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\begin{document}
\title[Schr\"odinger Operators in the Twenty-First Century]{Schr\"odinger
Operators \\ in the Twenty-First Century}
\author{Barry Simon}
\address{Division of Physics, Mathematics, and Astronomy, 253-37 \\
California Institute of Technology, Pasadena, CA 91125, USA \\
E-mail: bsimon@caltech.edu}
\thanks{This material is based upon work supported by the National Science
Foundation under Grant No.~DMS-9707661. The Government has certain rights
in this material.}
\thanks{To appear in Mathematical Physics 2000, Imperial College, London.}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction} \lb{s1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Yogi Berra is reputed to have said, ``Prediction is difficult, especially about the
future." Lists of open problems are typically lists of problems on which you expect
progress in a reasonable time scale and so they involve an element of prediction.
We have seen remarkable progress in the past fifty years in our understanding of
Schr\"odinger operators, as I discussed in Simon~\cite{jmp}. In this companion piece,
I present fifteen open problems. In 1984, I presented a list of open problem in
Mathematical Physics, including thirteen in Schr\"odinger operators. Depending
on how you count (since some are multiple), five have been solved.
We will focus on two main areas: anomalous transport (Section~\ref{s2}) where I
expect progress in my lifetime, and Coulomb energies where some of the problems
are so vast and so far from current technology that I do not expect them to be
solved in my lifetime. (There is a story behind the use of this phrase. I have
heard that when Jeans lectured in G\"ottingen around 1910 on his conjecture on the
number of nodes in a cavity, Hilbert remarked that it was an interesting problem
but it would not be solved in his lifetime. Two years later, Hilbert's own
student, Weyl, solved the problem using in part techniques pioneered by Hilbert.
So I figure the use of that phrase is a good jinx!)
In a final section, I present two other problems.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Quantum Transport and Anomalous Spectral Behavior} \lb{s2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the past twenty-five years, a major thrust has involved the study of
Schr\"odinger operators with ergodic potentials and unexpected spectral
behavior of Schr\"odinger operators in slowly decaying potentials. (This is
discussed in Sections~5 and 7 of Simon~\cite{jmp}.) The simplest models of ergodic
Schr\"odinger operators involve finite difference approximations. The first
is the prototypical random model and the second, the prototypical almost
periodic model.
\medskip
\begin{example} \lb{e2.1} \mbox{\rm (Anderson model)} Let $V_\omega (n)$ be a
multisequence of independent, identically distributed random variables with
distribution uniform on $[a,b]$. Here $n\in\bbZ^\nu$ is the multisequence
label and $\omega$ the stochastic label. On $\ell^2 (\bbZ^\nu)$, define
\[
(h_\omega u)(n) = \sum_{|j|=1} u(n+j) + V_\omega (n) u(n).
\]
\end{example}
\begin{example} \lb{e2.2} \mbox{\rm (Almost Mathieu equation)} On $\ell^2
(\bbZ)$, define
\[
(h_{\alpha, \lambda,\theta}u)(n) = u(n+1) + u(n-1) + \lambda\cos (\pi\alpha n +
\theta) u(n).
\]
Here $\alpha,\lambda$ are fixed parameters where $\alpha$ is usually required
to be irrational and $\lambda$ is a coupling constant. $\theta$ runs in $[0, 2\pi)$
and plays a role similar to the $\omega$ of Example~\ref{e2.1}.
\end{example}
\smallskip
It is known that the Anderson model has spectrum $[a-2\nu, b+2\nu]$ and that if
$\nu =1$, the spectrum is dense pure point with probability $1$, and if $\nu \geq
2$, this is true if $|b-a|$ is large enough (we will not try to recount
the history here; see Simon~\cite{van} for proofs of these facts and some history)
and also there is some pure point spectrum near the edges of the spectrum when $|b-a|$
is small.
\medskip
\noindent{\bf Problem 1.} (Extended states) Prove for $\nu\geq 3$ and suitable
values of $b-a$ that the Anderson model has purely absolutely continuous spectrum
in some energy range.
\smallskip
This is the big kahuna of this area, the problem whose solution will make a splash
outside the field. In fact, just proving that there is any a.c.~spectrum will cause
a big stir. The belief is that for $|b-a|$ small, there is a subinterval $(c,d)\subset
[a-2\nu, b+2\nu] = \sigma(H_\omega)$ on which the spectum is purely a.c.~and that
on the complement of this interval, the spectrum is dense pure point. As $|b-a|$
increases beyond a critical value, $|d-c|$ goes to zero.
\medskip
\noindent{\bf Problem 2.} (Localization in two dimensions) Prove that for $\nu =2$,
the spectrum of the Anderson model is dense pure point for all values of $b-a$.
\smallskip
This is the general belief among physicists, although the claims for this
model have fluctuated in time.
\medskip
\noindent{\bf Problem 3.} (Quantum diffusion) Prove that for $\nu \geq 3$ and values
of $|b-a|$ where there is a.c.~spectrum that $\sum_{n\in\bbZ^\nu} n^2 |e^{itH} (n,0)|^2$
grows as $c\,t$ as $t\to\infty$.
\smallskip
That is, $\langle x(t)^2\rangle^{1/2}\sim \tilde c\, t^{1/2}$. For scattering states,
of course, the a.c.~spectrum leads to ballistic behavior (i.e., $\langle x(t)^2
\rangle^{1/2} \sim c\, t$) rather than diffusive behavior. This problem is one of a large
number of issues concerning the long time dynamics of Schr\"odinger operators with
unusual spectral properties.
\medskip
An enormous amount is now known about the almost Mathieu model whose study is a
fascinating laboratory. I would mention three remaining problems about it:
\smallskip
\noindent{\bf Problem 4.} (Ten Martini problem) Prove for all $\lambda\neq 0$ and
all irrational $\alpha$ that $\spec(h_{\alpha,\lambda,\theta})$ (which is $\theta$
independent) is a Cantor set, that is, that it is nowhere dense.
\smallskip
The problem name comes from an offer of Mark Kac. Bellissard-Simon~\cite{bs} proved
the weak form of this for Baire generic pairs of $(\alpha,\lambda)$. It would be
interesting to prove this even just at the self-dual point $\lambda =2$.
\medskip
\noindent{\bf Problem 5.} Prove for all irrational $\alpha$ and $\lambda =2$ that $\spec
(h_{\alpha,\lambda,\theta})$ has measure zero.
\smallskip
This is known (Last~\cite{last1}) for all irrational $\alpha$'s whose continued fraction
expansion has unbounded entries. But it is open for $\alpha$ the golden mean which is
the value with the most numerical evidence! To prove this, one will need a new
understanding of the problem.
\medskip
\noindent{\bf Problem 6.} Prove for all irrational $\alpha$ and $\lambda <2$ that the
spectrum is purely absolutely continuous.
\smallskip
It is known (Last~\cite{last2}, Gesztesy-Simon~\cite{gs}) that the Lebesgue measure
of the a.c.~spectrum is the same as the typical Lebesgue measure of the spectrum for all
irrational $\alpha$ and $\lambda <2$. The result is known (Jitomirskaya~\cite{jito})
for all $\alpha$'s with good Diophantine properties but is open for other $\alpha$'s.
One will need a new understanding of a.c.~spectrum to handle the case of Louiville
$\alpha$'s.
\medskip
While we have focused on the almost Mathieu equation, the general almost periodic problem
needs more understanding. As for slowly decaying potentials, I will mention two problems:
\smallskip
\noindent{\bf Problem 7.} Do there exist potentials $V(x)$ on $[0,\infty)$ so that
$|V(x)| \leq C|x|^{-1/2-\vep}$ for some $\vep >0$ and so that $-\f{d^2}{dx^2}+V$ has
some singular continuous spectrum.
\smallskip
It is known that such models always have a.c.~spectrum on all of $[0,\infty)$
(Remling~\cite{rem}, Christ-Kiselev~\cite{ck}, Deift-Killip~\cite{dk}, Killip~\cite{kil}).
It is also known (Naboko~\cite{nab}, Simon~\cite{sim}) that such models can also have
dense point spectrum. Can they have singular continuous spectrum as well?
\medskip
\noindent{\bf Problem 8.} Let $V$ be a function on $\bbR^\nu$ which obeys
\[
\int |x|^{-\nu +1} |V(x)|^2\, d^\nu x < \infty.
\]
Prove that $-\Delta + V$ has a.c.~spectrum of infinite multiplicity on $[0,\infty)$ if
$\nu \geq 2$.
\smallskip
If $\nu =1$, this is the result of Deift-Killip~\cite{dk} (see also Killip~\cite{kil}).
Their result implies the conjecture in this problem for spherically symmetric potentials
(which is where the $|x|^{-\nu +1}$ comes from).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Coulomb Energies} \lb{s3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The past thirty-five years have seen impressive development in the study of energies
of Schr\"odinger operators with Coulomb potentials (see Sections 9 and 11 of
Simon~\cite{jmp} or the review of Lieb~\cite{lieb1}) of which the high points
were stability of matter, the three-term asymptotics of the total binding energy
of a large atom, and some considerable information on how many electrons a given
nucleus can bind.
While these results involve deep mathematics, except for stability of matter, they are
very remote from problems of real physics. Since one does not often fully ionize an
atom, total binding energies are not important, but rather single ionization energies
are. Understanding the binding energies of atoms and molecules is a huge task for
mathematical physics. The problems in this section may be signposts along the way.
As we progress, the problems will get less specific. We will deal throughout with
fermion electrons. $\calH^{(N)}_f$ will be the space of functions antisymmetric in
spin and space in $L^2 (\bbR^{3N};\bbC^{2N})$.
Define $H(N,Z)$ to be the Hamiltonian on $\calH_f$,
\[
\sum_{i=1}^N \biggl(-\Delta_i - \f{Z}{|x_i|}\biggr) + \sum_{i \max (L^{\qc}_{\gamma,\nu},
L^{\sob}_{\gamma,\nu})$ with strict inequality.
\medskip
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\end{thebibliography}
\end{document}