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\begin{document}
\title{The absolutely continuous spectrum of one-dimensional
Schr\"odinger operators with decaying potentials}
\author{Michael Christ\thanks{Department of Mathematics, 
University of California, Berkeley CA 94720, USA},
Alexander Kiselev\thanks{Mathematical Sciences Research Institute,
Berkeley CA 94720, USA, E-mail: akiselev@cco.caltech.edu},
and Christian Remling\thanks{Universit\"at Osnabr\"uck,
Fachbereich Mathematik/Informatik,
49069 Osnabr\"uck, GERMANY,
E-mail: cremling@mathematik.uni-osnabrueck.de}}
\maketitle
\noindent
1991 AMS Subject Classification: 34L40, 81Q10

\vspace{0.5cm}
In this announcement, we are interested in the spectral theory
of one-dimensional Schr\"odinger operators
\begin{equation}
\label{so}
H=-\frac{d^2}{dx^2}+V(x),
\end{equation}
acting on the Hilbert space $L_2(0,\infty)$. One also
needs a boundary condition at $x=0$ in order to obtain self-adjoint
operators. The operator (\ref{so}) describes the motion of a quantum
mechanical particle, and the spectral properties of $H$
are intimately connected to the physics of this system
(see, e.g., \cite{RS3}).

We will view $V$ as a perturbation of the free Hamiltonian
$H_0=-d^2/dx^2$. It is natural to expect that suitable
smallness assumptions on $V$ guarantee stability of the
absolutely continuous part of $H_0$. This problem
is one of the basic questions 
in quantum 
mechanics, and it has been studied extensively. It has been known for 
a long time that the spectrum of $H$ is purely absolutely 
continuous 
on $(0,\infty)$ if $V \in L_{1}.$ More information is available
for potentials satisfying certain additional assumptions.
We mention the classical result of Weidmann \cite{Weid} on potentials of 
bounded variation, 
a series of works on oscillating potentials of the type
$\frac{\sin x^{\alpha}}{x^{\beta}}$
\cite{Ben,BA,HiS,Mat,White}, and works on 
potentials satisfying additional conditions on the
derivatives, see, e.g., \cite{As,Bus,Hor}.
However, in all these results, the potential is required to
have further properties, in addition to decaying sufficiently
rapidly.
In fact, a result going back to von Neumann and Wigner says that
potentials
$V(x)=O(1/x)$ can already have positive eigenvalues
\cite{vNW}, and one can construct potentials with decay
arbitrarily close to $O(1/x)$ and dense point spectrum
in $(0,\infty)$ \cite{Na,Spp}. In other words, decay conditions
that are essentially weaker than $V\in L_1$ do not imply
{\it purely} absolutely continuous spectrum on $(0,\infty)$.

However, as was first noticed by one of us \cite{Kis1,Kis2},
it is still true that the absolutely continuous spectrum
is preserved if $V(x)=O(x^{-\alpha})\: (\alpha>2/3)$
(see \cite{Kis2}), although, according to the above
remarks, embedded singular spectrum can occur. Subsequently,
Molchanov presented an alternate proof of the same result
\cite{Mop}.

Our first
result is a sharp version of this theorem. Recall that
$S$ is called an essential support of the measure $\mu$
if $\mu({\Bbb R}\setminus S)=0$ and $\mu(T)>0$ for every
subset $T\subset S$ of positive Lebesgue measure. 
\begin{Theorem}[\cite{CK,Rem}]
\label{T1}
Suppose $|V(x)|\le C(1+x)^{-\alpha}$ with $\alpha>1/2$.
Then $\Sigma_{ac}=(0,\infty)$ is an essential support
of the absolutely continuous part of the spectral measure.
Moreover, for almost every $E>0$, one can find solutions
to the Schr\"odinger equation $Hy=Ey$ with WKB asymptotic behavior:
\[
y_{\pm}(x,E)=\exp\pm \left(i\sqrt{E}x-
\frac{i}{2\sqrt{E}}\int_0^x V(t)\, dt \right)(1+o(1))
\quad (x\to\infty).
\]
\end{Theorem}
This result is optimal, because the work on decaying
random potentials \cite{Del,Deletal,KLS,KU,S}
has shown that there are potentials $|V(x)|\le C(1+x)^{-1/2}$,
such that the corresponding Schr\"odinger operator has
purely singular spectrum.

We have independently found two different proofs of
Theorem \ref{T1}. These proofs will be given in two
separate publications \cite{CK,Rem}. The approach
of \cite{CK} is based on ideas developed in \cite{Kis1,Kis2}
and, in particular, on new norm estimates for certain
multilinear transformations which may be
of independent interest. The method of \cite{Rem} uses
ideas from both proofs of the $2/3$ result \cite{Kis2,Mop}.

Actually, our methods also yield a number of extensions
and generalizations of Theorem \ref{T1}. For example,
we do not really need a pointwise bound on $V$; we prove
the result under considerably more general assumptions
which allow, among other things, local singularities of
$V$. However, we do need a certain amount of regularity
in the decay of $V$; we are as yet unable to treat general
$L_p$ potentials (see also the open questions below).
We refer the reader to \cite{CK,Rem} for details.

We also obtain a general criterion for the stability
of the absolutely continuous spectrum of perturbed
Schr\"odinger operators. We now consider the following
situation: Given a Schr\"odinger operator $H_0=
-d^2/dx^2+U$ with absolutely continuous spectrum on
some set $S$, we ask under what conditions this spectrum
is stable under perturbations by $V$.
Again, we give the result
in the simplest form.
\begin{Hypothesis}
\label{H1}
Assume that for all $E\in S$, the Schr\"odinger equation
\begin{equation}
\label{se}
-y''+Uy=Ey
\end{equation}
has only bounded solutions.
Assume further that one can choose a solution $\theta(\cdot,E)
\:(E\in S)$ of (\ref{se}), such that the operator
$K: L_2((0,\infty),dx)\to L_2(S,dE)$, defined for
bounded functions $f$ of compact support by
\begin{equation}
\label{op}
(Kf)(E)= \int_0^{\infty} \theta(x,E)^2
\exp \left( \frac{i}{\mbox{{\rm Im }}\theta\overline{\theta}'}
\int_0^x V(t)|\theta(t,E)|^2\, dt \right)
f(x)\, dx,
\end{equation}
is norm bounded.
\end{Hypothesis}
Note that Hypothesis \ref{H1} in particular implies that
$\Sigma_{ac}(H_0)\supset S$ \cite{Sbdd,St}. The quantity
$\mbox{{\rm Im }}\theta\overline{\theta}'$ is independent
of $x$, since it is a multiple of the Wronskian of the
two solutions $\theta, \overline{\theta}$.
Moreover, it is non-zero precisely if $\theta$
and $\overline{\theta}$ are linearly independent.
We have the following result, first obtained by methods
of \cite{CK} (it can also be shown by methods of \cite{Rem}):
\begin{Theorem}
\label{T2}
Suppose that Hypothesis \ref{H1} holds. If $|V(x)|\le
C(1+x)^{-\alpha}$ with $\alpha>1/2$, then
$\Sigma_{ac}(H_0+V)\supset S$. Moreover, for almost
every $E\in S$, one can find solutions $y,\overline{y}$ to the
Schr\"odinger equation $(H_0+V)y=Ey$ with WKB type
asymptotic behavior:
\[
y(x,E)=\theta(x,E) \exp \left(
\frac{i}{2\, \mbox{{\rm Im }}\theta\overline{\theta}'}
\int_0^x V(t)|\theta(t,E)|^2\, dt \right)(1+o(1))
\quad (x\to\infty).
\]
\end{Theorem}

Hypothesis \ref{H1} can be
verified for $U=0$ and for periodic $U$ (see \cite{Kis2}).
Given this, it is clear that, in particular, Theorem \ref{T1}
follows from Theorem \ref{T2}.

We also have a result on decay conditions which
imply {\it purely} absolutely continuous spectrum
on $(0,\infty)$. 
This result improves the elementary
remark on $L_1$ potentials (on the power scale). 
This problem was brought to our attention
by S.~Molchanov, who has independently obtained
related (but weaker)
results using different methods \cite{Mop}.
\begin{Theorem}[\cite{Rem}]
\label{T3}
If  $C:=\limsup_{x\to\infty} x\, |V(x)|<\infty$,
then $H_{\alpha}$ is purely
absolutely continuous on $((2C/\pi)^2,\infty)$. In particular,
if $V(x)=o(1/x)$, then $H_{\alpha}$ is purely absolutely
continuous on $(0,\infty)$.
\end{Theorem}
The point of this Theorem is the absence of singular {\it
continuous} spectrum. That $E=(2C/\pi)^2$ is a (sharp)
bound on possible embedded
eigenvalues appears already in \cite[Section 3.2]{EK}.
See also \cite[Theorem 4.1]{KLS} for further information
on embedded eigenvalues.

We would like to conclude this paper with some open questions.
We think that these questions are interesting, but they
also look rather difficult at present.

1. Does Theorem \ref{T1} still hold under the assumption
$V\in L_2$ (or $V\in L_p$ for some $p<2$)? Currently, we can 
show 
that if $x^{\epsilon}V \in L_p$ for some $\epsilon>0$ and 
$p \leq 2,$ Theorem \ref{T1} holds. Still extending this 
result to $L_p$ seems hard.

2. Are there potentials $V(x)=O(x^{-\alpha}), \alpha>1/2$ with
embedded singular {\it continuous} spectrum? We expect that the
answer is yes. In this case, it would be interesting to
construct such potentials.

We would also like to point out that in recently constructed
examples with embedded singular continuous spectrum \cite{Mo,Remsc},
the essential support of the absolutely continuous part $\Sigma_{ac}$
does not have full measure in the absolutely continuous spectrum
$\sigma_{ac}$ (which is the essential closure of the set $\Sigma_{ac}$).

3. Formulate general conditions on $U$ which imply boundedness
of the integral operator from (\ref{op}).

These problems will be the subject of continuing research.

{\bf Acknowledgments:} We would like to thank S.\ Molchanov
for showing us his proof \cite{Mop} of the result of \cite{Kis2} 
and for stimulating discussions,
and B.\ Simon for useful advice. M.C.'s work has been supported
in part by NSF grant DMS96-23007.
A.K's work
at the MSRI was supported in part by NSF grant DMS 9022140.
C.R.\ would like to thank the
hospitality of Caltech, where most of this work
was done. He would also like to thank the
Deutsche Forschungsgemeinschaft for financial support.
 
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\end{document}
