\input amstex
\documentstyle{amsppt}
\TagsOnRight
\NoBlackBoxes
\topmatter
\title Operators with Singular Continuous Spectrum: \\
I. General Operators
\endtitle
\rightheadtext{Singular Continuous Spectrum: General Operators}
\author B.~Simon$^{*}$
\endauthor
\leftheadtext{B.~Simon}
\thanks ${*}$ 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
\thanks To appear in {\it Ann.~Math.}
\endthanks
\affil Division of Physics, Mathematics, and Astronomy \\
California Institute of Technology 253-37 \\ Pasadena, CA 91125
\endaffil
\endtopmatter
\document

\bigpagebreak
\flushpar {\bf \S 0. Introduction}

The Baire category theorem implies that the family, $\Cal F$, of dense 
sets $G_\delta$ in a fixed metric space, $X$, is a candidate for generic 
sets since it is closed under countable intersections; and if $X$ is 
perfect (has no isolated point), then $A\in\Cal F$ has uncountable 
intersections with any open ball in $X$.

There is a long tradition of soft arguments to prove that certain 
surprising sets are generic.  For example in $\Cal C[0, 1]$, a generic 
function is nowhere differentiable.  Closer to our concern here, 
Zamfirescu [20] has proven that a generic monotone function has purely 
singular continuous derivative, and Halmos [7]-Rohlin [14] have proven 
that a generic ergodic process is weak mixing but not mixing.  We will 
say a set $S\subset X$ is Baire typical if it is a dense $G_\delta$ and 
a set $S\subset X$ is Baire null if its complement is Baire typical.

Our goal is to look at generic sets of self-adjoint operators and show 
that their spectrum is quite often purely singular continuous.  Here 
are three of our results that give the flavor of what we will prove in 
\S 3 and \S 4.

Consider the sequence space, $[-a, a]^{\Bbb Z}$, of sequences $v_n$ 
with $|v_{n}|\leq a$.  Given any such $v$, we can define a Jacobi 
matrix $J(v)$ as the tridiagonal matrix with $J_{n, n\pm 1}=1$ 
and $J_{n,n}=v_n$.  View $J$ as a self-adjoint operator on 
$\ell^{2}(\Bbb Z)$.  It is known (e.g.~[4,17,16]) that if one puts a 
product of normalized Lebesgue measures on $[-a, a]^{\Bbb Z}$ (i.e., 
the $v_n$ are independent random variables each uniformly distributed in 
$[-a, a]$) then, $J(v)$ is a.e.~an operator with spectrum $[a-2, a+2]$ 
and the spectrum there is pure point.  So our first result is somewhat 
surprising.

\proclaim{Theorem 1}  View $[-a, a]^{\Bbb Z}$ in the product topology.  
Then $\{v\mid J(v)\text{\rom{ has spectrum }}\mathbreak [-a-2, a+2]
\text{\rom{ and the spectrum is purely singular continuous}}\}$ is 
Baire typical.
\endproclaim

We also have some results if $\Bbb Z$ is replaced by $\Bbb Z^{\nu}$ and 
the Jacobi matrix by the multidimensional discrete Schr\"odinger 
operator.  One might think that the weakness of the topology and the 
one dimension are critical.  They are not, as our second result shows.

For $V\in\Cal C(\Bbb R^{\nu})$, let $S(V)$ be the Schr\"odinger operator 
$-\Delta +V$ on $L^{2}(\Bbb R^{\nu})$.


\proclaim{Theorem 2}  Let $\Cal C_{\infty}(\Bbb R^{\nu})$ be the continuous 
functions vanishing at infinity in $\|\cdot\|_{\infty}$.  Then
$$
\{V\mid S(V)\text{\rom{ has purely singular continuous spectrum on }}
(0,\infty)\}
$$
is Baire typical.
\endproclaim

Note that for $V\in\Cal C_{\infty}(\Bbb R)$, the essential spectrum, 
$\text{spec}_{\text{\rom ess}}(S(V))=[0, \infty)$, so Theorem 2 says 
that generically, the singular continuous spectrum, $\text
{spec}_{\text{\rom sc}}(S(V))=[0, \infty)$, the absolutely continuous 
spectrum, $\text{spec}_{\text{\rom ac}}=\emptyset$, and the pure point 
spectrum, $\text{spec}_{\text{\rom pp}}(S(V))\subset (-\infty, 0]$.  For 
the discrete one-dimensional (Jacobi matrix) case, we will be able to say 
something about decay.  For example when $\nu=1$, a generic $v\in\ell^{p}$ 
($2<p<\infty$) has a $J(v)$ with purely singular continuous spectrum in 
$[-2, 2]$.  For $p=1$, we know $\text{spec}_{\text{\rom ac}}(J(v))=
[-2, 2]$ so the singular spectrum result doesn't extend to all $p$.  
$1<p\leq 2$ is open.

Our third example is related to the celebrated theorem of Weyl-von 
Neumann [18,19,8] that given any self-adjoint $A$ and any $\epsilon$, 
there exists a Hilbert-Schmidt operator $B$ with $\|B\|_{2}<\epsilon$ 
(where $\|C\|_{2}=\text{tr}(C^{*}C)^{1/2}$) so that $A+B$ has only 
point spectrum. That is not the generic situation.

\definition{Definition}  A self-adjoint operator, $C$, is called 
{\it usual} if and only if $\{\psi\mid C\psi=\lambda\psi$ and 
$\lambda\in\text{spec}_{\text{\rom disc}}(C)$, the discrete
spectrum of $C\}\cup\{\psi\mid d\mu^{C}_{\psi}(\lambda)$ is purely 
singular continuous$\}$ span the space $\Cal H$.  Here $d\mu^{C}_{\psi}$ 
is the spectral measure for $(C, \psi)$, that is
$$
\int e^{i\lambda t}d\mu^{C}_{\psi}(\lambda)=(\psi, e^{i\lambda C}\psi). 
\tag 0.1
$$
\enddefinition

$I_2$ is the Hilbert-Schmidt operators in $\|\cdot\|_2$ norm.

\proclaim{Theorem 3}  Let $A$ be a fixed self-adjoint operator.  Then 
$\{B\in I_{2}\mid A+B\text{ is usual}\}$ is Baire typical.
\endproclaim

For example, if spec$(A)=[-1, 1]$, generically $A+B$ has purely singular 
continuous spectrum in $(-1, 1)$.

In \S 1 we prove two results asserting that certain families of 
operators are always sets $G_\delta$.  We will use that to prove criteria 
for generic singular spectrum in \S 2. We then study general operators 
in \S 3 and Schr\"odinger/Jacobi operators in \S 4.

I would like to thank R.~del Rio and N.~Makarov for discussions which 
stimulated this work, and S.~Molchanov and A.~Teplyaev for telling me 
of [10].

\bigpagebreak
\flushpar {\bf \S 1. Soft Stuff}

A metric space, $X$, of (perhaps unbounded) self-adjoint operators on 
a separable Hilbert space, $\Cal H$, will be called {\it regular} if 
and only if:
\roster
\item"(1)" $X$ is complete.
\item"(2)" If $A_{n}\to A$ in the metric topology, then $A_{n}\to A$ in 
the strong resolvent sense.
\endroster

Our main technical results are three:

\proclaim{Theorem 1.1} Fix $C\subset\Bbb R$ closed and $X$ a regular 
metric space of operators.  Then
$$
\{A\mid A\text{\rom{ has no eigenvalues in }}C\}
$$
is a $G_\delta$ in $X$.
\endproclaim

\proclaim{Theorem 1.2}  Fix $U\subset\Bbb R$ open and $X$ a regular 
metric space of operators.  Then
$$
\{A\mid \text{\rom{ For any spectral measure for $A$, }} 
(\mu^{A}_{\psi})_{\text{\rom{ac}}}[U]=0\}
$$
is a $G_\delta$ in $X$.
\endproclaim

\remark{Remarks} 1.  Note the word ``dense'' does not appear before 
$G_\delta$.  That will hold sometimes, as we will analyze.

2. $\mu^{A}_{\psi}$ is defined in (0.1).  $(\nu)_{\text{\rom ac}}$ 
means the absolutely continuous component of $\nu$.
\endremark

\proclaim{Theorem 1.3} Fix $K\subset\Bbb R$ closed and $X$ a regular 
metric space of operators.  Then
$$
\{A\mid K\subset\text{\rom{spec}}(A)\}
$$
is a $G_\delta$.
\endproclaim

\proclaim{Lemma 1.4}  Let $A_n$ be a sequence of self-adjoint operators 
on $\Cal H$ so that $A_{n}\to A$ in strong resolvent sense for some 
self-adjoint $A$.  Let $K$ be a compact subset of $\Bbb R$; $\varphi$, 
a fixed vector in $\Cal H$, and $\epsilon>0$.  Suppose there exist 
eigenvectors $\eta_n$ of $A_n$:
$$
A_{n}\eta_{n}=\lambda_{n}\eta_{n}
$$
with $\|\eta_{n}\|=1$, $\lambda_{n}\in K$ and $|\langle\eta_{n}, 
\varphi\rangle|\geq\epsilon$.  Then $A$ has an eigenvector $\eta$ with
$$
A\eta = \lambda\eta
$$
with $\lambda\in K$, $\|\eta\|=1$ and $|\langle\eta, 
\varphi\rangle)\geq \epsilon$.
\endproclaim

\demo{Proof} $K$ is compact and $\{\psi\in\Cal H\mid \|\psi\|\leq 1\}$ 
is compact in the weak topology.  So we can pass to a subsequence and 
suppose $\eta_{n}\to\eta_{\infty}$ weakly and $\lambda_{n}\to\infty$.  
We will show that $\eta_{\infty}\in D(A)$ and $A\eta_{\infty}=\lambda
\eta_{\infty}$.  Since $|(\eta_{\infty}, \varphi)|\geq\epsilon$, we have 
$\eta_{\infty}\neq 0$ and so $\eta=\eta_{\infty}/\|\eta_{\infty}\|$ is 
the required vector.

Let $\psi\in\Cal H$ be arbitrary.  Then
$$\align
\left((A+i)^{-1}\eta_{\infty}, \psi\right)&=\left(\eta_{\infty}, 
(A-i)^{-1}\psi\right) \\
&= \lim_{n}\left(\eta_{n}, (A_{n}-i)^{-1}\psi\right) \tag 1.1 \\
&=\lim_{n}\left((A_{n}+i)^{-1}\eta_{n}, \psi\right) \\
&=\lim_{n}\left((\lambda_{n}+i)^{-1}\eta_{n}, \psi\right) \\
&=\left((\lambda+i)^{-1}\eta_{\infty}, \psi\right).
\endalign
$$
It follows that $\eta_{\infty}=(\lambda+i)$, $(A+i)^{-1}\eta_{\infty}
\in D(A_{\infty})$ and $A\eta_{\infty}=\lambda\eta_{\infty}$.  
(1.1) holds because $(A_{n}-i)^{-1}\psi$ converges to $(A-i)^{-1}\psi$ 
in norm and $\eta_{n}\to\eta_{\infty}$ weakly with $\|\eta_{n}\|\leq 
1$. \qed
\enddemo

\demo{Proof of Theorem \rom{1.1}} Fix $K\subset\Bbb R$ compact, 
$\epsilon>0$ and $\varphi\in\Cal H$.  Then Lemma 1.4 implies that
$$\multline
Q(K, \varphi, \epsilon)= \\
\{A\in X\mid\exists\eta\in D(A)\text{ with }\|\eta\|=1, |\langle\varphi, 
\eta\rangle|\geq \epsilon,\, A\eta=\lambda\eta \text{ for some } 
\lambda\in K\}
\endmultline
$$
is a closed subset of $X$.

Fix $\{\varphi_{l}\}^{\infty}_{l=1}$ an orthonormal basis of $\Cal H$.  
For $n, l, m\in\Bbb Z_+$, let
$$
Q_{n,l,m}=Q(C\cap [-n, n], \varphi_{l}, m^{-1}).
$$
Then
$$
\cup Q_{n,l,m}=\{A\mid A\text{ has an eigenvalue in }C\}
$$
is an $F_\sigma$, so its complement is a $G_\delta$ as claimed. \qed
\enddemo

\proclaim{Lemma 1.5}  Let $(a, b)$ be a fixed open interval in $\Bbb 
R^{n}$ and let $d\mu$ be a measure on $\Bbb R$.  Then $\mu$ is purely 
singular on $(a, b)$ if and only if for each $n>2$, there exists 
$\epsilon_{n}>0$ and $f_n$ obeying
\roster
\item"\rom{(1)}" $0\leq f_{n}\leq 1$,
\item"\rom{(2)}" $f_n$ is supported in $(a-\epsilon_{n}, 
b+\epsilon_{n})$,
\item"\rom{(3)}" $\int\limits^{\infty}_{-\infty} f_{n}(s)ds<2^{-n}$,
\item"\rom{(4)}" $\mu\left(\chi_{[a-\epsilon_{n}, b+\epsilon_{n}]}
-f_{n}\right)<2^{-n}$,
\item"\rom{(5)}" $\epsilon_{n}<2^{-n}$.
\endroster
\endproclaim

\demo{Proof}  Suppose such $\epsilon_n$ and $f_n$ exist.  Let
$$
C_{n}=\left\{x\mid f_{n}(x)>\frac12\right\}.
$$
Then (with $|\cdot|=$ Lebesgue measure):
$$\gathered
|C_{n}|<2^{-n+1} \\
\mu([a-\epsilon_{n}, b+\epsilon_{n}]\backslash C_{n})<2^{-n+1} 
\endgathered
$$
and
$$
C_{n}\subset [a-\epsilon_{n}, b+\epsilon_{n}].
$$
It follows that
$$
C=\bigcap_{m}\,\bigcup^{\infty}_{n=m} C_n
$$
obeys $|C|=0$ and $\mu([a, b]\backslash C)=0$.

Conversely, suppose that $\mu$ is purely singular continuous on $(a, 
b)$.  Find $C$ in $(a, b)$ so $|C|=0$ and $\mu((a, b)\backslash C)=0$.  
By adding $a$ and/or $b$ to $C$, we can suppose $C\subset [a, b]$ and 
$\mu([a, b]\backslash C)=0$.  Since $\lim\limits_{\epsilon\downarrow 0}
\mu([a-\epsilon, a))=0$ and $\lim\limits_{\epsilon\downarrow 0}
\mu((b, b+\epsilon])=0$, we can choose $\epsilon_{n}<2^{-n}$ so that
$$
\mu([a-\epsilon_{n}, a))+\mu((b, b+\epsilon_{n}])<2^{-n-1}.
$$

By regularity of measures, we can find $K_{n}\subset C\subset U_{n}
\subset (a-\epsilon_{n}, b+\epsilon_{n})$ so that $|U_{n}|<2^{-n}, 
\mu([a, b]\backslash K_{n})<2^{-n-1}$. By Urysohn's lemma, find $f$ 
continuous with $0\leq f\leq 1$, $f\equiv 1$ on $K_n$ and $\text{supp}
f\subset U_n$.  Then
$$
\int\limits^{\infty}_{-\infty} f_{n}(s)ds\leq|U_{n}|<2^{-n}
$$
while
$$
\mu\left(\chi_{[a-\epsilon_{n}, b+\epsilon_{n}]}-f_{n}\right)
\leq \mu([a-\epsilon_{n}, a))+\mu([a, b]\backslash K_{n})+
\mu((b, b+\epsilon_{n}])<2^{-n}
$$
as required. \qed
\enddemo

\demo{Proof of Theorem \rom{1.2}} Let $\varphi\in\Cal H$, $a, b\in\Bbb 
R$ and
$$
Q(\varphi, a, b)=\{A\mid d\mu^{A}_{\varphi}\text{ is purely singular on }
(a, b)\}.
$$
By Lemma 1.5
$$
Q(\varphi, a, b)=\bigcap^{\infty}_{n=2}\,\bigcup_{(f,\epsilon)\in B_{n}}
\Cal A_{n}(f, \epsilon; \varphi)
$$
where $B_n$ is the set of pairs $(f, \epsilon)$ obeying (1--3; 5) of 
Lemma 1.5 and
$$
\Cal A_{n}(f, \epsilon; \varphi)=\{A\mid (\varphi, [\chi_{[a-\epsilon,
b+\epsilon]}(A)-f(A)]\varphi)<2^{-n}\}.
$$

We claim each $A$ is open, equivalently that
$$
\Cal A^{c}_{n}(f, \epsilon; \varphi)=\{A\mid (\varphi, \chi_{[a-
\epsilon, b+\epsilon]}(A)-f(A)\varphi)\geq2^{-n}\}
$$
is closed.  For let $A_{l}\in\Cal A^{c}_{n}$ converge to $A$ in strong 
resolvent sense.  Then \linebreak $\lim(\varphi, f(A_{l})\varphi)
=(\varphi, f(A)\varphi)$ (see, e.g., [12]).  Let $h_m$ be continuous 
functions with $h_{m}\downarrow\chi_{[a-\epsilon, b+\epsilon]}$ 
monotonically.  Then $h_{m}(A)\to h_{m}(A)$ strongly, so
$$\align
(\varphi, \chi_{[a-\epsilon, b+\epsilon]}(A)\varphi)
&=\inf_{m}(\varphi, h_{m}(A)\varphi) \\
&=\inf_{m}[\lim_{n}(\varphi, h_{m}({}_{l})\varphi)] \\
&\geq \overline{\lim_{n}}(\varphi, \chi_{[a-\epsilon, 
b+\epsilon]}(A_{l})\varphi)
\endalign
$$
so the claim is proven.

Any open set $U$ is a countable union of open intervals $I_{n}=(a_{n}, 
b_{n})$. Let $\varphi_{l}$ be an orthonormal basis for $\Cal H$.  Then 
the set that the theorem asserts is a $G_\delta$ is just
$$
\bigcap^{\infty}_{n=1}\,\bigcap^{\infty}_{l=1} Q(\varphi_{l}, a_{n}, 
b_{n})
$$
which is indeed therefore a $G_\delta$. \qed
\enddemo

The following is an expression of the well-known fact of lower 
semicontinuity of the spectrum under strong limits.

\proclaim{Lemma 1.6}  If $A_{n}\to A$ in strong resolvent sense and 
$(a, b)\cap\text{\rom{spec}}(A_{n})=\emptyset$, then $(a, b)\cap
\text{\rom{spec}}(A)=\emptyset$.
\endproclaim

\demo{Proof}  Let $f$ be the function $f(x)=\text{dist}(x, \Bbb R
\backslash(a, b))$.  Then $(a, b)\cap\text{spec}(B)=\emptyset$ if and 
only if $f(B)=0$.  By the continuity of the functional calculus of 
$A_{n}\to A$ in strong resolvent sense, then $f(A)=\text{s-lim}\,
f(A_{n})=0$ if $(a, b)\cap\text{spec}(A_{n})=\emptyset$. \qed
\enddemo

\demo{Proof of Theorem \rom{1.3}}  Let $\lambda_n$ be a countable dense 
set in $K$. Then
$$
\{A\mid K\subset\text{spec}(A)\}=\bigcap_{n}\{A\mid\lambda_{n}\in
\text{spec}(A)\}
$$
so we need only consider the cases where $K=\{\lambda\}$.  But
$$
\{A\mid\lambda\notin\text{spec}(A)\}=\bigcup^{\infty}_{n=1}
\left\{A\mid\left(\lambda - \frac1n ,\, \lambda + \frac 1n\right)
\cap\text{spec}(A)=\emptyset\right\}
$$ 
is an $F_\sigma$ by Lemma 1.6.  Thus, its complement is a $G_\delta$. 
\qed
\enddemo

\bigpagebreak
\flushpar {\bf \S 2. Welcome to Wonderland}

The main point in the way to generate generic singular spectrum is

\proclaim{Theorem 2.1}  Let $X$ be a regular metric space of 
self-adjoint operators. Suppose that for some interval $(a, b)$, we have 
that
\roster
\item"\rom{(i)}" $\{A\mid A\text{\rom{ has purely continuous spectrum on 
$(a, b)$}}\}$ is dense in $X$.
\item"\rom{(ii)}"  $\{A\mid A\text{\rom{ has purely singular spectrum on 
$(a, b)$}}\}$ is dense in $X$.
\item"\rom{(iii)}" $\{A\mid A\text{\rom{ has $(a, b)$ in its spectrum}}\}$ 
is dense in $X$.
\endroster

Then $\{A\mid (a, b)\subset\text{\rom{spec}}_{\text{\rom{sc}}}(A), 
(a, b)\cap\text{\rom{spec}}_{\text{\rom{pp}}}(A)=\emptyset,
(a, b)\cap\text{\rom{spec}}_{\text{\rom{ac}}}(A)=\emptyset\}$ is a dense 
$G_\delta$.
\endproclaim

\demo{Proof}  Because $(a, b)$ is an $F_\delta$, each of the sets in 
(i)--(iii) is a $G_\delta$ by Theorems 1.1--3.  (For example, the set in 
(i) is the intersection of the same sets for $[a+\frac1n, b-\frac1n]$.)  
Thus, by hypothesis they are dense $G_\delta$'s.  By the Baire category 
theorem, their intersection is a dense $G_\delta$. \qed
\enddemo

\remark{Remarks}  1.  We pick an interval for definiteness.  In many 
cases, one can say things about other sets.

2.  We pick the same set $(a, b)$ for convenience.  In some examples 
later, we will take $(a, b)=\Bbb R$ in (ii), but replace $(a, b)$ by a 
closed set in (i).
\endremark

Here is a spectacular corollary, which we call the Wonderland Theorem:

\proclaim{The Wonderland Theorem}  Let $X$ be a regular metric space of 
operators.  Suppose
\roster
\item"\rom{(a)}" $\{A\mid A\text{\rom{ has purely absolutely continuous 
spectrum}}\}$ is dense in $X$;
\item"\rom{(b)}" $\{A\mid A\text{\rom{ has purely point spectrum}}\}$ 
is dense in $X$.
\endroster
Then Baire typically, $A$ has only singular continuous spectrum.
\endproclaim

\demo{Proof}  Strictly speaking, this is not a corollary of the 
theorem but of its proof, since we do not specify the spectrum.  By 
Theorem 1.1 and (a)
$$
\{A\mid A\text{ has purely continuous spectrum}\}
$$
is Baire typical.  Similarly, by Theorem 1.2 and (b)
$$
\{A\mid A\text{ has purely singular spectrum}\}
$$
is Baire typical.  So their intersection is Baire typical. \qed
\enddemo

\bigpagebreak
\flushpar {\bf \S 3. General Operators}

We apply the theory to general self-adjoint operators first.  Throughout, 
$\Cal H$ is a fixed separable Hilbert space.

\proclaim{Theorem 3.1}  Fix $a>0$.  Let $X=\{A\mid A\text
{\rom{ is self-adjoint}},\, \|A\|\leq a\}$ which is a complete 
metrizable space in the strong topology.  Then
$$
\{A\mid\text{\rom{spec}}(A)=[-a, a]; A\text{\rom{ has purely singular 
continuous spectrum}}\}
$$
is Baire typical.
\endproclaim

\remark{Remark} For example, if $\varphi_n$ is an orthonormal basis
$$
\rho (A, A')=\sum^{\infty}_{n=1} \min (2^{-n}, \|(A-A') \varphi_{n}\|)
$$
is a metric.
\endremark

\demo{Proof}  This will use the Wonderland Theorem.  By the 
Weyl-von Neumann theorem, the operators with point spectrum are norm 
dense, but there is a simpler argument since we only need strong 
density.  Since the same argument is needed for dense absolutely 
continuous spectrum, we give it.

Pick an orthonormal basis $\{\varphi_{n}\}^{\infty}_{n=-\infty}$ (this 
way of counting will be convenient) and let $P_N$ be the projection onto 
$\{\varphi_{n}\}_{|n|\leq N}$ so that $P_{N}\to 1$ strongly.  Let 
$\alpha_n$ be a counting of the rationals in $[-a, a]$ and let $B$ be 
the diagonal operator $B\varphi_{n}=\alpha_{n}\varphi_{n}$.  Then
$$
P_{N}AP_{N}+(1-P_{N})B(1-P_{N})\overset s\to\rightarrow A.
$$
The operator on the left has spectrum $[-a, a]$ and it is pure point.  
So we have two of the three hypotheses of the Wonderland Theorem.

To prove that absolutely continuous spectrum operators are dense, we 
need only prove that an operator $A$ with point spectrum and $\|A\|
\leq a-\epsilon$ can be approximated since we have just proven such 
operators are dense.  Let $\{ \varphi_{n}\}$ be the eigenvectors of $A$ 
(say, $A\varphi_{n}=\alpha_{n}\varphi_{n}$) and let $A_{N}=P_{N}AP_{N}$.  
Fix a sequence $\delta_N$ with $0<\delta_{N}<\frac{\epsilon}{2}$ and 
$\delta_{N}\to 0$.  Let $B_N$ be defined by
$$
B_{N}\varphi_{n}=\delta_{N}(\varphi_{n+(2N+1)}+\varphi_{n-(2N+1})
+\beta_{n}\varphi_n
$$
where $\beta_{n}=\alpha_j$ for the unique $j$ with $n\equiv j$ mod 
$(2N-1)$.  Then $\|B_{N}\|\leq a$ since $\delta_{N}\leq\frac{\epsilon}{2}$ 
and $B_{N}\to A$ strongly as $N\to\infty$.  Each $B_{N}$ is a direct sum 
of $N+1$ operators of the form
$$
\alpha_{n}\Bbb I + \delta_{N}J
$$
where $J$ is the tridiagonal operator with zeros on diagonal and 1 on 
the two principal off diagonals. $J$ has absolutely continuous spectrum 
and thus so does $\alpha_{n}\Bbb I + \delta_{N}J$ and $B_N$. \qed
\enddemo

Surprisingly, the strong topology is only relevant to be sure that the 
spectrum is $[-a, a]$:

\proclaim{Theorem 3.2}  Fix $a<b$.  Let $X=\{A\mid A\text{\rom{ is 
self-adjoint and spec}}(A)=[a, b]\}$ in the operator norm topology.  
\rom($X$ is closed in $\Cal L(\Cal H)$, so complete.\rom)  Then
$$
\{A\mid A\text{\rom{ has purely singular continuous spectrum}}\}
$$
is Baire typical.
\endproclaim

\demo{Proof}  We will use the Wonderland Theorem.  By the
Weyl-von Neumann theorem, given $A\in X$ and $\epsilon$, we can find 
$B_1$ so $\|B_{1}\|<\frac{\epsilon}2$ and $C_{1}\equiv A+B_1$ has pure 
point spectrum.  $C_1$ may have eigenvalues in $(a-\frac{\epsilon}2, a)
\cup (b, b+\frac{\epsilon}2)$ and so not be in $X$, but we can change 
those eigenvalues to $a$ or $b$ with an operator $B_2$ of norm at most 
$\frac{\epsilon}2$.  Then, $C_{2}=A+B_{1}+B_{2}\in X$ has pure point 
spectrum and $\|C_{2}-A\|<\epsilon$.

By the above, we need only show operators in $X$ with pure point spectrum 
can be approximated by operators with purely absolutely continuous spectrum.  
So, suppose $A\in X$ has pure point spectrum.

Let $c=b-a$.  Given $n$, let
$$
I_{1}=\left[a, a+\frac{c}{2^{n}}\right), I_{2}=\left[a+\frac{c}{2^{n}}, 
a+\frac{2c}{2^{n}}\right), \dots, I_{2^n}=\left[b-\frac{c}{2^{n}}, b \right].
$$
Let $\alpha_j$ be the midpoint of $I_j$.  Suppose
$$
A\varphi_{k}=\lambda_{k}\varphi_{k}
$$
is the orthonormal family of eigenvalues for $A$.  Define $B_n$ by
$$
B_{n}\varphi_{k}=\alpha_{j}\varphi_{k} \quad \text{if } \lambda_{k}\in 
I_{j}
$$
so $\|B_{n}-A\|\leq \frac{c}{2^{n+1}}$ and $B_n$ is a direct sum of 
$\alpha_{1}\Bbb I \oplus\cdots\oplus\alpha_{2^{n}}\Bbb I$ with each $\Bbb I$ 
an infinite dimensional identity.  Let $D$ be a self-adjoint operator with 
purely absolutely continuous spectrum on $[-1, 1]$ (e.g., the matrix with 
0 on diagonal and $\frac12$ on the two principal off diagonals).  Let
$$
C_{n}=(\alpha_{1}\Bbb I + \frac{c}{2^{n+1}}D)\oplus\cdots
\oplus(\alpha_{2^n}\Bbb I + \frac{c}{2^{n+1}}D).
$$
Then, $C\in X$, $C$ has purely absolutely continuous spectrum and $\|A-
C_{n}\|<\frac{c}{2^n}$. \qed
\enddemo

\proclaim{Theorem 3.3}  Let $A$ be a fixed self-adjoint operator.  Let 
$I_2$ be the Hilbert-Schmidt operators.  Then for a dense $G_\delta$ of 
$B$ in $I_2$:
\roster
\item"\rom{(1)}" $\text{\rom{spec}}_{\text{\rom{ac}}}(A+B)$ is empty.
\item"\rom{(2)}" $A+B$ has no eigenvalues on $\text{\rom{spec}}_
{\text{\rom{ess}}}(A+B)=\text{\rom{spec}}_{\text{\rom{ess}}}(A)$.
\endroster
\endproclaim

\remark{Remarks}  1.  This is equivalent to Theorem 3 of the 
introduction.

2.  Given Kuroda's extension of the Weyl-von Neumann theorem [11], 
this theorem extends to $I_p$ with $p>1$.  If $A$ has no absolutely 
continuous spectrum, one can take $p=1$.
\endremark

\demo{Proof}  By the Baire category theorem, it suffices to prove the 
set with (i), (ii) separately are given by dense $G_\delta$'s.  By 
Theorem 1.2, the set of operators $B$ with $\text{spec}_{\text{\rom{ac}}}
(A+B)$ empty is a $G_\delta$, and by the Weyl-von Neumann theorem, 
it is dense so (i) yields a dense $G_\delta$.

By Weyl-von Neumann and a simple additional argument, given 
$\epsilon$, we can find $B_0$ with $\|B_{0}\|_{2}<\frac{\epsilon}{2}$ so 
$A_{0}\equiv A+B_0$ has simple pure point spectrum.  Let $\varphi$ be a 
cyclic vector for $A_0$ and let $P_0$ be the projection onto $\{\alpha
\varphi\mid\alpha\in\Bbb C\}$.  By a theorem of [3], $A_{0}+\lambda
P_{0}$ has no eigenvalues in $\text{spec}(A_{0})$ for Baire typical 
$\lambda$ so we can find $|\lambda_{0}|<\frac{\epsilon}2$ so that $A_{0}+
\lambda_{0} P_{0}$ has no eigenvalues on $\text{spec}_{\text{\rom{ess}}}
(A)$.  Take $B=B_{0}+\lambda_{0} P_{0}$ so $\|B\|_{2}<\epsilon$.  This 
proves the density of the set in (ii).  It is a $G_\delta$ by Theorem 1.1. 
\qed
\enddemo

\bigpagebreak
\flushpar {\bf \S 4. Jacobi Matrices and Schr\"odinger Operators}

We will begin with the Jacobi matrix case and prove Theorem 1 of the 
introduction.

\proclaim{Theorem 4.1}  Fix $a>0$. Let $X$ be the set of Jacobi matrices 
on $\ell^{2}(\Bbb Z)$:
$$
Au_{n}=u_{n+1}+u_{n-1}+x_{n}u_{n}
$$
where $x_n$ is an arbitrary sequence with $|x_{n}|\leq a$.  Put the 
topology of pointwise convergence on $\{x_{n}\}$ \rom(so $X$ is a compact 
metrizable space\rom).  Then
$$
\{A\in X\mid\text{\rom{spec}}(A)=[-a-2, a+2],\, \text{\rom{spec}}(A)\text
{\rom{ is purely singular continuous}}\}
$$
is Baire typical.
\endproclaim

\demo{Proof} We use the Wonderland Theorem.  Let $d\mu$ be the product 
of Lebesgue measures $(2a)^{-1}dx_{n}$ so $\text{supp}(d\mu)=[-a, 
a]^{\Bbb Z}$.  Let $D=\{A\in X\mid\text{spec}(A)=[-a-2, a+2],\, 
\text{spec}(A)\text{ is pure point}\}$.  Then $\mu(X\backslash D)=0$ by 
Anderson localization (see, e.g., [14]).  $D$ is dense by the support result.

Given any $x_n$, let
$$\align
x^{j}_{n}&=x_{n} \quad |n|\leq j \\
&=\text{chosen to be periodic of period } 2j+1\ \quad \text{if } n>|j|.
\endalign
$$
Thus, $x^{(j)}\to x$ and the Jacobi matrix associated to $x^{(j)}$ has 
purely absolutely continuous spectrum. \qed
\enddemo

\remark{Remark}  We do not need the full proof of Anderson localization; 
it suffices that the Jacobi matrices associated to Lebesgue typical 
sequences have no a.c.~ spectrum and this is easier to prove.

For random Jacobi matrices in higher dimension, it is believed that there 
is sometimes a.c.~spectrum, but that is not so for the generic matrix.  
Let $\Bbb Z^\nu$ have the norms $|n|=\sum\limits^{\nu}_{j=1}|n_{j}|$ 
and $\|n\|=\sup\limits_{j}|n_{j}|$.
\endremark

\proclaim{Theorem 4.2} Fix $a>0$. Let $X$ be the set of Jacobi matrices 
on $\ell^{2}(\Bbb Z^{\nu})$
$$
Au_{n}=\sum_{|j|=1}u_{n+j}+x_{n}u_n
$$
where $x$ is an arbitrary multisequence with $|x_{n}|\leq n$. Put the 
topology of pointwise convergence on $\{x_{n}\}$.  Then
$$
\{A\in X\mid\text{\rom{spec}}(A)=[-a-2\nu, a+2\nu];\, \text{\rom{spec}}
(A)\text{\rom{ is purely singular continuous}}\}
$$
is Baire typical.
\endproclaim

We need a lemma which shows how ``loose'' generic really is:

\proclaim{Lemma 4.3} In the setup of Theorem {\rom{4.2}}, suppose that 
there is a single operator $A_{0}\in X$, $\text{\rom{spec}}_{\text{\rom
{ac}}}(A_{0})=\emptyset$.  Then, $\text{\rom{spec}}_{\text{\rom{ac}}}
(A)=\emptyset$ for a dense set of $A$ in $X$.
\endproclaim

\demo{Proof} Let $x^{(0)}_{n}$ be the multisequence defining $A_0$.  Given 
$B\in X$ with multisequence $x_n$, define $A_j$ by the multisequence 
$x^{(j)}_n$ where
$$\alignat2
x^{(j)}_{n}&=x_{n} &&\quad |n|\leq j \\
&=x^{(0)}_{n}&&\quad |n|> j.
\endalignat
$$
Since $x^{(j)}_{n}\to x_{n}$ pointwise, $A_{j}\to B$.  But $A_{j}\to A_0$ 
is finite rank, so $\text{spec}_{\text{\rom{ac}}}(A_{j})=\text{spec}_
{\text{\rom{ac}}}(A_{0})=\emptyset$. \qed
\enddemo

\demo{Proof of Theorem \rom{4.2}} We use the Wonderland Theorem. For any 
rational $q\in [-a, a]$, the set of potentials $x_n$ equal to $q$ if 
$|n|\geq j$ for some $j$ is dense. Such a potential yields an operator 
$A$ with $[q-2\nu, q+2\nu]\in\text{spec}(A)$, so generically 
$\operatornamewithlimits{\cup}\limits_{q} [q-2\nu, q+2\nu]$ is in ]
$\text{spec}(A)$.

As in the proof of Theorem 4.1, the periodic multisequences are dense and 
each yields an operator $A$ with no point spectrum, so the operators 
with no point spectrum are dense.

By the lemma, we need only find the operator $A$ in our space with no 
a.c.~spectrum.  Let $\{y_{i}\}_{i\in\Bbb Z}$ be a specific sequence in 
$[\frac{-a}\nu, \frac{a}\nu]^{\Bbb Z}$ whose one-dimensional Jacobi 
matrix $J_0$ has only dense point spectrum in $[-2-\frac{a}\nu, 2+
\frac{a}\nu]$.  Let
$$
x_{n}=y_{n_1}+\cdots + y_{n_\nu}
$$ 
so the corresponding $A$ has the form
$$
J\otimes 1\otimes\cdots\otimes 1+1\otimes J\otimes\cdots\otimes 
1+\cdots +1\otimes 1\otimes\cdots\otimes J
$$
in $\ell^{2}(\Bbb Z^{\nu})=\ell^{2}(\Bbb Z)\otimes\ell^{2}(\Bbb 
Z)\otimes\cdots\otimes\ell^{2}(\Bbb Z)$.  Then $\text{spec}(A)$ is 
also pure point. \qed
\enddemo

\proclaim{Theorem 4.4}  Let $c_0$ be the sequences $\{x_{n}\}_{n\in
\Bbb Z}$ with $|x_{n}|\to 0$.  For $x\in\ell^p$ or in $c_0$, let 
$J(x)$ be the corresponding Jacobi matrix on $\ell^{2}(\Bbb Z)$.  Then 
$\text{\rom{spec}}_{\text{\rom{ess}}}(J(x))=[-2, 2]$ and
$$
\{x\mid (x)\text{\rom{ has purely singular continuous spectrum on }}
[-2, 2]\}
$$
is Baire typical in $c_0$ and in each $\ell^{p}$ $(p>2)$ when these 
spaces are given the norm topology.
\endproclaim

\demo{Proof} Since $x_{n}\to 0$ at $\pm\infty$, the diagonal matrix is 
compact and $\text{spec}_{\text{\rom{ess}}}(J(x))=\text{spec}_{\text
{\rom{ess}}}(J(x=0))=[-2, 2]$. Thus, it suffices to find dense sets with 
no point spectrum in $[-2, 2]$ and with no a.c.~spectrum in $[-2, 2]$.  If 
$x$ has finite support, then any solution of $J(x)u=\lambda u$ with 
$\lambda\in (-2, 2)$ must be a plane wave outside a finite set and so is 
not in $\ell^2$. Since the sequences, $x$, of compact support are dense, 
we have the required density of operators without point spectrum.

As in the proof of the last theorem, we need only find one $x$ in our 
space with no a.c.~spectrum.  In [15], Simon showed that if $a_n$ is a 
typical random sequence, independent and uniformly distributed in 
$[-1, 1]$, then $x_{n}=(|n|+1)^{-\beta}a_n$ yields a $J(x)$ with pure 
point spectrum so long as $\beta<\frac12$.  This yields the required 
examples in $\ell^p$ or $c_0$. \qed
\enddemo

\remark{Remarks} 1. One could instead look at sequences $x_n$ with 
$\sup\left|(1+|n|)^{\beta}x_{n}\right|<\infty$ in the obvious norm and 
get the result so long as $\beta<\frac12$.

2. For $p=1$, or $\beta >1$ (in the language of Remark 1), $J(x)$ has 
lots of a.c.~spectrum, so the result requires some slow falloff 
hypothesis.  It is likely the result remains true for 
$1<\beta\leq\frac12$ and $1<p\leq 2$ but it is open.

3.  We are unable to extend this result to the higher dimensional 
$(\Bbb Z^{\nu})$ case because neither the method used in Theorem 4.2 
(taking $x_{n}=y_{n_1}+\cdots +y_{n_\nu}$) or Theorem 4.5 (spherical 
symmetry) works.
\endremark

We turn next to Schr\"odinger operators. We will begin with the case 
where $V\to 0$ at infinity.

\proclaim{Theorem 4.5}  Let $\Cal C_{\infty}(\Bbb R^{\nu})$ be the 
continuous function of $\Bbb R^\nu$\, which vanish at infinity in the 
uniform norm.  Then for a Baire typical set of $V$, $-\Delta +V$ has 
purely singular continuous spectrum on all of $(0, \infty)$.
\endproclaim

\demo{Proof} By general principles, (see, e.g., [13]), $\text{spec}
_{\text{\rom{ess}}}(-\Delta+V)=[0, \infty)$ so we need only show that 
for a dense set $\text{spec}_{\text{\rom{ac}}}(-\Delta+V)=\emptyset$ 
and for another dense set, $\text{spec}_{\text{\rom{pp}}}(-\Delta+V)
\subset (-\infty, 0]$.

If $V$ has compact support, it is well known [13] that $\text{spec}
_{\text{\rom{pp}}}(-\Delta+V)\subset (-\infty, 0]$, so we have that 
required dense set.

Suppose we find one $V\in\Cal C_{\infty}(\Bbb R^{\nu})$ with $\text{spec}
_{\text{\rom{ac}}}(-\Delta+V)=\emptyset$.  Suppose $W$ is another potential 
with $W(x)=V(x)$ for $|x|>R$ for some $R$.  Then $\text{spec}_{\text
{\rom{ac}}}(-\Delta+W)=\emptyset$ by using Dirichlet decoupling as in 
Deift-Simon [2].  Any $W_{0}\in\Cal C_{\infty}(\Bbb R^{\nu})$ is a limit 
of functions equal to $V$ outside of some ball, so we get the required 
density.  Thus we need only find one $V$\!.

To find the required $V$, we choose $V$ spherically symmetric and given 
by a typical potential in the analysis of Kotani-Ushiroya [10].  These go 
to zero at infinity and are known to have $\text{spec}(-\frac{d^2}{dx^2}+
V(r))$ pure point.  Each partial wave Hamiltonian $-\frac{d^2}{dr^2} +
\frac{c}{r^2}+V(r)$ also has no a.c.~spectrum by trace class theory, so 
$-\Delta+V$ has no a.c.~spectrum. \qed
\enddemo

\remark{Remark}  By looking carefully at [10], the result extends to 
$L^{p}(\Bbb R^{\nu})$, $p>2n$.
\endremark

Here is a typical example for random Schr\"odinger operators.

\proclaim{Theorem 4.6} For $v\in [-a, a]^{\Bbb Z^{\nu}}$, define $V$ 
on $\Bbb R^\nu$ by
$$
V(x)=v(i(x))
$$
where $i(x)$ is defined by
$$
i(x)=j \quad \text{\rom{if }} j_{\alpha}\leq x_{\alpha}<j_{\alpha}+1.
$$
Then for a Baire typical $v$, $-\Delta+V$ has spectrum $[-a, \infty)$ and 
is purely singular continuous there.
\endproclaim

\demo{Proof} By using periodic $v$, we see that Baire typically $V$ has 
no point spectrum.  As in the last theorem, we need only find a single $v$ 
with no a.c.~spectrum.  Take $v(i)=\tilde{v}_{i_1}+\cdots +\tilde{v}_{i_n}$ 
for a one-dimensional $\tilde v$.  If $-\frac{d^2}{dx^2}+\tilde{V}$ has 
point spectrum, so does $-\Delta +V$.  Thus localization in the 
one-dimensional case [5,9] completes the proof. \qed
\enddemo

Finally, we want to say something about the almost periodic case with 
a series of remarks.

1.  Consider the almost Mathieu equation, the Jacobi matrix with $v, 
\lambda\cos(\pi\alpha n+\theta)$ for $\lambda, \theta$ fixed.  For 
$\alpha$ rational, the potential is periodic and there is no point 
spectrum. It follows that for Baire typical $\alpha$, there is no 
point spectrum either.  This is a soft version of Gordon's theorem 
(Gordon [6], Avron-Simon [1]).

2.  Fix $\lambda, \alpha$ in the almost Mathieu equation with $\alpha$ 
irrational.  Suppose that there is a single $\theta_0$ leading to 
purely s.c.~spectrum.  Then its translates are dense and so Baire 
typically, there will be only s.c.~spectrum.  It may well happen that 
for $\alpha$ with good Diophantine properties and $\lambda >2$, we have 
pure point spectrum for Lebesgue typical $\theta$ and purely 
s.c.~spectrum for Baire typical $\theta$.

3.  The argument in Remark 1 applies to generic potentials, $v$, in spaces of 
limit periodic potentials.

\vskip 0.5in
\Refs
\widestnumber\key{20}
\ref\key 1
\by J.~Avron and B.~Simon \paper Singular continuous spectrum 
for a class of almost periodic Jacobi matrices \jour Bull.~AMS 
\vol 6 \yr 1982 \pages 81--85
\endref
\ref\key 2
\by P.~Deift and B.~Simon \paper On the decoupling of the finite 
singularities from the question of asymptotic completeness in two-body 
quantum systems \jour J.~Func.~Anal. \vol 23 \yr 1976 \pages 218--238
\endref
\ref\key 3
\by R.~del Rio, N.~Makarov, and B.~Simon \paper Operators with 
singular continuous spectrum, II.~Rank one perturbations 
\jour Commun.~Math.~Phys. \toappear
\endref                                   
\ref\key 4
\by F.~Delyon, H.~Kunz, and B.~Souillard \paper One-dimensional 
wave equations in disordered media \jour J.~Phys.~A \vol 16 
\yr 1983 \pages 25--42
\endref
\ref\key 5
\by I.~Goldsheid, S.~Molchanov, and L.~Pastur \paper A pure 
point spectrum of the stochastic one-dimensional Schr\"odinger 
equation \jour Func.~Anal.~Appl. \vol 11 \yr 1977 \pages 1--10
\endref
\ref\key 6
\by A.~Gordon, \jour Usp.~Mat.~Nauk \vol 31 \yr 1976 \page 257
\endref
\ref\key 7
\by P.~Halmos \paper In general a measure preserving transformation 
is mixing \jour Ann.~Math. \vol 45 \yr 1944 \pages 786--792
\endref
\ref\key 8
\by T.~Kato \book Perturbation Theory for Linear Operators
\bookinfo 2nd ed. \publ Springer-Verlag \yr 1980
\endref
\ref\key 9
\by S.~Kotani and B.~Simon \paper Stochastic Schr\"odinger 
operators and Jacobi matrices on the strip \jour Commun.~Math.~Phys. 
\vol 119 \yr 1988 \pages 403--429
\endref
\ref\key 10
\by S.~Kotani and N.~Ushiroya \paper One-dimensional Schr\"odinger 
operators with random decaying potentials \jour Commun.~Math.~Phys. 
\vol 115 \yr 1988 \pages 247--266
\endref
\ref\key 11
\by S.~Kuroda \paper On a theorem of Weyl-von Neumann, 
\jour Proc.~Japan Acad. \vol 34 \yr 1958 \pages 11--15
\endref
\ref\key 12
\by M.~Reed and B.~Simon \book Methods of Modern Mathematical 
Physics, I.~ Functional Analysis \bookinfo rev.~ed. 
\publ Academic Press \yr 1980
\endref
\ref\key 13
\bysame \book Methods of Modern Mathematical Physics, IV.~Analysis 
of Operators \publ Academic Press \yr 1978
\endref
\ref\key 14
\by V.~Rohlin \paper A ``general'' measure-preserving transformation 
is not mixing \jour Dokl.~Akad. Nauk SSSR (N.S.) \vol 60 
\yr 1948 \pages 349--351
\endref
\ref\key 15
\by B.~Simon \paper Some Jacobi matrices with decaying potential and 
dense point spectrum \jour Commun.~Math.~Phys. \vol 87 \yr 1982 
\pages 253--258
\endref
\ref\key 16 
\bysame \paper Localization in general one-dimensional random systems, 
I.~Jacobi matrices \jour Commun.~Math.~Phys. \vol 102 \yr 1985 
\pages 327--336
\endref
\ref\key 17              
\by B.~Simon and T.~Wolff \paper Singular continuous spectrum under 
rank one perturbations and localization for random Hamiltonians, 
\jour Commun.~Pure Appl.~Math. \vol 39 \yr 1986 \pages 75--90
\endref
\ref\key 18
\by J.~von Neumann  \book Charakterisierung des Spektrums eines 
Integraloperators \bookinfo Actualit\'e Sci. Indust.~No.~229 
\publ Hermann \publaddr Paris \yr 1935
\endref
\ref\key 19
\by H.~Weyl \paper\"Uber beschr\"ankte quadratische Formen, 
deren Differenz vollstetig ist \jour Rend.~Circ. Mat.~Palermo 
\vol 27 \yr 1909 \pages 373--392
\endref
\ref\key 20
\by T.~Zamfirescu \paper Most monotone functions are singular
\jour Amer.~Math.~Monthly \vol 88 \yr 1981 \pages 47--49
\endref
\endRefs

\enddocument