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\begin{titlepage}
\Large\title {What Is Localization?}
\large\author { R. del Rio\\IIMAS-UNAM \\Apdo.~Postal 20-726,
Admon.~No.~20 \\ 01000 Mexico, D.F., Mexico \and
S. Jitomirskaya\thanks{This material is based upon work supported by the
National Science Foundation under Grant No. DMS-9208029. The
Government has certain rights in this material.}\\ Department of
Mathematics\\ University of California at Irvine\\
Irvine, CA 92717 \and Y. Last and
B.~Simon\thanks{ This material is based upon work supported by the
National Science Foundation under Grant No.~DMS-9401491. The
Government has certain rights in this material.}\\Division of Physics,
Mathematics, and Astronomy \\
California Institute of Technology 253-37 \\
Pasadena, CA 91125}
\end{titlepage}
\maketitle
\begin{abstract}
\normalsize
We examine various issues relevant to localization in the
Anderson model. We show there is more to localization than
exponentially localized states by presenting an example with such
states but where $\langle x(t)^{2}\rangle /t^{2-\delta}$ is unbounded for
any $\delta >0$. We show that the recently discovered instability of
localization under rank one perturbations is only a weak instability.
\end{abstract}
\vfill
PACS: 72.15.Rn
\eject
Localization in random media is basic to a variety of physical
situations. We wish to report here on a number of rigorous
mathematical results that shed light on the phenomenon of localization
in the Anderson model. Mathematically complete proofs of our results
will appear elsewhere$^{(1)}$. Our goal here is to describe the ideas
behind the results.
Throughout, we'll consider the Anderson model, that is, the
Hamiltonian $H_\omega$ on $\ell^2(Z^d)$ (namely, on the
$d$-dimensional cubic lattice)
\begin{equation}
(H_{\omega} u)(n)=\sum_{|j|=1} u(n+j)+V_{\omega}(n)u(n)
\end{equation}
where the potentials, $V_{\omega}$, are identically distributed independent
random variables with distribution$^{(2)}$
$$
\frac{1}{2\eta}\,\chi_{[-\eta,\eta]}(x)\,dx
$$
with $\chi_{[-\eta, \eta]}$ the characteristic function of the interval
$[-\eta,\eta]$.
Many of the claimed proofs of localization show that for almost all
$\omega$, an Anderson model Hamiltonian $H_\omega$ has a complete set
of normalized eigenvectors$^{(3)}$
$\{\varphi_{\omega, m}\}_{m=1}^\infty$ obeying
\begin{equation}
|\varphi_{\omega, m}(n)|\leq C_{\omega, m} e^{-A|n-n_{\omega, m}|}
\end{equation}
where $A$ is fixed, the $n_{\omega, m}$'s are some centers of localization,
and the $C_{\omega, m}$'s are constants depending on $\omega$ and $m$.
Our first result is an example that shows that mere ``exponential
localization'' of eigenfunctions in the form (2) need not have very
strong consequences for the dynamics. We can construct a non-random
potential $V$ in one dimension so that
\begin{description}
\item{(i) } $H$ has a complete set of normalized eigenvectors obeying
Eqn.~(2).
\item{(ii) } Let $\langle x^{2}\rangle (t)$ denote
$\langle e^{-itH}\delta_{0}, x^2 e^{-itH}\delta_{0}\rangle$; then for any
$\delta >0$, $\langle x^{2}(t)\rangle /t^{2-\delta}$ is unbounded as
$t\to\pm\infty$$^{(4)}$.
\end{description}
The potential $V$ for this example is
$$
V(n)=3\cos(2\pi\alpha n+\theta)+\lambda\delta_{n0},
$$
which we consider on the positive half of the lattice ($n \geq 0$),
with a Dirichlet (or any other) boundary condition at the origin.
The 3 in front of the cosine can be replaced by any number larger than 2,
and is chosen so that when $\lambda=0$, the problem has a positive Lyapunov
exponent$^{(5)}$. The $\alpha$ is an irrational, which is specially chosen
so that for suitable time scales $T_n\to\infty$, $V$ is so close to
periodic that we can show $\langle x^{2}(T_n)\rangle$ is large compared
to $T_n^{2-\delta}$. The local perturbation $\lambda\delta_{n0}$ pushes the
spectrum to be pure point and forces Eqn.~(2) to hold.
While $V$ is very far from random, it illustrates that Eqn.~(2) is not
enough to restrict dynamics. The main failing in (2) is the total
freedom given to the constants $C_{\omega, m}$. Indeed, when one thinks
of ``localization'', one usually thinks of the eigenvectors as being
confined, at least roughly, within some typical length-scale. If the
$C_{\omega, m}$'s are allowed to grow
arbitrarily as $m$ changes, it means that eigenvectors are allowed to
be ``extended'' over arbitrarily large length-scales.
We have shown that a correct condition,
which does give correspondence between eigenvector localization and
dynamical localization, is what
we call semi-uniformly localized eigenvectors
({{SULE}}): There are sites $n_{\omega, m}$ so that for
each $\epsilon>0$, there is $C_{\omega, \epsilon}$ for which
\begin{equation}
|\varphi_{\omega, m}(n)|\leq C_{\omega, \epsilon}e^{\epsilon|n_{\omega, m}|}
e^{-A|n-n_{\omega, m}|}
\end{equation}
(3) says that the constants $C_{\omega, m}$ of (2)
are allowed to grow at a rate which is
less than exponential in the distance of the $n_{\omega, m}$'s from the
origin.
{{SULE}} is closely related to a dynamical condition, which
we call semi-uniform dynamical localization ({{SUDL}})
\begin{equation}
\sup\limits_{t}|e^{-itH_{\omega}}(n,\ell)|\leq\tilde
C_{\omega,\epsilon}e^{\epsilon|\ell|}e^{-\tilde{A}|n-\ell|}
\end{equation}
We have proven that (3) implies (4) with $A$ arbitrarily close to
$\tilde A$, and that if
$H_\omega$ has simple eigenvalues$^{(6)}$, then (4) implies (3) with
$A=\frac{1}{2}\tilde A$$^{(7)}$. (4) is sufficient to show that $\langle
x^{2}(t)\rangle$ (or any other positive moment of $x$) is bounded.
By standard probability arguments, (4) is implied by
\begin{equation}
\mbox{E}\biggl(\sup\limits_{t}\,|e^{-itH_{\omega}}(n,\ell)|
\biggr)\leq Ce^{-\tilde{A}|n-\ell|}
\end{equation}
where $\mbox{E}(\cdot)$ denotes expectation over
realizations. (5) has been proven by Delyon et
al.$^{(8)}$ in the one-dimensional case and by Aizenman$^{(9)}$ in
multidimensional cases at large coupling$^{(10)}$.
{\it A priori,} one may wish to consider a more restrictive condition than
(3), which is to consider (2), but with $C_\omega$ independent of $m$
instead of $C_{\omega, m}$. We call this condition uniformly
localized eigenvectors ({{ULE}}). Indeed, {{ULE}}
is related to the dynamical condition (which we call uniform
dynamical localization ({{UDL}}))
\begin{equation}
\sup\limits_{t}|e^{-itH_{\omega}}(n,\ell)|\leq\tilde
C_\omega e^{-\tilde{A}|n-\ell|}
\end{equation}
in essentially the same way that {SULE} is related
to (4). The problem is that {ULE} does not occur:
We have shown$^{(1),(11)}$ that
{ULE} can't occur for a large class of models, and,
in particular, it can't occur for the Anderson model in any dimension.
It is an open question, in fact, whether there is {\it any} Schr\"odinger
operator with {ULE}. The Maryland model$^{(12)}$, which has
an unbounded quasiperiodic potential, exhibits a weak form of
{{ULE}} in the sense that for any finite energy interval, there
exists a uniform constant for all eigenvectors with energies in this
interval. The Almost Mathieu operator$^{(13)}$, however, does not have
{{ULE}}$^{(1),(11)}$. For the Anderson model, or any
other random Hamiltonian obeying Eqn.~(5), one can actually show
stronger uniformity than what is given by Eqn.~(3). That is, one
can get an explicit (dimension dependent) polynomial bound on the
growth of the $C_{\omega, m}$'s. Our definition of {SULE} by
(3) attempts to get, at least roughly, a minimal uniformity
requirement that would still have a two-way relationship with
corresponding dynamical localization.
Our second set of results concerns the following discovery of
Gordon$^{(14)}$ and del Rio et al.$^{(15)}$: Let $H_\omega$ be an
Anderson Hamiltonian in the localized regime and let$^{(16)}$
$H_{\omega}(\lambda)=H_{\omega}+\lambda|0\rangle\langle 0|$.
Then for a set $S$
of couplings $\lambda$, which is dense and locally uncountable$^{(17)}$,
$H_{\omega}(\lambda)$ has purely singular continuous
spectrum$^{(18)}$. In particular, for $\lambda\in S$,
$\langle x^{2}(t)\rangle$ is unbounded$^{(19)}$. So, the strong dynamical
localization discussed above can be destroyed by an arbitrarily
small perturbation of the potential at a single point: a disturbing
fact.
We have found$^{(1)}$ that this instability is a mild one in the
following senses:
\begin{description}
\item{(i) } For {\it{all}} $\lambda$, $\langle x^{2}(t)\rangle \leq
C(\ln|t|)^{2}$ for $t$ large.
\item{(ii) } The spectral measures in the singular continuous case are
supported on a set of zero Hausdorff dimension$^{(20)}$. In fact, this
follows from (i) by a result of Last$^{(21)}$, which is based on
ideas originally due to Guarneri$^{(22)}$.
\item{(iii) } $S$ is contained in a set of coupling constants $\tilde
S$ so that $\tilde S$ has zero Hausdorff dimension, and so that if
$\lambda\notin\tilde S$, $H_{\omega}(\lambda)$ has pure point
spectrum.
\end{description}
\vskip 0.4in
\centerline{\bf Acknowledgments}
\vskip 0.1in
We would like to thank J.~Avron and A.~Klein for useful discussions.
\vskip 0.4in
\centerline{\bf Notes}
\begin{enumerate}
\item R.~del Rio, S.~Jitomirskaya, Y.~Last, and B.~Simon,
{\it{Operators with singular continuous spectrum, IV.~Hausdorff dimension,
rank one perturbations, and localization}}, to be submitted to
J.~d'Analyse Math.
\item The results hold for much more general distributions;
see ref.~1.
\item Or complete in some energy range.
\item For $\delta =0$, it is known that the limit is zero in
the region of pure point spectrum. See B.~Simon, Commun.~Math.~Phys.
{\bf 134}, 209--212 (1990).
\item S.~Aubrey and G.~Andre, Ann.~Israel Phys.~Soc. {\bf 3},
133--164 (1980); J.~Avron and B.~Simon, Duke Math.~J. {\bf 50}, 369--391
(1983).
\item This is known in the Anderson localization regime;
see B.~Simon, Rev.~Math. Phys. {\bf 6}, 1183--1185 (1994).
\item The idea behind the proof that (4) implies (3) is to use
simplicity of the spectrum and (4) to show that
$$
|\varphi_{\omega, m}(n)|\,|\varphi_{\omega, m}(\ell)|\leq\tilde
C_{\omega, \epsilon}e^{\epsilon|\ell|}e^{-\tilde{A}|n-\ell|}\;.
$$
Take $n_{\omega, m}$ to be a point where
$\varphi_{\omega, m}(\,\cdot\,)$ takes
its maximum value, then
$$
|\varphi_{\omega, m}(n)|^{2}\leq
|\varphi_{\omega, m}(n)|\,|\varphi_{\omega, m}(n_{\omega, m})|\leq\tilde
C_{\omega, \epsilon}
e^{\epsilon|n_{\omega, m}|}e^{-\tilde{A}|n-n_{\omega, m}|}\;.
$$
\item F.~Delyon, H.~Kunz, and B.~Souillard, J.~Phys.~{\bf A 16},
25 (1983).
\item M.~Aizenman, Rev.~Math.~Phys. {\bf 6}, 1163--1182 (1994).
For related results, also see F.~Martinelli and E.~Scoppola, {\it
Introduction to the mathematical theory of Anderson localization},
Rivista del Nuovo Cimento {\bf 10}, N. 10 (1987).
\item If the potential is unbounded (e.g., Gaussian), there
are also results at small coupling and large energy.
\item S.~Jitomirskaya, {\it Singular continuous spectrum and uniform
localization for ergodic Schr\"{o}dinger operators}, preprint.
\item The Maryland model is the discrete Schr\"odinger operator
in one dimension with potential $V(n)=\lambda\tan(2\pi\alpha n + \theta)$.
See D.R.~Grempel, S.~Fishman, and R.E.~Prange, Phys.\ Rev.\ Lett. {\bf 49},
833 (1982); R.E.~Prange, D.R.~Grempel, and S.~Fishman, Phys.\ Rev.\
{\bf B 29}, 6500--6512 (1984); A.~Figotin and L.~Pastur, Commun.~Math.~Phys.
{\bf 95}, 401--425 (1984); B.~Simon, Ann.~Phys. {\bf 159}, 157--183 (1985).
\item The Almost Mathieu operator is the discrete Schr\"odinger
operator in one dimension with potential
$V(n)=\lambda\cos(2\pi\alpha n + \theta)$. For large $\lambda$, it is
known to exhibit localization in the sense of Eqn.~(2); see
Ya.~Sinai, J.~Statist.~Phys. {\bf 46}, 861--909 (1987);
J.~Fr\"{o}hlich, T.~Spencer, and P.~Wittwer, Commun.~Math.~Phys.
{\bf 132}, 5--25 (1990);
S.~Jitomirskaya, Commun.~Math.~Phys. {\bf 165}, 49--57 (1994).
\item A.~Gordon, Commun.~Math.~Phys. {\bf 164}, 489--505 (1994).
\item R.~del Rio, N.~Makarov, and B.~Simon,
Commun.~Math.~Phys. {\bf{165}}, 59--67 (1994).
\item $H_{\omega}(\lambda)$ has the potential modified at zero
by a small amount $\lambda$. For a general theory on such perturbations,
see B.~Simon, {\it Spectral analysis of rank one perturbations and
applications}, Lecture notes given at the Vancouver Summer School in
Mathematical Physics, August 1993.
\item Technically, $S$ is a dense $G_\delta$.
\item Singular continuous spectrum is sometimes called
``critical states'' in physics literature.
\item It follows from the {{RAGE}} theorem,
see M.~Reed and B.~Simon, {\it Methods of Modern Mathematical Physics,
III.~Scattering Theory}, London: Academic Press 1979, that if
$\langle x^{2}(t)\rangle$ is bounded, then the spectrum is pure point.
Also see ref.~21.
\item For a discussion of Hausdorff dimension, see K.J.~Falconer,
{\it Fractal Geometry}, Chichester: Wiley 1990.
\item Y.~Last, {\it Quantum dynamics and decompositions of
singular continuous spectra}, preprint.
\item I.~Guarneri, Europhys.~Lett. {\bf 10}, 95--100 (1989).
\end{enumerate}
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