$\Bbb R$ since supp$(d \rho)$ might itself have interior empty in $\Bbb 
R$.
\endremark
\demo{Proof}  The following are fundamental facts about Borel-Stieltjes 
transforms and their relation to $d \rho$ (see [3]):
\roster
\item $\lim\limits_{\epsilon\downarrow 0}F(E+i\epsilon)\equiv F(E+i0)$ 
exists and is finite for Lebesgue a.e.~$E$.
\item $d \rho_{\text{ac}}$ is supported on $\{E|\text{Im}\ 
F(E+i0)>0\}$.
\item $d \rho_{\text{sing}}$ is supported on 
$\{E|\lim\limits_{\epsilon\downarrow 0}\ \text{Im}\ F(E+i0)=\infty\}$.
\endroster
\flushpar If $G(y)<\infty$, it is easy to see that 
$\lim\limits_{\epsilon\downarrow 0}\ F(E+i0)$ exists is finite and real.  
Thus, if $G(y)<\infty$ on an interval $(\alpha, \beta)\subset\Bbb R$, 
then by (2), (3), $d \rho(\alpha, \beta)=0$, that is, $(\alpha, 
\beta)\cap \text{supp}(d \rho)=\emptyset$.  Thus, $\{ y|G(y)=\infty\}$ 
is dense in supp$(d \rho)$.

That $\{y|G(y)=\infty\}$ is a $G_\delta$ follows from the fact that 
$G$ is lower semicontinuous.  To be explicit, let
$$G_{m}(y)=\int \frac {d \rho(x)}{(x-y)^{2}+(m^{-1})^2}$$
which is a $C^\infty$ function by (1.2) and $G(y)=\sup\limits_m G_{m}(y)$.  
Thus
$$
\align
\{y|G(y)=\infty\}&=\{y|\forall n, \exists mG_{m}(y)>n\}\\
 &=\bigcap_{n} \bigcup_{m} \{y|G_{m}(y)>n\}
\endalign
$$
is a $G_\delta$. \qed
\enddemo
\bigpagebreak

\flushpar {\bf \S 3.  The Main Technical Lemma}
\smallpagebreak

In this section we'll prove
\proclaim{Lemma 3.1}  Let $d \rho$ obey \rom{(1.2)}.  Then
$$\{F(y)|G(y)<\infty \ \text{\rm and}\ y\in\text{\rm supp}(d \rho)\}$$
is a countable union of nowhere dense subsets of  \thinspace $\Bbb R$.
\endproclaim

Note that $G(y)<\infty$ implies the integral defining $F(y)$ is 
absolutely convergent and $F(y)$ is real.  The proof will depend 
critically on the fact that $F$ is the boundary value of an analytic 
function.  That such considerations must enter is seen by

\example{Example}  Let $A\subset[0, 1]$ be a nowhere dense set of 
positive measure (e.g., remove the middle open $\frac {1}{4}$ from 
$[0, 1]$, the middle $\frac {1}{9}$ from the remaining two pieces, the 
middle $\frac {1}{16}$ \dots , $\frac {1}{n^2}$ at the $(n-1)$st 
step).  Let
$$\tilde F(y)=\left|A\cap[0, y]\right|$$
where $|\ \cdot \ |$ is Lebesgue measure.  Then $\tilde F$ is Lipschitz; indeed, 
if $x<y$, $\left| \tilde F(x)-\tilde F(y)\right| \allowmathbreak =
\left|A\cap[x, y]\right|\le |x-y|$.  But $\tilde F[A]=[0, |A|]$ has
non-empty interior.  Thus for our $F$, we need more than just Lipschitz
properties (our $F$ is certainly not Lipschitz but
$F\restriction \{y|G(y)<\alpha\}$ is the restriction of a Lipschitz
function to that set).
\endexample
\bigpagebreak

The idea of the proof will be to break up $\{y|G(y)<\infty,\ 
y\in\text{supp}(d \rho)\}$ into a countable union of nowhere dense 
sets, $A_n$, so that $F$ is a homeomorphism on each of those sets.  On 
each $A_n$, $G$ will be close to constant. We'll use:

\proclaim{Lemma 3.2}  Let $B\subset\Bbb R$ be a nowhere dense set and 
let $F:B\to\Bbb R$ be a function obeying for $x<y$, with $x, y\in B$:
$$\alpha(y-x)<F(y)-F(x)<\beta(y-x) \tag3.1$$
for fixed $\alpha, \beta>0$.  Then $F[B]$ is nowhere dense.
\endproclaim

\demo{Proof}  By (3.1) $F$ has a unique continuum extension to $\bar 
B$ obeying (3.1).  $\Bbb R\setminus\bar B$ is a union of intervals 
$(x_{i}, y_{i})$ with $x_{i}, y_{i}\in\bar B$.  Extend $F$ to the 
interval by linear interpolation using slope $\frac {1}{2}(\alpha + 
\beta)$ on any semi-infinite subintervals of $\Bbb R\setminus\bar B$.  
The extended $F$ also obeys (3.1) and so defines a homeomorphism of 
$\Bbb R$ to $\Bbb R$.  As a homeomorphism, it takes nowhere dense sets 
to nowhere dense sets. \qed
\enddemo

\demo{Proof of Lemma 3.1}  We first break $A=\{y\in\text{supp}(d 
\rho)|G(y)<\infty\}$ into a countable family of sets $A_n$ so that for 
each $n$, there is $a_{n}>0$, $\delta_{n}>0$ so that
\roster
\item"\rom {(i)}" for $y\in A_n$, $\frac {8a_n}{9}<G(y)\le a_{n}$;
\item"\rom {(ii)}" for $y\in A_n$, $\int\limits_{|x-y|<\delta_{n}} \frac {d 
\rho(x)}{|x-y|^2} \le \frac {a_n}{21}$;
\item"\rom {(iii)}" $\bigcup\limits_{y\in A_n} [y-\beta\delta_{n}, y+\beta 
\delta_{n}]$ is connected where $\beta=\frac {1}{18}\left(\frac 
34\right)^{3/2}$.
\endroster

Such a breakup exists for we can first break $\Bbb R$ into intervals 
$\left(\left(\frac 89\right)^{m+1}, \left(\frac 
89\right)^m\right]$ and pigeonhole $G$ by its values.  Since $G(y)<\infty$ 
implies $\lim\limits_{\delta\downarrow 0} \int\limits_{|x-y|<\delta} \frac 
{d \rho(x)}{(x-y)^2}=0$, we can break each such set into countably 
many sets where (ii) holds.  Then we can break each such set into 
countably many sets so that (iii) holds by looking for gaps of size 
longer than $\delta_{n}\beta$.

Each $A_n$ is nowhere dense by Thm. 2 and we'll show that on $A_n$, $y>x$ 
implies that
$$\frac {1}{3}\ a_{n}(y-x)<F(y)-F(x)<\frac {5}{3}\ a_{n}(y-x) \tag3.2$$
so that the lemma follows from Lemma 3.2.
\enddemo

Define $\ell_{n}=\beta\delta_n$ and $\epsilon_{n}=\sqrt{\frac 34} \ 
\ell_n$.  For $y\in A_n$, let $\bigtriangleup_{n}(y)$ be the triangle in 
$\Bbb C$ (see Fig.~3.1):
$$\bigtriangleup_{n}(y)=\left\{z|0<\text{Im}\ z\le \epsilon_{n}, 
\left|\text{arg}(z-y)-\frac \pi2 \right| \le \frac \pi6 \right\}.$$
This is the equilateral triangle of side $\ell_n$ with one side 
parallel to the real axis at distance $\epsilon_n$ from that axis and 
the opposite vertex at $y$.

For $z\in\bigtriangleup_{n}(y)$, define
$$G(z)=\int \frac {d \rho (x)}{(x-z)^2}=\frac {dF}{dz}.$$
We claim that for $z\in\bigtriangleup_n$
$$\left|G(z)-a_{n}\right|\le \frac {a_n}{3}.\tag3.3$$

Accepting (3.3) for the moment, let us prove (3.2).  By the 
fundamental theorem of calculus, (3.3) implies for $z, z^\prime \in 
\bigtriangleup_{n}(y)$:
$$\left|F(z)-F(z^\prime)-a_{n}(z-z^\prime)\right| \le \frac {a_n}3\ |z-
z^\prime|.\tag3.4$$

By hypothesis (iii) on $A_n$, $\bigcup\limits_{y\in A_n} \bigtriangleup_{n}(y)$ 
is connected and so, given $y<y^\prime \in A_n$, we can find a finite 
sequence $y_{0}=y<y_{1}<\dots<y_{n}=y^\prime$ and $z_{1},\dots , z_n$ 
so that (see Fig. 3.2)
$$z_j \in \bigtriangleup_{n}(y_{j-1})\cap \bigtriangleup_{n}(y_j) \qquad 
\text{and} \qquad \left|z_{j}-y_{j-1}\right|=\left|z_{j}-
y_{j}\right|=\left|y_{j}-y_{j-1}\right|. \tag3.5$$
By (3.4) and (3.5)
$$\left|F(y)-F(y^\prime)-a_{n}(y-y^\prime)\right| \le \frac {2a_n}3\ (y-
y^\prime)$$
which is (3.2).

Thus we need only prove (3.3)  We write
$$\left|G(z)-a_n\right| \le 
\left|b_0\right|+\left|b_1\right|+\left|b_2\right|+\left|b_3\right|$$
where
$$\align b_0 &=G(y)-a_n \\
 b_1 &= \int\limits_{|x-y|<\delta_n} \frac {d \rho(x)}{|x-z|^2} \\
 b_2 &= \int\limits_{|x-y|<\delta_n} \frac {d \rho(x)}{(x-y)^2} \\
 b_3 &= \tilde G(z)-\tilde G(y)
\endalign$$
with
$$\tilde G(z)=\int\limits_{|x-y| \ge \delta_n} \frac {d \rho(x)}{(x-
z)^2}.$$

By hypothesis (i) on $A_n$, $\left|b_0\right| \le \frac {a_n}9$.

By hypothesis (ii) on $A_n$, $\left|b_2\right| \le \frac {a_n}{21}$.  
By elementary trigonometry,
$$z\in\bigtriangleup (y) \qquad \text{and} \qquad x\in\Bbb R 
\Rightarrow |z-y| \ge \sqrt {\frac 34}\ |x-y|. \tag3.6$$
Thus
$$\left|b_1\right| \le \frac 43 \left|b_2\right|$$
so
$$\left|b_1\right|+\left|b_2\right| \le \frac 73 \frac {a_n}{21}=\frac 
{a_n}9.$$

Finally, using the fundamental theorem of calculus and (3.6)
$$\align
\left|b_3\right| &\le 2|z-y| \left(\frac 43\right)^{3/2} \delta^{-1}_n 
\int\limits_{|x-y|\ge \delta_n} \frac {d \rho(x)}{(x-y)^2} \\
 &\le 2\delta^{-1}_n \ell_n \left(\frac 43\right)^{3/2} a_n \\
 &= 2\beta \left(\frac 43\right)^{3/2} a_n=\frac {a_n}9
\endalign$$
by definition of the constant $\beta$.  Thus (3.3) holds. \qed
\bigpagebreak

\flushpar {\bf \S 4.  Proof of the Main Theorem}
\smallpagebreak

Our goal here is to prove Thm.~3 (from \S1) and derive some simple 
abstract consequences of it.

\demo{Proof of Thm.~3}  The maps $M_1:\Bbb R\setminus\{0\}\to\Bbb R\setminus
\{0\}$ by $M_1(\lambda)=-\lambda^{-1}$ and $M_2:[0, \pi)\to\Bbb 
R\cup\{\infty\}$ by $M_2(\theta)=\cot(\theta-\theta_0)$ are 
homeomorphisms.  Thus, by Lemma 3.1,
$$\{\lambda|\exists E \text{ s.t. } G(E)<\infty, E\in\text{spec}(A_0), 
F(E)=-\lambda^{-1}\}$$
and
$$\{\theta|\exists E \text{ s.t. } G(E)<\infty, E\in\text{spec}(A_0), 
F(E)=\cot(\theta-\theta_1)\}$$
are countable unions of nowhere dense sets.  Its complement is thus a 
dense set by the Baire category theorem.  But by Thm.~1, this is 
precisely $\{\lambda|A_\lambda$ has no eigenvalues on spec$(A_0)\}$, 
which we conclude is dense.  By general principles [12], it is also a 
$G_\delta$. \qed
\enddemo
\smallpagebreak
Here are some simple corollaries of Thm.~3.  We state them in the 
rank one case but they hold in the $\cot(\theta-\theta_0)$ B.C.~case 
also.

\proclaim{Corollary 4.1}  Suppose that $A_0$ is an operator with no 
a.c.~spectrum and $P$ is a rank one projection whose range is cyclic 
for $A$.  Then for a dense $G_\delta$ of $\lambda$'s, 
$A_\lambda=A+\lambda P$ has only singular continuous spectrum in 
\text{\rm spec}$(A_0)^{\text{int}}$.
\endproclaim

\demo{Proof}  $A_\lambda$ has no a.c.~spectrum since the a.c.~spectrum 
is left invariant by finite rank perturbations.  spec$(A_0)$ has no 
eigenvalues for a dense $G_\delta$ of $\lambda$.  There can be 
eigenvalues on $\Bbb R \setminus\text{spec}(A_0)$ and so point 
spectrum on $\partial(\text{spec}(A_0))$.  But there cannot be point 
spectrum in spec$(A_0)^{\text{int}}$.
\enddemo

\proclaim{Corollary 4.2}  Suppose that $A_0$ is an operator with no 
a.c.~spectrum and an interval $[\alpha, \beta]\subset\text{\rm spec}(A_0)$.
Let $P$ be a rank one projection whose range is a cyclic vector for
$A_0$. Then for a dense $G_\delta$ of $\lambda$'s, $A_0+\lambda P$
has singular continuous spectrum on all of $(\alpha, \beta)$ and
only singular continuous spectrum there.
\endproclaim

\demo{Proof}  A direct consequence of Cor.~4.1. \qed
\enddemo
\bigpagebreak

\flushpar {\bf \S 5.  Examples and Consequences}
\smallpagebreak

Our first class of examples are half-line Schr\"odinger operators:

\proclaim{Theorem 5.1}  Let $V(x)$ be a locally $L_1$ function on $[0, 
\infty)$ and let $H_\theta=-\frac {d^2}{dx^2} +V(x)$ with $\theta$ 
boundary conditions.  Suppose there is some $\theta_0$ and 
$\alpha<\beta$ so that
\roster
\item"\rom{(i)}"  $[\alpha, \beta]\subset\text{\rm spec}(H_{\theta_0})$.
\item"\rom{(ii)}"  for Lebesgue a.e., $E_0\in[\alpha, \beta]$, there 
exists a function $\varphi_{E_0}$ obeying
$$-\varphi''(x)+V(x)\varphi(x)=E_0\varphi(x) \tag5.1$$
$$\int\limits_0^\infty \left|\varphi(x)\right|^2 dx<\infty.\tag5.2$$
\endroster
Then:
\roster
\item"\rom{(a)}"  For a dense $G_\delta$ of $E$'s in $[\alpha, 
\beta]$, there is no solution of \rom{(5.1)} obeying \rom{(5.2)}.
\item"\rom{(b)}"  For Lebesgue a.e.~$\theta$, $H_\theta$ has only 
point spectrum in $(\alpha, \beta)$.
\item"\rom{(c)}"  For a dense $G_\delta$ of $\theta$, $H_\theta$ has 
only singular continuous spectrum in $(\alpha, \beta)$.
\endroster
\endproclaim

\remark{Remarks}  1.  (b), (c) say that there are disjoint sets both 
locally uncountable where $H_\theta$ shifts between purely pure point 
and purely singular continuous spectrum on $[\alpha, \beta]$.

2.  It may well happen that there are $\theta$'s with spectrum of 
mixed type.

3.  For precursors of (a), see del Rio [4].
\endremark

\demo{Proof}  If $E$ is such that (5.1) has a solution obeying (5.2), 
then $\varphi_E$ obeys some boundary condition at $x=0$ and so $E$ is 
an eigenvalue of some $H_\theta$.  Thus (a) follows from Thm.~2.

To prove (b), note that if $E_0$ has a solution obeying (5.1--2) and 
$E_0$ is not an eigenvalue of $H_{\theta_0}$, then 
$\lim\limits_{\epsilon\downarrow 0}\ \int\limits_0^\infty \left|G(0, 
x; E+i\epsilon)\right|^2 dx<\infty$.  Now apply the ideas of
Kotani [11] and Simon-Wolff [13].

(c) follows from Thm.~3. \qed
\enddemo

\example{Example 5.2}  Suppose that $[a, b]\subset\text{spec}
(-\frac {d^2}{dx^2} +V(x))$ and that for a.e.~$E\subset [a, b],
\lim\limits_{x\to\infty} \frac 1{|x|} \text{ln}\|T_E(x)\|=\gamma(E)$
and is positive.  Here $T$ is the standard transfer matrix, that is,
$$T_E(x)=\left( \matrix
\varphi_1(x) & \varphi_2(x) \\
\varphi'_1(x) & \varphi'_2(x)
\endmatrix \right)
$$
where $\varphi_i$ obeys $-u''+Vu=Eu$ with $\varphi_1(0)=\varphi'_2(0)=1$ and 
$\varphi'_1(0)=\varphi_2(0)=0$.  Then (a) implies there must be a dense 
$G_\delta$ of $E$ where either $\lim \frac 1{|x|}\|T_E(x)\|$ fails to exist or 
is zero.  Thus, a positive limit can {\it never} exist for all $E$ in an
interval. Results of this genre have been found previously by Goldsheid [6]
and Carmona [2].
\endexample

\example{Example 5.3}  Consider a one-dimensional random model with
localization, for example, the GMP model [7,2].  Then for almost every
$E$ in $[\alpha, \infty)$, one knows $\gamma (E)$ exists and is positive.
It follows from Thm.~5.1 that for a locally uncountable set of boundary
conditions (a Lebesgue typical set), one has pure point spectrum, while
for a distinct set of locally uncountable boundary conditions (a Baire
typical set), one has singular spectrum.  Each spectral type is unstable
to change to the other spectral type.
\endexample

\example{Example 5.4} Let $H=- \frac {d^2}{dx^2}+ \cos(\sqrt x)$ on $L^2(0, 
\infty)$, a model studied by Stolz [14].  As proven by him for any boundary 
condition $\theta$:
$$\text{spec}(H_\theta)=[-1, \infty).$$
spec$(H_\theta)$ is purely absolutely continuous on $(1, \infty)$. Kirsch
et al.~[10] prove that for a.e.~$\theta$, $H_\theta$ has pure point spectrum
in $[-1, 1]$ only.  Our results show that for a dense $G_\delta$ of $\theta$,
the spectrum is purely singular continuous.  Once again you have intertwined
purely pure point and purely singular continuous spectrum.
\endexample

\bigpagebreak
Finally, we consider the case of the Anderson model:

\proclaim{Theorem 5.5}  Let $V_\omega$ be a $\nu$-dimensional model with 
uniform distribution on $[a, b]$ at each set.  Suppose that the corresponding 
Jacobi matrix $h_\omega$ has point spectrum in $[\alpha, \beta]$ for 
a.e.~$V_\omega$.  Fix $V_\omega(n)$ typical.  Then for a.e., choice 
$V_\omega(0)\in [a, b]$, $h_\omega$ has pure point spectrum in $[\alpha, 
\beta]$; but for a dense $G_\delta$ of values of $V_\omega(0)$, $h_\omega$ has 
purely singular spectrum in $[\alpha, \beta]$.
\endproclaim

This follows from Thm.~3.

\bigpagebreak
\example{Acknowledgment} We would like to thank H. Kalf for useful
conversations.
\endexample

\bigpagebreak
\Refs
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of singular Sturm-Liouville equations \jour Am. J. Math. \vol 79 
\yr 1957 \pages 597--610
\endref
\ref \key 2 \by R. Carmona \paper Exponential localization in one-dimensional
disordered systems \jour Duke Math. J. \vol 49 \yr 1982 
\pages 191--213
\endref
\ref \key 3 \by R. Carmona and J. Lacroix \book Spectral theory of random
Schr\"odinger operators \publ Birkh\"auser 
\publaddr Boston \yr 1990
\endref
\ref \key 4 \by R. del Rio \paper A forbidden set for embedded 
eigenvalues \toappear\ Proc. AMS
\endref
\ref \key 5 \by W. Donoghue \paper On the perturbation of spectra \jour 
Comm. Pure App. Math. \vol 18 \yr 1965 \pages 559--579
\endref
\ref \key 6 \by I. Goldsheid \paper Asymptotics of the product of 
random matrices depending on a parameter \jour Soviet Math. Dokl. \vol 
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\enddocument