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\topmatter
\title Singular Continuous Spectrum Is Generic
\endtitle
\author R. del Rio$^{1,3,5}$, S. Jitomirskaya$^{2}$, N. 
Makarov$^{3,4}$, and B. Simon$^{3,5}$
\endauthor
\leftheadtext {del Rio, Jitomirskaya, Makarov, and Simon}
\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
\thanks $^{2}$ Department of Mathematics, University of California,
Irvine, CA 92717.
\endthanks
\thanks $^{3}$ Division of Physics, Mathematics and Astronomy, 253-37, 
California Institute of Technology, Pasadena, CA  91125.
\endthanks
\thanks $^{4}$  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
\thanks $^{5}$ 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 {\it To be submitted to Bull.~Amer.~Math.~Soc.}
\endthanks
\abstract In a variety of contexts, we prove that singular continuous 
spectrum is generic in the sense that for certain natural complete metric 
spaces of operators, those with singular spectrum are a dense $G_\delta$.
\endabstract
\endtopmatter

\bigpagebreak

In the spectral analysis of various operators of mathematical physics, 
a key step, often the hardest, is to prove that the operator has no 
continuous singular spectrum, that is, that the spectral measures for 
the operators have only pure point and absolutely continuous parts.  
Examples are the absence of such spectrum for $N$-body Schr\"odinger 
operators [3,19] and for the one-dimensional random models 
[12,7,8,24,18].

Our goal here is to announce results that show that singular 
continuous spectrum is lying quite close to many operators by proving 
it is often generic in Baire sense.  Detailed proofs and further 
results will appear in three papers: one for general operators [22], 
one for rank one perturbations [6], and one for almost periodic 
Schr\"odinger operators [23].

Precursors of our results include work on generic ergodic processes 
[15,21], on special energies for Sch\"rodinger operators/Jacobi matrices 
[11,4,5].  Gordon [13,14] has independently (and presumably, before 
us) proven Theorem 5.  His method of proof is very different from ours.

Recall that the Baire category theorem implies that if $X$ is a 
complete metric space, a countable intersection of dense $G_\delta$ 
is still a dense $G_\delta$ and if $X$ is perfect, then any dense 
$G_\delta$ has uncountable intersection with any open ball.

Our first two results are for one-body Schr\"odinger operators and for 
the ``generic Anderson model.''

\proclaim{Theorem 1 ([22])}  Let $C_{\infty}(\Bbb R^{\nu})$ denote the 
continuous functions on $\Bbb R^{\nu}$ vanishing at infinity in the $\|\cdot 
\|_{\infty}$ norm.  Then for a dense $G_\delta$ of $V\in C_{\infty}
(\Bbb R^{\nu})$, $-\Delta+V$ has purely singular continuous spectrum on 
$(0, \infty)$.
\endproclaim

\remark{Remarks}  1.  If $V(x)=0(|x|^{-1-\epsilon})$ at infinity, it is 
known [1,20] that $-\Delta+V$ has absolutely continuous spectrum on 
$(0, \infty)$ with a possible set of eigenvalues.

2.  There is a similar result ([22]) for $\{V\mid (1+x^{2})^{\alpha/2} 
V\in C(\Bbb R^{\nu})\}$ with norm $|||V|||=\|(1+x^{2})^{\alpha/2} 
V\|_{\infty}$ so long as $\alpha <\frac{1}{2}$.

3.  In one dimension, there is a similar result for Jacobi matrices [22].
\endremark

\proclaim{Theorem 2 ([22])}  Let $a<b$ be fixed and let $\Omega=
[a, b]^{\Bbb Z^{\nu}}$, functions $v:\Bbb Z^{\nu}\to [a,b]$ with the (compact 
metrizable) Tychonoff topology.  Given any $v$, let $h(v)$ be the Jacobi 
matrix on $l^2 (\Bbb Z^{\nu})$ by 
$$
(h(v)u)(n)=\sum_{|j|=1}u(n+j)+v(n)u(n). \tag 1
$$
Then for a dense $G_\delta$ in $\Omega$, $h(u)$ has spectrum $[a-2\nu, 
b+2\nu]$ and the spectrum is purely singular continuous.
\endproclaim

\remark{Remark}  If $\nu=1$ or if $\nu$ is arbitrary and $b-a$ is large, 
it is known that if $\Omega$ is given the product measure 
$\operatornamewithlimits{X}\limits_{j\in\Bbb Z^{\nu}} (b-a)^{-1}\, dx_{j}$, 
then for a.e.~$v$, $h(v)$ has only pure point spectrum [7,9,8,24,2].  
So the generic Baire and generic Lebesgue behavior are very different.
\endremark

These results are not limited to Schr\"odinger operators:

\proclaim{Theorem 3 ([22)]}  Let $X_a$ be the family of all self-adjoint 
operators, $A$, on a fixed separable Hilbert space, $\Cal H$ with 
$\|A\|\leq a$.  Give $X_a$ the metrizable topology of strong 
convergence.  Then $X_a$ has a dense $G_\delta$ of operators with 
$\text{spec}(A)=[-a, a]$ and the spectrum purely singular continuous.
\endproclaim

\proclaim{Theorem 4 ([22])}  Let $A$ be a fixed self-adjoint operator.  
Let $\Cal I_2$ be the Hilbert-Schmidt operators with Hilbert-Schmidt 
norm.  Then for a dense $G_\delta$ of $C$'s in $\Cal I_2$, the set of 
vectors
$$\multline
\{\psi\mid d\mu^{\psi}_{(A+C)}\text{ is purely singular continuous}\} \\
\cup\{\psi\mid (A+C)\psi=E\psi; E\in\text{\rom{spec}}_{\text{disc}}(A+C)\}
\endmultline
$$
span $\Cal H$.
\endproclaim   

\remark{Remarks}  1.  $d\mu_{D}^\psi$ is the spectral measure 
$(\psi, e^{isD}\psi)=\int e^{isE}\,d\mu_{D}^{\psi}(E)$.

2.  If $\text{spec}(A)$ is thin, for example, $A\equiv 0$, the vectors are 
the discrete eigenvectors.  But if $\text{spec}(A)$ contains an interval, 
$\text{spec}(A+C)$ will have lots of singular continuous spectrum.

3.  This is to be distinguished from the Weyl-von Neumann theorem 
[26,27,17] that there are $C$'s with $\|C\|_{2}$ arbitrarily small so that 
$A+C$ has only point spectrum.  Here we see that generically there 
will be singular continuous spectrum (for $A$ suitable).

4.  The result holds if $\Cal I_2$ is replaced by $\Cal I_p$ with 
$p>1$.  If $A$ has no a.c.~spectrum, it even holds for $\Cal I_1$.
\endremark

These four theorems are rather soft with no hard estimates.  More 
subtle is the case of rank one perturbations.  We'll consider two 
closely related cases:
\roster
\item"{(a)}"  $A$ is a self-adjoint operator with cyclic vector $\varphi$; let 
$P$ be the projection onto $\varphi$ and let $A_\lambda =A+\lambda P$.
\item"{(b)}"  Let $H$ be the differential operator $-\frac{d^2}{dx^2}+V(x)$ on 
$[0, \infty)$ assumed to be limit point at infinity.  $H_\theta$ is the 
self-adjoint operator with boundary condition $\cos\theta u(0)+\sin\theta 
u'(0)=0$.
\endroster

\proclaim{Theorem 5 ([6])}  {\rm (a)}  Suppose $A$ has an interval $[a, b]$ 
in its spectrum and the spectrum there has no a.c.~component.  Then
\roster
\item"\rom{(i)}"  There is a dense $G_\delta$, $C$, in $[a, b]$, so that if
$E\in C$, then $E$ is not an eigenvalue of any $A_\lambda$.
\item"\rom{(ii)}"  For a dense $G_\delta$, $L$, of $\Bbb R$, $A_\lambda$ has 
purely singular spectrum in $[a, b]$ if $\lambda\in L$.
\endroster
{\rm (b)}  Suppose for some $\theta _0$, $H_{\theta_0}$ has an interval 
$[a, b]$ in its spectrum and the spectrum there has no a.c.~component, then
\roster
\item"\rom{(i)}"  There is a dense $G_\delta$, $C$, in $[a, b]$, so that if
$E\in C$, then $E$ is not an eigenvalue of any $H_\theta$.
\item"\rom{(ii)}"  For a dense $G_\delta$, $L$, of $[0, \pi)$, 
$H_\theta$ has purely singular spectrum in $[a, b]$ if $\theta\in L$.
\endroster
\endproclaim

\remark{Remarks}  1.  Case (i) implies that under the hypothesis of $E\in C$, 
either $\lim\limits _{x\to\infty}\frac{1}{x}\mathbreak \ln\|T_{E}(x)\|$ fails 
to exist or is $0$ when $T(E)$ is the fundamental matrix for the problem.  
This means that for many cases where one can only prove Lyapunov behavior for 
a.e.~$E$, there really is a set where the Lyapunov behavior fails [11,4,5].

2.  There are also results for general $A$ without any hypothesis on 
$\text{spec}(A)$ or absolute continuous spectrum.

3.  These results imply that in the Anderson model in the localized 
regime, varying $V(0)$ a little can produce singular spectrum.  Indeed, there 
are disjoint, locally uncountable sets with purely pure point spectrum when 
$V(0)$ is in one set and pure singular continuous spectrum when $V(0)$ is 
in the other set!
\endremark

Another subtle class are almost periodic Schr\"odinger operators.  
We'll consider functions $V$ on $\Bbb R$ or $\Bbb Z$ that are even and almost 
periodic (typical examples are $V(n)=\lambda\cos(\pi\alpha n)$ in the 
$\Bbb Z$ case and $V(x)=\lambda\cos(\pi x)+\mu\cos(\pi \alpha x)$ in 
the $\Bbb R$ case with $\alpha$ irrational) and define
$$\alignat2
H_\omega &=-\frac{d^2}{dx^2}+V_{\omega}(x) &&\qquad\Bbb R\text{ case} \\
({H_\omega}u)(n) &=u(n+1)+u(n-1)+V_{\omega}(n)u(n) &&\qquad\Bbb Z\text{ case}
\endalignat
$$
where $\omega$ is a point in the hull, $\Omega$, of $V$ and $V_\omega$ the 
corresponding potential (in the typical cases above, $\Omega=S^1$ and 
$S^{1}\times S^1$ with $V_{\theta}(n)=\lambda\cos(\pi\alpha n+\theta)$ and 
$V_{\theta,\psi}(x)=\lambda\cos(\pi x+\theta)+\mu\cos(\pi\alpha x+\psi)$).  
$\Omega$ is a compact metric space in the Bohr topology.

\proclaim{Theorem 6 ([23])}  Let $V$ be an even almost periodic 
potential on $\Bbb R$ or $\Bbb Z$.  Then
\roster
\item"\rom{(a)}"  For a dense $G_\delta$ in the hull, $H_\omega$ has no point 
spectrum.
\item"\rom{(b)}"  If for some point $\omega_0$ in the hull, $H_{\omega_{0}}$ 
has no a.c.~spectrum, then for a dense $G_\delta$ in the hull, $H_\omega$ 
has purely singular continuous spectrum.
\endroster
\endproclaim

\example{Example ([23])}  In the $\Bbb Z$ case, if $V=\lambda\cos 
(\pi\alpha n+\theta)$ with $\lambda\geq 2$, and $\alpha$ irrational, then 
it follows that $H_\theta$ has purely singular spectrum for a dense 
$G_\delta$ of $\theta$.  When $\lambda$ is large [25,10,16], it is known 
that we have pure point spectrum only for a set of $\theta$ of full 
Lebesgue measure.  Once again, we have locally uncountable sets of 
parameters with point spectrum for one parameter set and singular continuous 
in the other.
\endexample

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